From 6866930697dcfb450087ee58ea1d6d37a7bc6531 Mon Sep 17 00:00:00 2001 From: A-Larsen Date: Wed, 14 Oct 2026 21:08:58 -0400 Subject: [PATCH] first commit --- README.adoc | 420 ++++++ docinfo.conf | 3 + docs/chatgpt-dump/Makehtml | 15 + docs/chatgpt-dump/camera.html | 1122 +++++++++++++++ docs/chatgpt-dump/camera.md | 987 +++++++++++++ docs/chatgpt-dump/euler-angle.html | 460 ++++++ docs/chatgpt-dump/euler-angle.md | 443 ++++++ docs/chatgpt-dump/quaternion-rotation.png | Bin 0 -> 676423 bytes docs/chatgpt-dump/quaternions.html | 809 +++++++++++ docs/chatgpt-dump/quaternions.md | 982 +++++++++++++ docs/chatgpt-dump/rendered.html | 468 +++++++ example.html | 78 ++ index.html | 9 + index.js | 186 +++ look-at(getViewMatrix).png | Bin 0 -> 76666 bytes notes.adoc | 6 + reference.js | 264 ++++ rendered.html | 1538 +++++++++++++++++++++ 18 files changed, 7790 insertions(+) create mode 100644 README.adoc create mode 100644 docinfo.conf create mode 100755 docs/chatgpt-dump/Makehtml create mode 100644 docs/chatgpt-dump/camera.html create mode 100644 docs/chatgpt-dump/camera.md create mode 100644 docs/chatgpt-dump/euler-angle.html create mode 100644 docs/chatgpt-dump/euler-angle.md create mode 100644 docs/chatgpt-dump/quaternion-rotation.png create mode 100644 docs/chatgpt-dump/quaternions.html create mode 100644 docs/chatgpt-dump/quaternions.md create mode 100644 docs/chatgpt-dump/rendered.html create mode 100644 example.html create mode 100644 index.html create mode 100644 index.js create mode 100644 look-at(getViewMatrix).png create mode 100644 notes.adoc create mode 100644 reference.js create mode 100644 rendered.html diff --git a/README.adoc b/README.adoc new file mode 100644 index 0000000..ac1ab78 --- /dev/null +++ b/README.adoc @@ -0,0 +1,420 @@ += Basic 3D Example + +click https://nyquistdev.net/git/?p=basic-3d-example.git;a=blob_plain;f=example.html;hb=HEAD"[here] for an interactive example + +A good explaination of how 3D works. + +NOTE: This started off as code from Tsoding, I used chatgpt to change + specific things, and also made some of my own changes. This is a learning + tool! not something I wrote off the top of my head + +== Graphics Pipeline + +---- +vertex + -> rotate cube + -> translate cube (object position) + -> apply camera (inverse transform) + -> project + -> screen +---- + +NOTE: Some parts of these pipeline may be left out if that part of the pipeline + is not needed + +== Basic Projection Formula + +Projection formula for a vector latexmath:[[x,y,z]]: + +[latexmath] +++++ +\[ +\begin{aligned} +x' &= x/z \\ +y' &= y/z +\end{aligned} +\] +++++ + +== Vertex Rotation Formulas +click https://nyquistdev.net/git/?p=basic-3d-example.git;a=blob_plain;f=example.html;hb=refs/heads/simple-rotation[here] for an interactive example + +Rotate vector formula for x-axis (y-z plane): + +[latexmath] +++++ +\[ +\begin{aligned} +x' &= x \\ +y' &= y\cos\theta - z\sin\theta \\ +z' &= y\sin\theta + z\cos\theta +\end{aligned} +\] +++++ + +Rotate vector formula for y-axis (x-z plane): + +[latexmath] +++++ +\[ +\begin{aligned} +x' &= x\cos\theta - z\sin\theta \\ +y' &= y \\ +z' &= x\sin\theta + z\cos\theta +\end{aligned} \] +++++ + +Rotate vector formula for z-axis (x-y plane): + +[latexmath] +++++ +\[ +\begin{aligned} +x' &= x\cos\theta - y\sin\theta \\ +y' &= x\sin\theta + y\cos\theta \\ +z' &= z +\end{aligned} +\] +++++ + +== 2. Single-axis rotation matrices (Euler Angles) + +=== Rotation about X-axis (roll) + +[latexmath] +++++ +\boxed{ +R_x(\alpha) = +\begin{bmatrix} +1 & 0 & 0 \\ +0 & \cos\alpha & -\sin\alpha \\ +0 & \sin\alpha & \cos\alpha +\end{bmatrix} +} +++++ + +=== Rotation about Y-axis (pitch) + +[latexmath] +++++ +\boxed{ +R_y(\beta) = +\begin{bmatrix} +\cos\beta & 0 & \sin\beta \\ +0 & 1 & 0 \\ +-\sin\beta & 0 & \cos\beta +\end{bmatrix} +} +++++ + +=== Rotation about Z-axis (yaw) + +[latexmath] +++++ +\boxed{ +R_z(\gamma) = +\begin{bmatrix} +\cos\gamma & -\sin\gamma & 0 \\ +\sin\gamma & \cos\gamma & 0 \\ +0 & 0 & 1 +\end{bmatrix} +} +++++ + + +Euler angles will always have gimbal lock. You can switch which axis has gimbal +lock by switching the order. Most application use quaternions instead of eular +angles. + +== Quaternions +click https://nyquistdev.net/git/?p=basic-3d-example.git;a=blob_plain;f=example.html;hb=refs/heads/quaternions[here] for an interactive example + +Quaternions allow you to rotate smoothly around multiple axes without gimbal +lock. Chat-gpt rewrote the code to use quaternions. Here are some of the +replys about https://nyquistdev.net/git/?p=basic-3d-example.git;a=blob_plain;f=docs/chatgpt-dump/quaternions.html;hb=HEAD[quaternions]. + +=== 1. What is a quaternion (in graphics)? + +A quaternion is a 4-component number: + +[latexmath] +++++ +q = (w, x, y, z) +++++ + +In 3D graphics, we use **unit quaternions** to represent **rotations**. + +A rotation by: + +* angle latexmath:[\theta] +* around a **unit axis** latexmath:[\vec{u} = (u_x, u_y, u_z)] + +is encoded as: + +[latexmath] +++++ +\boxed{ +q = \left( +\cos\frac{\theta}{2}, +\; u_x \sin\frac{\theta}{2}, +\; u_y \sin\frac{\theta}{2}, +\; u_z \sin\frac{\theta}{2} +\right) +} +++++ + +This is the **axis–angle -> quaternion** formula. + +==== Why half-angle? +Because quaternion rotation comes from complex numbers and Euler’s formula — +doubling happens naturally in the math. + +=== 2. quatFromAxisAngle + +[source, js] +---- +function quatFromAxisAngle(axis, angle) { + const h = angle / 2; + const s = Math.sin(h); + return [Math.cos(h), axis.x*s, axis.y*s, axis.z*s]; +} +---- + +You’re directly implementing: + +[latexmath] +++++ +q = (\cos(\theta/2), \vec{u}\sin(\theta/2)) +++++ + +* latexmath:[h = angle / 2] -> half-angle +* latexmath:[cos(h)] → scalar part (w) +* latexmath:[axis * sin(h)] → vector part latexmath:[(x,y,z)] + +IMPORTANT: `axis` **must be normalized** (length = 1), otherwise rotation will + scale or skew. + +=== 3. Quaternion multiplication (rotation composition) + +[source, js] +---- +function quatMultiply(a, b) { + const [w1,x1,y1,z1] = a; + const [w2,x2,y2,z2] = b; + return [ + w1*w2 - x1*x2 - y1*y2 - z1*z2, + w1*x2 + x1*w2 + y1*z2 - z1*y2, + w1*y2 - x1*z2 + y1*w2 + z1*x2, + w1*z2 + x1*y2 - y1*x2 + z1*w2 + ]; +} +---- + +formula: + +[latexmath] +++++ +q_1 = (w_1, \vec{v}_1), \quad q_2 = (w_2, \vec{v}_2) +++++ + +Quaternion multiplication is: + +[latexmath] +++++ +\boxed{ +q_1 q_2 = +\left( +w_1 w_2 - \vec{v}_1 \cdot \vec{v}_2,\; +w_1 \vec{v}_2 + w_2 \vec{v}_1 + \vec{v}_1 \times \vec{v}_2 +\right) +} +++++ + +Expanded component-wise -> exactly your code. + +==== Why this matters + +* Quaternion multiplication **combines rotations** +* Order matters: + +[latexmath] +++++ +q_{\text{final}} = q_{\text{new}} \cdot q_{\text{current}} +++++ + +This avoids **gimbal lock**, unlike Euler angles. + +=== 4. Quaternion conjugate + +[source, js] +---- +function quatConjugate(q) { + return [q[0], -q[1], -q[2], -q[3]]; +} +---- + +For: + +[latexmath] +++++ +q = (w, x, y, z) +++++ + +The conjugate is: + +[latexmath] +++++ +\boxed{ +q^* = (w, -x, -y, -z) +} +++++ + +=== Why this works + +If (q) is a **unit quaternion**: + +[latexmath] +++++ +q^{-1} = q^* +++++ + +So conjugate = **inverse rotation**. + +This is why cameras use conjugates — moving the world opposite to the camera +rotation. + +=== 5. Rotating a vector using quaternions + +[source, js] +---- +function quatRotateVector(q, v) { + const qv = [0, v.x, v.y, v.z]; + const r = quatMultiply(quatMultiply(q, qv), quatConjugate(q)); + return { x: r[1], y: r[2], z: r[3] }; +} +---- + +The key formula + +To rotate a vector latexmath:[\vec{v}]: + +1. Convert it to a **pure quaternion** + +[latexmath] +++++ +v_q = (0, v_x, v_y, v_z) +++++ + +2. Apply: + +[latexmath] +++++ +\boxed{ +v' = q \; v_q \; q^{-1} +} +++++ + +3. Extract the vector part + +==== Why this works + +Quaternion rotation encodes: + +* axis +* angle +* orientation + +The sandwich product rotates without: + +* scaling +* skewing +* drifting + +=== Visual intuition + +A rotation quaternion represents: + +[latexmath] +++++ +q = \cos(\theta/2) + \sin(\theta/2)(u_x i + u_y j + u_z k) +++++ + +This is the 3D analog of: + +[latexmath] +++++ +e^{i\theta} = \cos\theta + i\sin\theta +++++ + +== Camera + +chat-gpt replys about https://nyquistdev.net/git/?p=basic-3d-example.git;a=blob_plain;f=docs/chatgpt-dump/camera.html;hb=HEAD[camera]. + +== Simple clickable object + +click https://nyquistdev.net/git/?p=basic-3d-example.git;a=blob_plain;f=example.html;hb=refs/heads/clickable[here] for an interactive example + +This does not account for the rotation of the object but it does check to see if +your mouse is within the rectangular bounds of the mesh. + +1. finding the min and max values for x and y of the mesh: + +[source, js] +---- +function computeBounds(mesh) { + let min = {x: Infinity, y: Infinity} + let max = {x: -Infinity, y: -Infinity} + + for (const v of mesh) { + min.x = Math.min(min.x, v.x) + min.y = Math.min(min.y, v.y) + + max.x = Math.max(max.x, v.x) + max.y = Math.max(max.y, v.y) + } + return {min, max} +} +---- + +2. Define the box corners of the rect for the mesh + +[source, js] +---- +function boundingBoxCorners({min, max}) { + return { + bottom: {left: {x: min.x, y: max.y}, right: {x: max.x, y: max.y}}, + top: {left: {x: min.x, y: min.y}, right: {x: max.x, y: min.y}}, + + } +} +---- + +3. Check to see if the mouse is in the rect in the game loop. If it is, and the + mouse button is down, then you can move the object + +[source, js] +---- +inObject = ( + mousePos.x > rect.top.left.x + && + mousePos.x < rect.top.right.x + && + mousePos.y > rect.top.left.y + && + mousePos.y < rect.bottom.right.y +) + +if (mouseDown && inObject) canMoveObject = true + +if (!mouseDown) canMoveObject = false +---- + +== Links + +* https://www.youtube.com/watch?v=qjWkNZ0SXfo[One Formula That demystifies 3D Graphics] +* https://www.youtube.com/watch?v=EZufiIwwqFA[Rotation matrix derivation (step-by-step prove)] +* https://matthew-brett.github.io/teaching/rotation_2d.html[rotation 2D by mathew] +* https://www.youtube.com/watch?v=zc8b2Jo7mno[gimbal lock visual representation] +* https://www.youtube.com/watch?v=zjMuIxRvygQ[3Blue1Brown | Quaternions and 3d rotation, explained interactively] +* https://www.youtube.com/watch?v=d4EgbgTm0Bg&t=17s[3Blue1Brown | Visualizing the 4d numbers Quanternions] +* https://phas.ubc.ca/~berciu/TEACHING/PHYS206/LECTURES/FILES/euler.pdf[euler.pdf] diff --git a/docinfo.conf b/docinfo.conf new file mode 100644 index 0000000..fc4596c --- /dev/null +++ b/docinfo.conf @@ -0,0 +1,3 @@ +#!/usr/bin/env bash +HOSTNAME=nyquistdev.net +DOCINFODIR=/usr/local/share/asciidoctor-styles/dark1/ diff --git a/docs/chatgpt-dump/Makehtml b/docs/chatgpt-dump/Makehtml new file mode 100755 index 0000000..615bffe --- /dev/null +++ b/docs/chatgpt-dump/Makehtml @@ -0,0 +1,15 @@ +#!/usr/bin/env bash +set -e + +PANDOC_OPTS=( + --from markdown+smart + --to html + --standalone + --metadata charset=utf-8 + --mathjax +) + +for md in *.md; do + html="${md%.md}.html" + pandoc "$md" "${PANDOC_OPTS[@]}" -o "$html" --mathjax +done diff --git a/docs/chatgpt-dump/camera.html b/docs/chatgpt-dump/camera.html new file mode 100644 index 0000000..833aff7 --- /dev/null +++ b/docs/chatgpt-dump/camera.html @@ -0,0 +1,1122 @@ + + + + + + + camera + + + +

Yes 👍 You’re very close already. What you currently have is +object movement, not a camera. We’ll add a real +camera while changing as little of your code as possible.

+

I’ll do this in small, precise steps, and I’ll +clearly mark what to add and what to +replace.

+
+

1️⃣ What “adding a +camera” means in your code

+

Right now your pipeline is:

+
vertex
+ → rotate cube
+ → translate cube (position)
+ → project
+ → screen
+

With a camera, it becomes:

+
vertex
+ → rotate cube
+ → translate cube (object position)
+ → apply camera (inverse transform)
+ → project
+ → screen
+

So we need:

+ +
+

2️⃣ Add a camera object (NEW)

+

Add this near your position variable:

+
let camera = {
+    position: { x: 0, y: 0, z: 0 },
+    rotation: [1, 0, 0, 0] // identity quaternion
+};
+
+

3️⃣ Camera helper functions +(NEW)

+

Add these below your quaternion functions:

+
function quatConjugate(q) {
+    return [q[0], -q[1], -q[2], -q[3]];
+}
+
+function applyCamera(p, cam) {
+    // inverse translation
+    let v = {
+        x: p.x - cam.position.x,
+        y: p.y - cam.position.y,
+        z: p.z - cam.position.z
+    };
+
+    // inverse rotation
+    const invRot = quatConjugate(cam.rotation);
+    return quatRotateVector(invRot, v);
+}
+

Why this works:

+ +
+

4️⃣ Modify your render loop +(IMPORTANT)

+

❌ OLD code (inside +frame()):

+
line(
+    screen(project(translate(quatRotateVector(q, a), position))),
+    screen(project(translate(quatRotateVector(q, b), position)))
+);
+

✅ REPLACE with this:

+
let ra = translate(quatRotateVector(q, a), position);
+let rb = translate(quatRotateVector(q, b), position);
+
+ra = applyCamera(ra, camera);
+rb = applyCamera(rb, camera);
+
+line(
+    screen(project(ra)),
+    screen(project(rb))
+);
+

That’s the entire camera integration.

+
+

5️⃣ Move the camera (instead +of the cube)

+

Right now your scroll code moves position. We’ll change +it to move the camera instead.

+
+

5.1 Forward / backward (scroll)

+

❌ OLD:

+
position.z += .03;
+position.z -= .03;
+

✅ REPLACE with:

+
camera.position.z += .03;
+camera.position.z -= .03;
+
+

5.2 Left / right (shift + scroll)

+

❌ OLD:

+
position.x += .004;
+position.x -= .004;
+

✅ REPLACE with:

+
camera.position.x += .04;
+camera.position.x -= .04;
+

Also remove this loop (it does nothing useful):

+
for (const v of vs) { ... }
+

You should never modify vertices for movement.

+
+

6️⃣ (Optional but +recommended) Rotate the camera

+

Add this helper:

+
function rotateCamera(axis, angle) {
+    const q = quatFromAxisAngle(axis, angle);
+    camera.rotation = quatMultiply(q, camera.rotation);
+}
+

Add keyboard look:

+
window.addEventListener("keydown", e => {
+    const rot = 0.03;
+
+    if (e.key === "ArrowLeft")  rotateCamera({x:0,y:1,z:0},  rot);
+    if (e.key === "ArrowRight") rotateCamera({x:0,y:1,z:0}, -rot);
+    if (e.key === "ArrowUp")    rotateCamera({x:1,y:0,z:0},  rot);
+    if (e.key === "ArrowDown")  rotateCamera({x:1,y:0,z:0}, -rot);
+});
+

Now you have true camera yaw & pitch.

+
+

7️⃣ What you have now +(important milestone)

+

You now have:

+

✔ Object-local rotation ✔ Object world position ✔ Camera position ✔ +Camera rotation ✔ World → view → projection pipeline

+

This is literally the same architecture used by:

+ +

You’re writing a software 3D renderer.

+
+

8️⃣ One key mental rule +(remember this)

+
+

Objects move themselves. Cameras move the world.

+
+

If something feels “backwards”, it usually means you forgot the +inverse.

+
+ +

1️⃣ FPS-style movement (WASD relative to camera) 2️⃣ Proper near-plane +clipping 3️⃣ Field-of-view control 4️⃣ Depth sorting / painter’s algorithm +5️⃣ Backface culling

+

If you want, say which one, and I’ll guide you +through it step by step.

+

Yes — below is a clean, complete rewrite of your entire +snippet that:

+ +

I’ve kept it very close to your original style, just +corrected architecturally.

+
+

✅ FULL WORKING CAMERA-BASED +VERSION

+
<canvas id="game"></canvas>
+<script>
+/* ===================== SETUP ===================== */
+const game = document.getElementById("game");
+game.width = 400;
+game.height = 400;
+
+const ctx = game.getContext("2d");
+
+const BACKGROUND = "#101010";
+const FOREGROUND = "#50FF50";
+const TEXT = "#FFFFFF";
+
+const FPS = 60;
+const NEAR = 0.1;
+
+/* ===================== GEOMETRY ===================== */
+const vertices = [
+    {x:  0.25, y:  0.25, z:  0.25},
+    {x: -0.25, y:  0.25, z:  0.25},
+    {x: -0.25, y: -0.25, z:  0.25},
+    {x:  0.25, y: -0.25, z:  0.25},
+    {x:  0.25, y:  0.25, z: -0.25},
+    {x: -0.25, y:  0.25, z: -0.25},
+    {x: -0.25, y: -0.25, z: -0.25},
+    {x:  0.25, y: -0.25, z: -0.25},
+];
+
+const edges = [
+    [0,1,2,3],
+    [4,5,6,7],
+    [0,4],
+    [1,5],
+    [2,6],
+    [3,7],
+];
+
+/* ===================== OBJECT ===================== */
+let objectPosition = { x: 0, y: 0, z: 2 };
+let angle = 0;
+let activeAxes = ["x"];
+
+/* ===================== CAMERA ===================== */
+let camera = {
+    position: { x: 0, y: 0, z: 0 },
+    rotation: [1,0,0,0] // quaternion
+};
+
+/* ===================== QUATERNIONS ===================== */
+function quatFromAxisAngle(axis, angle) {
+    const h = angle / 2;
+    const s = Math.sin(h);
+    return [Math.cos(h), axis.x*s, axis.y*s, axis.z*s];
+}
+
+function quatMultiply(a, b) {
+    const [w1,x1,y1,z1] = a;
+    const [w2,x2,y2,z2] = b;
+    return [
+        w1*w2 - x1*x2 - y1*y2 - z1*z2,
+        w1*x2 + x1*w2 + y1*z2 - z1*y2,
+        w1*y2 - x1*z2 + y1*w2 + z1*x2,
+        w1*z2 + x1*y2 - y1*x2 + z1*w2
+    ];
+}
+
+function quatConjugate(q) {
+    return [q[0], -q[1], -q[2], -q[3]];
+}
+
+function quatRotateVector(q, v) {
+    const qv = [0, v.x, v.y, v.z];
+    const inv = quatConjugate(q);
+    const t = quatMultiply(q, qv);
+    const r = quatMultiply(t, inv);
+    return { x: r[1], y: r[2], z: r[3] };
+}
+
+/* ===================== TRANSFORMS ===================== */
+function translate(p, t) {
+    return { x: p.x + t.x, y: p.y + t.y, z: p.z + t.z };
+}
+
+function applyCamera(p) {
+    let v = {
+        x: p.x - camera.position.x,
+        y: p.y - camera.position.y,
+        z: p.z - camera.position.z
+    };
+    return quatRotateVector(quatConjugate(camera.rotation), v);
+}
+
+/* ===================== PROJECTION ===================== */
+function project(p) {
+    return { x: p.x / p.z, y: p.y / p.z };
+}
+
+function screen(p) {
+    return {
+        x: (p.x + 1) * 0.5 * game.width,
+        y: (1 - (p.y + 1) * 0.5) * game.height
+    };
+}
+
+/* ===================== DRAW ===================== */
+function clear() {
+    ctx.fillStyle = BACKGROUND;
+    ctx.fillRect(0, 0, game.width, game.height);
+
+    ctx.fillStyle = TEXT;
+    ctx.font = "14px monospace";
+    ctx.fillText("Click: change cube rotation axes", 10, 18);
+    ctx.fillText("Scroll: move camera forward/back", 10, 36);
+    ctx.fillText("Shift+Scroll: strafe camera", 10, 54);
+    ctx.fillText("Arrow keys: look around", 10, 72);
+    ctx.fillText("Axes: " + activeAxes.join(","), 10, 90);
+}
+
+function line(a, b) {
+    ctx.strokeStyle = FOREGROUND;
+    ctx.lineWidth = 2;
+    ctx.beginPath();
+    ctx.moveTo(a.x, a.y);
+    ctx.lineTo(b.x, b.y);
+    ctx.stroke();
+}
+
+/* ===================== LOOP ===================== */
+function frame() {
+    angle += Math.PI / FPS;
+
+    let q = [1,0,0,0];
+    for (const ax of activeAxes) {
+        if (ax === "x") q = quatMultiply(quatFromAxisAngle({x:1,y:0,z:0}, angle), q);
+        if (ax === "y") q = quatMultiply(quatFromAxisAngle({x:0,y:1,z:0}, angle), q);
+        if (ax === "z") q = quatMultiply(quatFromAxisAngle({x:0,y:0,z:1}, angle), q);
+    }
+
+    clear();
+
+    for (const e of edges) {
+        for (let i = 0; i < e.length; i++) {
+            let a = vertices[e[i]];
+            let b = vertices[e[(i+1)%e.length]];
+
+            a = translate(quatRotateVector(q, a), objectPosition);
+            b = translate(quatRotateVector(q, b), objectPosition);
+
+            a = applyCamera(a);
+            b = applyCamera(b);
+
+            if (a.z <= NEAR || b.z <= NEAR) continue;
+
+            line(screen(project(a)), screen(project(b)));
+        }
+    }
+
+    setTimeout(frame, 1000/FPS);
+}
+
+frame();
+
+/* ===================== INPUT ===================== */
+game.onclick = () => {
+    const modes = [["x"],["y"],["z"],["x","y"],["y","z"],["x","z"],["x","y","z"]];
+    activeAxes = modes[(modes.findIndex(m => m+"" === activeAxes+"") + 1) % modes.length];
+};
+
+game.addEventListener("wheel", e => {
+    e.preventDefault();
+    if (e.shiftKey) camera.position.x += e.deltaY * 0.002;
+    else camera.position.z += e.deltaY * 0.01;
+}, { passive:false });
+
+window.addEventListener("keydown", e => {
+    const r = 0.04;
+    if (e.key === "ArrowLeft")  camera.rotation = quatMultiply(quatFromAxisAngle({x:0,y:1,z:0},  r), camera.rotation);
+    if (e.key === "ArrowRight") camera.rotation = quatMultiply(quatFromAxisAngle({x:0,y:1,z:0}, -r), camera.rotation);
+    if (e.key === "ArrowUp")    camera.rotation = quatMultiply(quatFromAxisAngle({x:1,y:0,z:0},  r), camera.rotation);
+    if (e.key === "ArrowDown")  camera.rotation = quatMultiply(quatFromAxisAngle({x:1,y:0,z:0}, -r), camera.rotation);
+});
+</script>
+
+

🎯 What this version gives you

+

✔ True camera system ✔ Object rotation separate from +camera ✔ Proper world → view → projection pipeline ✔ No +vertex mutation ✔ No fake camera tricks ✔ Clean mental model identical +to real engines

+
+

🚀 You are now at the level +where:

+ +

If you want next, I strongly recommend: 👉 FPS-style movement +(WASD relative to camera direction) 👉 +Mouse-look 👉 Perspective FOV math

+

Just say which one 👍

+

Absolutely — this is the correct next step 👍 “Fly +mode” means:

+ +

Below is a full rewrite of the previous camera-based +code, upgraded to a true fly camera. Nothing is faked, +and the math matches real engines.

+
+

✅ FULL FLY-CAMERA +VERSION (WASD + Mouse Look)

+
<canvas id="game"></canvas>
+<script>
+/* ===================== SETUP ===================== */
+const game = document.getElementById("game");
+game.width = 400;
+game.height = 400;
+const ctx = game.getContext("2d");
+
+const BACKGROUND = "#101010";
+const FOREGROUND = "#50FF50";
+const TEXT = "#FFFFFF";
+
+const FPS = 60;
+const NEAR = 0.1;
+
+/* ===================== GEOMETRY ===================== */
+const vertices = [
+    {x:  0.25, y:  0.25, z:  0.25},
+    {x: -0.25, y:  0.25, z:  0.25},
+    {x: -0.25, y: -0.25, z:  0.25},
+    {x:  0.25, y: -0.25, z:  0.25},
+    {x:  0.25, y:  0.25, z: -0.25},
+    {x: -0.25, y:  0.25, z: -0.25},
+    {x: -0.25, y: -0.25, z: -0.25},
+    {x:  0.25, y: -0.25, z: -0.25},
+];
+
+const edges = [
+    [0,1,2,3],
+    [4,5,6,7],
+    [0,4],[1,5],[2,6],[3,7]
+];
+
+/* ===================== OBJECT ===================== */
+let objectPos = { x: 0, y: 0, z: 2 };
+let angle = 0;
+let activeAxes = ["x","y"];
+
+/* ===================== CAMERA ===================== */
+let camera = {
+    position: { x: 0, y: 0, z: 0 },
+    rotation: [1,0,0,0], // quaternion
+    speed: 0.05,
+    sensitivity: 0.002
+};
+
+/* ===================== INPUT ===================== */
+const keys = {};
+window.addEventListener("keydown", e => keys[e.key.toLowerCase()] = true);
+window.addEventListener("keyup",   e => keys[e.key.toLowerCase()] = false);
+
+/* ===================== QUATERNIONS ===================== */
+function quatFromAxisAngle(axis, angle) {
+    const h = angle / 2;
+    const s = Math.sin(h);
+    return [Math.cos(h), axis.x*s, axis.y*s, axis.z*s];
+}
+
+function quatMultiply(a, b) {
+    const [w1,x1,y1,z1] = a;
+    const [w2,x2,y2,z2] = b;
+    return [
+        w1*w2 - x1*x2 - y1*y2 - z1*z2,
+        w1*x2 + x1*w2 + y1*z2 - z1*y2,
+        w1*y2 - x1*z2 + y1*w2 + z1*x2,
+        w1*z2 + x1*y2 - y1*x2 + z1*w2
+    ];
+}
+
+function quatConjugate(q) {
+    return [q[0], -q[1], -q[2], -q[3]];
+}
+
+function quatRotateVector(q, v) {
+    const qv = [0, v.x, v.y, v.z];
+    const r = quatMultiply(quatMultiply(q, qv), quatConjugate(q));
+    return { x: r[1], y: r[2], z: r[3] };
+}
+
+/* ===================== CAMERA MATH ===================== */
+function cameraForward() {
+    return quatRotateVector(camera.rotation, {x:0,y:0,z:1});
+}
+function cameraRight() {
+    return quatRotateVector(camera.rotation, {x:1,y:0,z:0});
+}
+function cameraUp() {
+    return quatRotateVector(camera.rotation, {x:0,y:1,z:0});
+}
+
+function applyCamera(p) {
+    let v = {
+        x: p.x - camera.position.x,
+        y: p.y - camera.position.y,
+        z: p.z - camera.position.z
+    };
+    return quatRotateVector(quatConjugate(camera.rotation), v);
+}
+
+/* ===================== PROJECTION ===================== */
+function project(p) {
+    return { x: p.x / p.z, y: p.y / p.z };
+}
+function screen(p) {
+    return {
+        x: (p.x + 1) * 0.5 * game.width,
+        y: (1 - (p.y + 1) * 0.5) * game.height
+    };
+}
+
+/* ===================== DRAW ===================== */
+function clear() {
+    ctx.fillStyle = BACKGROUND;
+    ctx.fillRect(0,0,game.width,game.height);
+    ctx.fillStyle = TEXT;
+    ctx.font = "14px monospace";
+    ctx.fillText("WASD: move | QE: up/down", 10, 18);
+    ctx.fillText("Mouse: look | Click canvas to lock", 10, 36);
+}
+
+function line(a,b) {
+    ctx.strokeStyle = FOREGROUND;
+    ctx.lineWidth = 2;
+    ctx.beginPath();
+    ctx.moveTo(a.x,a.y);
+    ctx.lineTo(b.x,b.y);
+    ctx.stroke();
+}
+
+/* ===================== LOOP ===================== */
+function frame() {
+    angle += Math.PI / FPS;
+
+    // cube rotation
+    let q = [1,0,0,0];
+    for (const ax of activeAxes) {
+        if (ax==="x") q = quatMultiply(quatFromAxisAngle({x:1,y:0,z:0}, angle), q);
+        if (ax==="y") q = quatMultiply(quatFromAxisAngle({x:0,y:1,z:0}, angle), q);
+        if (ax==="z") q = quatMultiply(quatFromAxisAngle({x:0,y:0,z:1}, angle), q);
+    }
+
+    // camera movement
+    if (keys["w"]) {
+        const f = cameraForward();
+        camera.position.x += f.x * camera.speed;
+        camera.position.y += f.y * camera.speed;
+        camera.position.z += f.z * camera.speed;
+    }
+    if (keys["s"]) {
+        const f = cameraForward();
+        camera.position.x -= f.x * camera.speed;
+        camera.position.y -= f.y * camera.speed;
+        camera.position.z -= f.z * camera.speed;
+    }
+    if (keys["a"]) {
+        const r = cameraRight();
+        camera.position.x -= r.x * camera.speed;
+        camera.position.y -= r.y * camera.speed;
+        camera.position.z -= r.z * camera.speed;
+    }
+    if (keys["d"]) {
+        const r = cameraRight();
+        camera.position.x += r.x * camera.speed;
+        camera.position.y += r.y * camera.speed;
+        camera.position.z += r.z * camera.speed;
+    }
+    if (keys["q"]) {
+        const u = cameraUp();
+        camera.position.x -= u.x * camera.speed;
+        camera.position.y -= u.y * camera.speed;
+        camera.position.z -= u.z * camera.speed;
+    }
+    if (keys["e"]) {
+        const u = cameraUp();
+        camera.position.x += u.x * camera.speed;
+        camera.position.y += u.y * camera.speed;
+        camera.position.z += u.z * camera.speed;
+    }
+
+    clear();
+
+    for (const e of edges) {
+        for (let i=0;i<e.length;i++) {
+            let a = vertices[e[i]];
+            let b = vertices[e[(i+1)%e.length]];
+
+            a = quatRotateVector(q, a);
+            b = quatRotateVector(q, b);
+
+            a = { x:a.x+objectPos.x, y:a.y+objectPos.y, z:a.z+objectPos.z };
+            b = { x:b.x+objectPos.x, y:b.y+objectPos.y, z:b.z+objectPos.z };
+
+            a = applyCamera(a);
+            b = applyCamera(b);
+
+            if (a.z<=NEAR || b.z<=NEAR) continue;
+            line(screen(project(a)), screen(project(b)));
+        }
+    }
+
+    setTimeout(frame, 1000/FPS);
+}
+frame();
+
+/* ===================== MOUSE LOOK ===================== */
+game.onclick = () => game.requestPointerLock();
+
+window.addEventListener("mousemove", e => {
+    if (document.pointerLockElement !== game) return;
+
+    const yaw   = quatFromAxisAngle({x:0,y:1,z:0}, -e.movementX * camera.sensitivity);
+    const pitch = quatFromAxisAngle({x:1,y:0,z:0}, -e.movementY * camera.sensitivity);
+
+    camera.rotation = quatMultiply(yaw, camera.rotation);
+    camera.rotation = quatMultiply(camera.rotation, pitch);
+});
+</script>
+
+

🎯 What you now have

+

✔ True fly camera ✔ Movement relative to view +direction ✔ Mouse-look yaw + pitch ✔ Quaternion-based, no gimbal lock ✔ +Identical behavior to Unity / Unreal fly mode

+

This is real engine-level camera control.

+
+ +

1️⃣ Clamp pitch to avoid flipping 2️⃣ Add roll (airplane-style) 3️⃣ Add +acceleration / inertia 4️⃣ Add multiple objects + scene graph 5️⃣ Replace +projection with real FOV math

+

If you want any of those, say which one — you’re doing excellent +work.

+

Yes — below is a full rewrite of the fly-camera +version that keeps the ability to change which axes the +cube rotates on, exactly like your earlier versions.

+

Nothing is removed:

+ +
+

✅ FULL FLY-CAMERA ++ ROTATION-AXIS TOGGLE VERSION

+
<canvas id="game"></canvas>
+<script>
+/* ===================== SETUP ===================== */
+const game = document.getElementById("game");
+game.width = 400;
+game.height = 400;
+const ctx = game.getContext("2d");
+
+const BACKGROUND = "#101010";
+const FOREGROUND = "#50FF50";
+const TEXT = "#FFFFFF";
+
+const FPS = 60;
+const NEAR = 0.1;
+
+/* ===================== GEOMETRY ===================== */
+const vertices = [
+    {x:  0.25, y:  0.25, z:  0.25},
+    {x: -0.25, y:  0.25, z:  0.25},
+    {x: -0.25, y: -0.25, z:  0.25},
+    {x:  0.25, y: -0.25, z:  0.25},
+    {x:  0.25, y:  0.25, z: -0.25},
+    {x: -0.25, y:  0.25, z: -0.25},
+    {x: -0.25, y: -0.25, z: -0.25},
+    {x:  0.25, y: -0.25, z: -0.25},
+];
+
+const edges = [
+    [0,1,2,3],
+    [4,5,6,7],
+    [0,4],[1,5],[2,6],[3,7]
+];
+
+/* ===================== OBJECT ===================== */
+let objectPos = { x: 0, y: 0, z: 2 };
+let angle = 0;
+
+const AXIS_MODES = [
+    ["x"], ["y"], ["z"],
+    ["x","y"], ["y","z"], ["x","z"],
+    ["x","y","z"]
+];
+let axisModeIndex = 0;
+let activeAxes = AXIS_MODES[axisModeIndex];
+
+/* ===================== CAMERA ===================== */
+let camera = {
+    position: { x: 0, y: 0, z: 0 },
+    rotation: [1,0,0,0],
+    speed: 0.05,
+    sensitivity: 0.002
+};
+
+/* ===================== INPUT ===================== */
+const keys = {};
+window.addEventListener("keydown", e => keys[e.key.toLowerCase()] = true);
+window.addEventListener("keyup",   e => keys[e.key.toLowerCase()] = false);
+
+/* ===================== QUATERNIONS ===================== */
+function quatFromAxisAngle(axis, angle) {
+    const h = angle / 2;
+    const s = Math.sin(h);
+    return [Math.cos(h), axis.x*s, axis.y*s, axis.z*s];
+}
+
+function quatMultiply(a, b) {
+    const [w1,x1,y1,z1] = a;
+    const [w2,x2,y2,z2] = b;
+    return [
+        w1*w2 - x1*x2 - y1*y2 - z1*z2,
+        w1*x2 + x1*w2 + y1*z2 - z1*y2,
+        w1*y2 - x1*z2 + y1*w2 + z1*x2,
+        w1*z2 + x1*y2 - y1*x2 + z1*w2
+    ];
+}
+
+function quatConjugate(q) {
+    return [q[0], -q[1], -q[2], -q[3]];
+}
+
+function quatRotateVector(q, v) {
+    const qv = [0, v.x, v.y, v.z];
+    const r = quatMultiply(quatMultiply(q, qv), quatConjugate(q));
+    return { x: r[1], y: r[2], z: r[3] };
+}
+
+/* ===================== CAMERA VECTORS ===================== */
+function cameraForward() {
+    return quatRotateVector(camera.rotation, {x:0,y:0,z:1});
+}
+function cameraRight() {
+    return quatRotateVector(camera.rotation, {x:1,y:0,z:0});
+}
+function cameraUp() {
+    return quatRotateVector(camera.rotation, {x:0,y:1,z:0});
+}
+
+function applyCamera(p) {
+    const v = {
+        x: p.x - camera.position.x,
+        y: p.y - camera.position.y,
+        z: p.z - camera.position.z
+    };
+    return quatRotateVector(quatConjugate(camera.rotation), v);
+}
+
+/* ===================== PROJECTION ===================== */
+function project(p) {
+    return { x: p.x / p.z, y: p.y / p.z };
+}
+function screen(p) {
+    return {
+        x: (p.x + 1) * 0.5 * game.width,
+        y: (1 - (p.y + 1) * 0.5) * game.height
+    };
+}
+
+/* ===================== DRAW ===================== */
+function clear() {
+    ctx.fillStyle = BACKGROUND;
+    ctx.fillRect(0,0,game.width,game.height);
+
+    ctx.fillStyle = TEXT;
+    ctx.font = "14px monospace";
+    ctx.fillText("WASD: move | QE: up/down", 10, 18);
+    ctx.fillText("Mouse: look | Click canvas to lock", 10, 36);
+    ctx.fillText("Click canvas again: change cube axes", 10, 54);
+    ctx.fillText("Cube axes: " + activeAxes.join(","), 10, 72);
+}
+
+function line(a,b) {
+    ctx.strokeStyle = FOREGROUND;
+    ctx.lineWidth = 2;
+    ctx.beginPath();
+    ctx.moveTo(a.x,a.y);
+    ctx.lineTo(b.x,b.y);
+    ctx.stroke();
+}
+
+/* ===================== LOOP ===================== */
+function frame() {
+    angle += Math.PI / FPS;
+
+    // --- cube rotation quaternion ---
+    let q = [1,0,0,0];
+    for (const ax of activeAxes) {
+        if (ax==="x") q = quatMultiply(quatFromAxisAngle({x:1,y:0,z:0}, angle), q);
+        if (ax==="y") q = quatMultiply(quatFromAxisAngle({x:0,y:1,z:0}, angle), q);
+        if (ax==="z") q = quatMultiply(quatFromAxisAngle({x:0,y:0,z:1}, angle), q);
+    }
+
+    // --- camera movement ---
+    if (keys["w"]) {
+        const f = cameraForward();
+        camera.position.x += f.x * camera.speed;
+        camera.position.y += f.y * camera.speed;
+        camera.position.z += f.z * camera.speed;
+    }
+    if (keys["s"]) {
+        const f = cameraForward();
+        camera.position.x -= f.x * camera.speed;
+        camera.position.y -= f.y * camera.speed;
+        camera.position.z -= f.z * camera.speed;
+    }
+    if (keys["a"]) {
+        const r = cameraRight();
+        camera.position.x -= r.x * camera.speed;
+        camera.position.y -= r.y * camera.speed;
+        camera.position.z -= r.z * camera.speed;
+    }
+    if (keys["d"]) {
+        const r = cameraRight();
+        camera.position.x += r.x * camera.speed;
+        camera.position.y += r.y * camera.speed;
+        camera.position.z += r.z * camera.speed;
+    }
+    if (keys["q"]) {
+        const u = cameraUp();
+        camera.position.x -= u.x * camera.speed;
+        camera.position.y -= u.y * camera.speed;
+        camera.position.z -= u.z * camera.speed;
+    }
+    if (keys["e"]) {
+        const u = cameraUp();
+        camera.position.x += u.x * camera.speed;
+        camera.position.y += u.y * camera.speed;
+        camera.position.z += u.z * camera.speed;
+    }
+
+    clear();
+
+    for (const e of edges) {
+        for (let i=0;i<e.length;i++) {
+            let a = vertices[e[i]];
+            let b = vertices[e[(i+1)%e.length]];
+
+            a = quatRotateVector(q, a);
+            b = quatRotateVector(q, b);
+
+            a = { x:a.x+objectPos.x, y:a.y+objectPos.y, z:a.z+objectPos.z };
+            b = { x:b.x+objectPos.x, y:b.y+objectPos.y, z:b.z+objectPos.z };
+
+            a = applyCamera(a);
+            b = applyCamera(b);
+
+            if (a.z<=NEAR || b.z<=NEAR) continue;
+            line(screen(project(a)), screen(project(b)));
+        }
+    }
+
+    setTimeout(frame, 1000/FPS);
+}
+frame();
+
+/* ===================== INTERACTION ===================== */
+game.onclick = () => {
+    game.requestPointerLock();
+    axisModeIndex = (axisModeIndex + 1) % AXIS_MODES.length;
+    activeAxes = AXIS_MODES[axisModeIndex];
+};
+
+window.addEventListener("mousemove", e => {
+    if (document.pointerLockElement !== game) return;
+
+    const yaw   = quatFromAxisAngle({x:0,y:1,z:0}, -e.movementX * camera.sensitivity);
+    const pitch = quatFromAxisAngle({x:1,y:0,z:0}, -e.movementY * camera.sensitivity);
+
+    camera.rotation = quatMultiply(yaw, camera.rotation);
+    camera.rotation = quatMultiply(camera.rotation, pitch);
+});
+</script>
+
+

🧠 What this version +teaches (important)

+ +

This separation is exactly how real engines +work.

+
+ +

1️⃣ Clamp camera pitch (avoid flipping) 2️⃣ Add roll for airplane +controls 3️⃣ Add multiple objects 4️⃣ Add face culling 5️⃣ Add real +perspective FOV

+

If you want any of those, tell me which — you’re doing real 3D +graphics now.

+ + diff --git a/docs/chatgpt-dump/camera.md b/docs/chatgpt-dump/camera.md new file mode 100644 index 0000000..5be4a45 --- /dev/null +++ b/docs/chatgpt-dump/camera.md @@ -0,0 +1,987 @@ +Yes 👍 +You’re *very close already*. What you currently have is **object movement**, not a camera. +We’ll add a **real camera** while changing as little of your code as possible. + +I’ll do this in **small, precise steps**, and I’ll clearly mark **what to add** and **what to replace**. + +--- + +# 1️⃣ What “adding a camera” means in your code + +Right now your pipeline is: + +``` +vertex + → rotate cube + → translate cube (position) + → project + → screen +``` + +With a camera, it becomes: + +``` +vertex + → rotate cube + → translate cube (object position) + → apply camera (inverse transform) + → project + → screen +``` + +So we need: + +* a **camera position** +* a **camera rotation (quaternion)** +* a function that applies the **inverse camera transform** + +--- + +# 2️⃣ Add a camera object (NEW) + +Add this near your `position` variable: + +```js +let camera = { + position: { x: 0, y: 0, z: 0 }, + rotation: [1, 0, 0, 0] // identity quaternion +}; +``` + +--- + +# 3️⃣ Camera helper functions (NEW) + +Add these **below your quaternion functions**: + +```js +function quatConjugate(q) { + return [q[0], -q[1], -q[2], -q[3]]; +} + +function applyCamera(p, cam) { + // inverse translation + let v = { + x: p.x - cam.position.x, + y: p.y - cam.position.y, + z: p.z - cam.position.z + }; + + // inverse rotation + const invRot = quatConjugate(cam.rotation); + return quatRotateVector(invRot, v); +} +``` + +Why this works: + +* Camera movement = **world moves opposite** +* Camera rotation = **world rotates inverse** + +--- + +# 4️⃣ Modify your render loop (IMPORTANT) + +### ❌ OLD code (inside `frame()`): + +```js +line( + screen(project(translate(quatRotateVector(q, a), position))), + screen(project(translate(quatRotateVector(q, b), position))) +); +``` + +### ✅ REPLACE with this: + +```js +let ra = translate(quatRotateVector(q, a), position); +let rb = translate(quatRotateVector(q, b), position); + +ra = applyCamera(ra, camera); +rb = applyCamera(rb, camera); + +line( + screen(project(ra)), + screen(project(rb)) +); +``` + +That’s the **entire camera integration**. + +--- + +# 5️⃣ Move the camera (instead of the cube) + +Right now your scroll code moves `position`. +We’ll change it to move **the camera instead**. + +--- + +## 5.1 Forward / backward (scroll) + +### ❌ OLD: + +```js +position.z += .03; +position.z -= .03; +``` + +### ✅ REPLACE with: + +```js +camera.position.z += .03; +camera.position.z -= .03; +``` + +--- + +## 5.2 Left / right (shift + scroll) + +### ❌ OLD: + +```js +position.x += .004; +position.x -= .004; +``` + +### ✅ REPLACE with: + +```js +camera.position.x += .04; +camera.position.x -= .04; +``` + +Also remove this loop (it does nothing useful): + +```js +for (const v of vs) { ... } +``` + +You should **never modify vertices for movement**. + +--- + +# 6️⃣ (Optional but recommended) Rotate the camera + +Add this helper: + +```js +function rotateCamera(axis, angle) { + const q = quatFromAxisAngle(axis, angle); + camera.rotation = quatMultiply(q, camera.rotation); +} +``` + +Add keyboard look: + +```js +window.addEventListener("keydown", e => { + const rot = 0.03; + + if (e.key === "ArrowLeft") rotateCamera({x:0,y:1,z:0}, rot); + if (e.key === "ArrowRight") rotateCamera({x:0,y:1,z:0}, -rot); + if (e.key === "ArrowUp") rotateCamera({x:1,y:0,z:0}, rot); + if (e.key === "ArrowDown") rotateCamera({x:1,y:0,z:0}, -rot); +}); +``` + +Now you have **true camera yaw & pitch**. + +--- + +# 7️⃣ What you have now (important milestone) + +You now have: + +✔ Object-local rotation +✔ Object world position +✔ Camera position +✔ Camera rotation +✔ World → view → projection pipeline + +This is **literally the same architecture** used by: + +* OpenGL +* WebGL +* Unity +* Unreal + +You’re writing a **software 3D renderer**. + +--- + +# 8️⃣ One key mental rule (remember this) + +> Objects move themselves. +> Cameras move the world. + +If something feels “backwards”, it usually means you forgot the inverse. + +--- + +## 🔥 Next things you can add (recommended order) + +1️⃣ FPS-style movement (WASD relative to camera) +2️⃣ Proper near-plane clipping +3️⃣ Field-of-view control +4️⃣ Depth sorting / painter’s algorithm +5️⃣ Backface culling + +If you want, say **which one**, and I’ll guide you through it step by step. + +Yes — below is a **clean, complete rewrite of your entire snippet** that: + +* Uses a **real camera** (position + quaternion rotation) +* Keeps your **cube rotation via quaternions** +* Uses **scroll + shift+scroll** for camera movement +* Uses **arrow keys to rotate the camera** +* Uses a **proper 3D pipeline** (object → world → view → projection) + +I’ve kept it **very close to your original style**, just corrected architecturally. + +--- + +## ✅ FULL WORKING CAMERA-BASED VERSION + +```html + + +``` + +--- + +## 🎯 What this version gives you + +✔ True **camera system** +✔ Object rotation separate from camera +✔ Proper **world → view → projection** pipeline +✔ No vertex mutation +✔ No fake camera tricks +✔ Clean mental model identical to real engines + +--- + +### 🚀 You are now at the level where: + +* WebGL will suddenly make sense +* Matrices will feel intuitive +* Game engines won’t feel “magical” anymore + +If you want next, I strongly recommend: +👉 **FPS-style movement (WASD relative to camera direction)** +👉 **Mouse-look** +👉 **Perspective FOV math** + +Just say which one 👍 + +Absolutely — this is the **correct next step** 👍 +“Fly mode” means: + +* **W / S** → move forward / backward **relative to where the camera is facing** +* **A / D** → strafe left / right +* **Q / E** → move down / up +* **Mouse** → look around (yaw + pitch) +* The cube stays in the world; **the camera flies** + +Below is a **full rewrite** of the previous camera-based code, upgraded to a **true fly camera**. +Nothing is faked, and the math matches real engines. + +--- + +## ✅ FULL FLY-CAMERA VERSION (WASD + Mouse Look) + +```html + + +``` + +--- + +## 🎯 What you now have + +✔ **True fly camera** +✔ Movement relative to view direction +✔ Mouse-look yaw + pitch +✔ Quaternion-based, no gimbal lock +✔ Identical behavior to Unity / Unreal fly mode + +This is *real engine-level camera control*. + +--- + +### 🔥 Next upgrades (strongly recommended) + +1️⃣ Clamp pitch to avoid flipping +2️⃣ Add roll (airplane-style) +3️⃣ Add acceleration / inertia +4️⃣ Add multiple objects + scene graph +5️⃣ Replace projection with real FOV math + +If you want any of those, say which one — you’re doing excellent work. + +Yes — below is a **full rewrite of the fly-camera version** that **keeps the ability to change which axes the cube rotates on**, exactly like your earlier versions. + +Nothing is removed: + +* ✅ Fly camera (WASD + QE + mouse look) +* ✅ Quaternion camera +* ✅ Quaternion cube rotation +* ✅ **Click to cycle cube rotation axes** +* ✅ Clean, correct 3D pipeline + +--- + +## ✅ FULL FLY-CAMERA + ROTATION-AXIS TOGGLE VERSION + +```html + + +``` + +--- + +## 🧠 What this version teaches (important) + +* Cube rotation is **object-local** +* Camera rotation is **view-space** +* Both use **quaternions**, but for different purposes +* Axis toggling affects **only the cube**, never the camera + +This separation is **exactly** how real engines work. + +--- + +## 🚀 Strong next steps (recommended) + +1️⃣ Clamp camera pitch (avoid flipping) +2️⃣ Add roll for airplane controls +3️⃣ Add multiple objects +4️⃣ Add face culling +5️⃣ Add real perspective FOV + +If you want any of those, tell me which — you’re doing *real 3D graphics now*. + diff --git a/docs/chatgpt-dump/euler-angle.html b/docs/chatgpt-dump/euler-angle.html new file mode 100644 index 0000000..5f3ed29 --- /dev/null +++ b/docs/chatgpt-dump/euler-angle.html @@ -0,0 +1,460 @@ + + + + + + + euler-angle + + + + + +

Absolutely. I’ll give you the actual math formulas, +show the rotation matrices, explain the order +issue, and relate them directly to what you’ve been doing with +quaternions.

+
+

1. What are Euler angles?

+

Euler angles describe a 3D rotation as three +sequential rotations about coordinate axes.

+

Typically:

+ +

We’ll denote:

+ +

⚠️ Order matters. XYZ is not +the same as ZYX.

+
+

2. Single-axis rotation +matrices

+

Rotation about X-axis (roll)

+

[ & & \end{bmatrix} }]

+
+

Rotation about Y-axis (pitch)

+

[ & 0 & \end{bmatrix} }]

+
+

Rotation about Z-axis (yaw)

+

[ & 0 & 1 \end{bmatrix} }]

+
+

3. Combined Euler rotation +(XYZ order)

+

If you rotate X → then Y → then Z:

+

[ ]

+

Matrix multiplication is right to left (apply X +first).

+

Expanded matrix:

+

[ ]

+

Where:

+ +
+

4. Rotating a vector with +Euler angles

+

Given a vector:

+

[ = ]

+

Rotated vector:

+

[ ]

+
+

5. Euler angles as equations +(explicit)

+

If using yaw–pitch–roll (Z–Y–X):

+

[ ]

+

This is what people usually implement directly (and +regret later).

+
+

6. Why Euler angles +cause problems (gimbal lock)

+

If: [ = ^]

+

Then:

+ +

This is gimbal lock.

+

Your quaternion code avoids this entirely.

+
+

7. Euler → Quaternion +conversion

+

Given Euler angles ( (, , ) ):

+

[ _z &= - \end{aligned} }]

+

(This is why quaternions feel scary — but they’re actually simpler in +practice.)

+
+

8. When Euler angles are +still useful

+

Euler angles are fine for:

+ +

But not for:

+ +
+

9. Key mental takeaway

+
+

Euler angles describe how you rotate. Quaternions +describe what rotation you have.

+
+

That’s why engines store quaternions internally but expose Euler +angles in editors.

+
+

10. Great references

+ +
+

If you want next:

+ +

Just say the word.

+

Great question — this is exactly where rotation matrices stop being +“symbols on a page” and start making sense.

+

I’ll walk you through how to actually do the math, +step by step, for rotation about the X-axis, and show +why the formula looks the way it does.

+
+

1. The X-axis rotation matrix

+

Rotation by angle ( ) around the X-axis:

+

[ R_x() = ]

+

This matrix rotates points in the YZ-plane, leaving +X unchanged.

+
+

2. The vector you’re rotating

+

Let your 3D point be:

+

[ = ]

+

To rotate it:

+

[ ]

+
+

3. Matrix multiplication +rule (important)

+

When multiplying a matrix by a vector:

+[ +\[\begin{bmatrix} +a & b & c \ +d & e & f \ +g & h & i +\end{bmatrix}\] +g & h & i \end{bmatrix} +\[\begin{bmatrix} +x \ y \ z +\end{bmatrix}\] +

=============

+\[\begin{bmatrix} +ax + by + cz \ +dx + ey + fz \ +gx + hy + iz +\end{bmatrix}\] +


+gx + hy + iz \end{bmatrix} ]

+

Each row produces one output component.

+
+

4. Multiply (R_x()) by the +vector

+

First row → (x’)

+

[ x’ = 1x + 0y + 0z = x]

+

✔ X stays the same

+
+

Second row → (y’)

+

[ y’ = 0x + y + (-)z]

+

[ ]

+
+

Third row → (z’)

+

[ z’ = 0x + y + z]

+

[ ]

+
+

5. Final result (important)

+

[ ’ &= y+ z \end{aligned} }]

+

This is the explicit math behind X-axis +rotation.

+
+

6. Why this makes geometric +sense

+

Imagine looking down the +X-axis

+

You see the YZ-plane as a flat 2D plane.

+

The matrix:

+

[ ]

+

is just the 2D rotation matrix, applied to +(y, z).

+

So rotating around X in 3D = rotating (y, z) in 2D.

+
+

7. Connection to your +quaternion code

+

This line in your earlier code:

+
y = y*c - z*s
+z = y*s + z*c
+

is exactly this matrix multiplication.

+

Quaternions just wrap this logic in a more stable form.

+
+

8. Quick numeric example

+

Rotate point (0, 1, 0) by 90° (π/2) +around X:

+

[ (90)=0,(90)=1]

+

[ ]

+

Result:

+

[ (0,0,1)]

+

✔ Point moved from Y-axis to Z-axis

+
+

9. Key mental model

+
+

A rotation matrix does not move axes. It +re-expresses the point in a rotated coordinate system.

+
+
+

If you want next, I can show:

+ +

Just tell me 👍

+ + diff --git a/docs/chatgpt-dump/euler-angle.md b/docs/chatgpt-dump/euler-angle.md new file mode 100644 index 0000000..1e3d20a --- /dev/null +++ b/docs/chatgpt-dump/euler-angle.md @@ -0,0 +1,443 @@ +Absolutely. I’ll give you the **actual math formulas**, show the **rotation matrices**, explain the **order issue**, and relate them directly to what you’ve been doing with quaternions. + +--- + +## 1. What are Euler angles? + +**Euler angles** describe a 3D rotation as **three sequential rotations** about coordinate axes. + +Typically: + +* **Roll** → rotation about X +* **Pitch** → rotation about Y +* **Yaw** → rotation about Z + +We’ll denote: + +* ( \alpha ) = rotation about X (roll) +* ( \beta ) = rotation about Y (pitch) +* ( \gamma ) = rotation about Z (yaw) + +⚠️ **Order matters**. +`XYZ` is *not* the same as `ZYX`. + +--- + +## 2. Single-axis rotation matrices + +### Rotation about X-axis (roll) + +[ +\boxed{ +R_x(\alpha) = +\begin{bmatrix} +1 & 0 & 0 \ +0 & \cos\alpha & -\sin\alpha \ +0 & \sin\alpha & \cos\alpha +\end{bmatrix} +} +] + +--- + +### Rotation about Y-axis (pitch) + +[ +\boxed{ +R_y(\beta) = +\begin{bmatrix} +\cos\beta & 0 & \sin\beta \ +0 & 1 & 0 \ +-\sin\beta & 0 & \cos\beta +\end{bmatrix} +} +] + +--- + +### Rotation about Z-axis (yaw) + +[ +\boxed{ +R_z(\gamma) = +\begin{bmatrix} +\cos\gamma & -\sin\gamma & 0 \ +\sin\gamma & \cos\gamma & 0 \ +0 & 0 & 1 +\end{bmatrix} +} +] + +--- + +## 3. Combined Euler rotation (XYZ order) + +If you rotate **X → then Y → then Z**: + +[ +\boxed{ +R = R_z(\gamma) ; R_y(\beta) ; R_x(\alpha) +} +] + +Matrix multiplication is **right to left** (apply X first). + +Expanded matrix: + +[ +\begin{bmatrix} +c_\gamma c_\beta & +c_\gamma s_\beta s_\alpha - s_\gamma c_\alpha & +c_\gamma s_\beta c_\alpha + s_\gamma s_\alpha \ + +s_\gamma c_\beta & +s_\gamma s_\beta s_\alpha + c_\gamma c_\alpha & +s_\gamma s_\beta c_\alpha - c_\gamma s_\alpha \ + +-s_\beta & +c_\beta s_\alpha & +c_\beta c_\alpha +\end{bmatrix} +] + +Where: + +* ( c_x = \cos(x) ) +* ( s_x = \sin(x) ) + +--- + +## 4. Rotating a vector with Euler angles + +Given a vector: + +[ +\vec{v} = +\begin{bmatrix} +x \ y \ z +\end{bmatrix} +] + +Rotated vector: + +[ +\boxed{ +\vec{v}' = R \vec{v} +} +] + +--- + +## 5. Euler angles as equations (explicit) + +If using **yaw–pitch–roll (Z–Y–X)**: + +[ +\begin{aligned} +x' &= x(\cos\gamma\cos\beta) ++ y(\cos\gamma\sin\beta\sin\alpha - \sin\gamma\cos\alpha) ++ z(\cos\gamma\sin\beta\cos\alpha + \sin\gamma\sin\alpha) \ +y' &= x(\sin\gamma\cos\beta) ++ y(\sin\gamma\sin\beta\sin\alpha + \cos\gamma\cos\alpha) ++ z(\sin\gamma\sin\beta\cos\alpha - \cos\gamma\sin\alpha) \ +z' &= x(-\sin\beta) ++ y(\cos\beta\sin\alpha) ++ z(\cos\beta\cos\alpha) +\end{aligned} +] + +This is what people usually implement **directly** (and regret later). + +--- + +## 6. Why Euler angles cause problems (gimbal lock) + +If: +[ +\beta = \pm 90^\circ +] + +Then: + +* X and Z axes align +* You lose one degree of freedom +* Rotations become unstable + +This is **gimbal lock**. + +Your quaternion code avoids this entirely. + +--- + +## 7. Euler → Quaternion conversion + +Given Euler angles ( (\alpha, \beta, \gamma) ): + +[ +\boxed{ +\begin{aligned} +q_w &= \cos\frac{\alpha}{2}\cos\frac{\beta}{2}\cos\frac{\gamma}{2} ++ \sin\frac{\alpha}{2}\sin\frac{\beta}{2}\sin\frac{\gamma}{2} \ +q_x &= \sin\frac{\alpha}{2}\cos\frac{\beta}{2}\cos\frac{\gamma}{2} +- \cos\frac{\alpha}{2}\sin\frac{\beta}{2}\sin\frac{\gamma}{2} \ +q_y &= \cos\frac{\alpha}{2}\sin\frac{\beta}{2}\cos\frac{\gamma}{2} ++ \sin\frac{\alpha}{2}\cos\frac{\beta}{2}\sin\frac{\gamma}{2} \ +q_z &= \cos\frac{\alpha}{2}\cos\frac{\beta}{2}\sin\frac{\gamma}{2} +- \sin\frac{\alpha}{2}\sin\frac{\beta}{2}\cos\frac{\gamma}{2} +\end{aligned} +} +] + +(This is why quaternions feel scary — but they’re actually simpler in practice.) + +--- + +## 8. When Euler angles are still useful + +Euler angles are fine for: + +* UI sliders +* Editing tools +* Debugging +* Serialization + +But **not** for: + +* Camera motion +* Free rotation +* Animation interpolation + +--- + +## 9. Key mental takeaway + +> Euler angles describe **how** you rotate. +> Quaternions describe **what rotation you have**. + +That’s why engines store quaternions internally but expose Euler angles in editors. + +--- + +## 10. Great references + +* Wikipedia (math-heavy but correct): + [https://en.wikipedia.org/wiki/Euler_angles](https://en.wikipedia.org/wiki/Euler_angles) +* LearnOpenGL (practical): + [https://learnopengl.com/Getting-started/Camera](https://learnopengl.com/Getting-started/Camera) +* 3Blue1Brown (intuition): + [https://www.youtube.com/watch?v=zc8b2Jo7mno](https://www.youtube.com/watch?v=zc8b2Jo7mno) + +--- + +If you want next: + +* 🔁 Euler ↔ Quaternion code +* 🎥 Camera math explained geometrically +* 🧠 Why order matters visually +* 🧩 Gimbal lock demo + +Just say the word. + +Great question — this is exactly where rotation matrices stop being “symbols on a page” and start making sense. + +I’ll walk you through **how to actually do the math**, step by step, for **rotation about the X-axis**, and show **why the formula looks the way it does**. + +--- + +# 1. The X-axis rotation matrix + +Rotation by angle ( \alpha ) around the **X-axis**: + +[ +R_x(\alpha) = +\begin{bmatrix} +1 & 0 & 0 \ +0 & \cos\alpha & -\sin\alpha \ +0 & \sin\alpha & \cos\alpha +\end{bmatrix} +] + +This matrix rotates points in the **YZ-plane**, leaving X unchanged. + +--- + +# 2. The vector you’re rotating + +Let your 3D point be: + +[ +\vec{v} = +\begin{bmatrix} +x \ y \ z +\end{bmatrix} +] + +To rotate it: + +[ +\boxed{ +\vec{v}' = R_x(\alpha)\vec{v} +} +] + +--- + +# 3. Matrix multiplication rule (important) + +When multiplying a matrix by a vector: + +[ +\begin{bmatrix} +a & b & c \ +d & e & f \ +g & h & i +\end{bmatrix} +\begin{bmatrix} +x \ y \ z +\end{bmatrix} +============= + +\begin{bmatrix} +ax + by + cz \ +dx + ey + fz \ +gx + hy + iz +\end{bmatrix} +] + +Each **row** produces one output component. + +--- + +# 4. Multiply (R_x(\alpha)) by the vector + +### First row → (x') + +[ +x' = 1\cdot x + 0\cdot y + 0\cdot z = x +] + +✔ X stays the same + +--- + +### Second row → (y') + +[ +y' = 0\cdot x + \cos\alpha \cdot y + (-\sin\alpha)\cdot z +] + +[ +\boxed{ +y' = y\cos\alpha - z\sin\alpha +} +] + +--- + +### Third row → (z') + +[ +z' = 0\cdot x + \sin\alpha \cdot y + \cos\alpha \cdot z +] + +[ +\boxed{ +z' = y\sin\alpha + z\cos\alpha +} +] + +--- + +# 5. Final result (important) + +[ +\boxed{ +\begin{aligned} +x' &= x \ +y' &= y\cos\alpha - z\sin\alpha \ +z' &= y\sin\alpha + z\cos\alpha +\end{aligned} +} +] + +This is the **explicit math** behind X-axis rotation. + +--- + +# 6. Why this makes geometric sense + +### Imagine looking down the X-axis + +You see the **YZ-plane** as a flat 2D plane. + +The matrix: + +[ +\begin{bmatrix} +\cos\alpha & -\sin\alpha \ +\sin\alpha & \cos\alpha +\end{bmatrix} +] + +is just the **2D rotation matrix**, applied to `(y, z)`. + +So rotating around X in 3D = rotating `(y, z)` in 2D. + +--- + +# 7. Connection to your quaternion code + +This line in your earlier code: + +```js +y = y*c - z*s +z = y*s + z*c +``` + +is **exactly** this matrix multiplication. + +Quaternions just wrap this logic in a more stable form. + +--- + +# 8. Quick numeric example + +Rotate point `(0, 1, 0)` by **90° (π/2)** around X: + +[ +\cos(90^\circ)=0,\quad \sin(90^\circ)=1 +] + +[ +\begin{aligned} +x' &= 0 \ +y' &= 1\cdot0 - 0\cdot1 = 0 \ +z' &= 1\cdot1 + 0\cdot0 = 1 +\end{aligned} +] + +Result: + +[ +(0,0,1) +] + +✔ Point moved from Y-axis to Z-axis + +--- + +# 9. Key mental model + +> A rotation matrix **does not move axes**. +> It re-expresses the point in a rotated coordinate system. + +--- + +If you want next, I can show: + +* Y-axis and Z-axis derivations (they follow the same logic) +* Why the sine signs flip +* How to derive rotation matrices from scratch +* How this turns into quaternion math + +Just tell me 👍 diff --git a/docs/chatgpt-dump/quaternion-rotation.png b/docs/chatgpt-dump/quaternion-rotation.png new file mode 100644 index 0000000000000000000000000000000000000000..06eddd8455275e3d33fb3b68a1d9840041cffa38 GIT binary patch literal 676423 zcmeFabyQT}_cwlL7`g=MMnD>e25FE6Dd|wUyFsM{B?Xj5x=WF6K}wX6mXrv`h9s^ED9_D0B{u*?r8!58h8l< zFreU9(M?oh0H}XtZYr&Z__p(OGy@;gfZrD8JT9gH0dOM_07?L%G*=Yl0{{>>{nJYR zx|QR9w(@jyx{GLe*TR&a-^vQk&1nkf6XxXOv4C@$3Yv3sT5$7OSel#h^YQXof$xH} z{~3<;dN@<}|K+7zrta>RZXUMIPVQg_VC6qLm^r)sHyzB8t+~8-5FM(`!A9BErq>UH zbFHqg91zpwGIz5y^{}*Xba%G$@HTa`l(M#T@@TVgGIg}Hl2?Dg$>)6KYGG?_>tX7k 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font-weight: bold; font-style: italic; } /* Warning */ + </style> +</head> +<body> +<p>Absolutely! I can show you how to incorporate +<strong>quaternions</strong> into your code to allow smooth rotation +around <strong>multiple axes at once</strong>, avoiding the limitations +of your current <code>rotate</code> function.</p> +<p>Let’s go step by step.</p> +<hr /> +<h3 id="why-quaternions-here">1. <strong>Why Quaternions +Here</strong></h3> +<p>Currently, you rotate around a single axis per frame:</p> +<div class="sourceCode" id="cb1"><pre class="sourceCode js"><code class="sourceCode javascript"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="fu">rotate</span>(a<span class="op">,</span> rotationAxis<span class="op">,</span> angle)</span></code></pre></div> +<p>With quaternions, you can <strong>combine rotations around multiple +axes</strong> seamlessly. For example, you could rotate simultaneously +around X <strong>and</strong> Y.</p> +<hr /> +<h3 id="quaternion-basics">2. <strong>Quaternion Basics</strong></h3> +<p>A quaternion can be represented as:</p> +<p>[ q = w + xi + yj + zk]</p> +<p>For 3D rotation:</p> +<div class="sourceCode" id="cb2"><pre class="sourceCode js"><code class="sourceCode javascript"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a>q <span class="op">=</span> [w<span class="op">,</span> x<span class="op">,</span> y<span class="op">,</span> z]</span></code></pre></div> +<ul> +<li><code>w</code> = cos(angle/2)</li> +<li><code>x, y, z</code> = axis vector * sin(angle/2)</li> +</ul> +<p>To rotate a point <code>v</code>:</p> +<ol type="1"> +<li>Convert <code>v</code> to a quaternion +<code>[0, vx, vy, vz]</code>.</li> +<li>Apply <code>v' = q * v * q⁻¹</code>.</li> +</ol> +<hr /> +<h3 id="quaternion-utilities-in-js">3. <strong>Quaternion Utilities in +JS</strong></h3> +<div class="sourceCode" id="cb3"><pre class="sourceCode js"><code class="sourceCode javascript"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="kw">function</span> <span class="fu">quatFromAxisAngle</span>(axis<span class="op">,</span> angle) {</span> +<span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a> <span class="kw">const</span> half <span class="op">=</span> angle <span class="op">/</span> <span class="dv">2</span></span> +<span id="cb3-3"><a href="#cb3-3" aria-hidden="true" tabindex="-1"></a> <span class="kw">const</span> s <span class="op">=</span> <span class="bu">Math</span><span class="op">.</span><span class="fu">sin</span>(half)</span> +<span id="cb3-4"><a href="#cb3-4" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> [</span> +<span id="cb3-5"><a href="#cb3-5" aria-hidden="true" tabindex="-1"></a> <span class="bu">Math</span><span class="op">.</span><span class="fu">cos</span>(half)<span class="op">,</span> <span class="co">// w</span></span> +<span id="cb3-6"><a href="#cb3-6" aria-hidden="true" tabindex="-1"></a> axis<span class="op">.</span><span class="at">x</span> <span class="op">*</span> s<span class="op">,</span> <span class="co">// x</span></span> +<span id="cb3-7"><a href="#cb3-7" aria-hidden="true" tabindex="-1"></a> axis<span class="op">.</span><span class="at">y</span> <span class="op">*</span> s<span class="op">,</span> <span class="co">// y</span></span> +<span id="cb3-8"><a href="#cb3-8" aria-hidden="true" tabindex="-1"></a> axis<span class="op">.</span><span class="at">z</span> <span class="op">*</span> s <span class="co">// z</span></span> +<span id="cb3-9"><a href="#cb3-9" aria-hidden="true" tabindex="-1"></a> ]</span> +<span id="cb3-10"><a href="#cb3-10" aria-hidden="true" tabindex="-1"></a>}</span> +<span id="cb3-11"><a href="#cb3-11" aria-hidden="true" tabindex="-1"></a></span> +<span id="cb3-12"><a href="#cb3-12" aria-hidden="true" tabindex="-1"></a><span class="kw">function</span> <span class="fu">quatMultiply</span>(q1<span class="op">,</span> q2) {</span> +<span id="cb3-13"><a href="#cb3-13" aria-hidden="true" tabindex="-1"></a> <span class="kw">const</span> [w1<span class="op">,</span>x1<span class="op">,</span>y1<span class="op">,</span>z1] <span class="op">=</span> q1</span> +<span id="cb3-14"><a href="#cb3-14" aria-hidden="true" tabindex="-1"></a> <span class="kw">const</span> [w2<span class="op">,</span>x2<span class="op">,</span>y2<span class="op">,</span>z2] <span class="op">=</span> q2</span> +<span id="cb3-15"><a href="#cb3-15" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> [</span> +<span id="cb3-16"><a href="#cb3-16" aria-hidden="true" tabindex="-1"></a> w1<span class="op">*</span>w2 <span class="op">-</span> x1<span class="op">*</span>x2 <span class="op">-</span> y1<span class="op">*</span>y2 <span class="op">-</span> z1<span class="op">*</span>z2<span class="op">,</span></span> +<span id="cb3-17"><a href="#cb3-17" aria-hidden="true" tabindex="-1"></a> w1<span class="op">*</span>x2 <span class="op">+</span> x1<span class="op">*</span>w2 <span class="op">+</span> y1<span class="op">*</span>z2 <span class="op">-</span> z1<span class="op">*</span>y2<span class="op">,</span></span> +<span id="cb3-18"><a href="#cb3-18" aria-hidden="true" tabindex="-1"></a> w1<span class="op">*</span>y2 <span class="op">-</span> x1<span class="op">*</span>z2 <span class="op">+</span> y1<span class="op">*</span>w2 <span class="op">+</span> z1<span class="op">*</span>x2<span class="op">,</span></span> +<span id="cb3-19"><a href="#cb3-19" aria-hidden="true" tabindex="-1"></a> w1<span class="op">*</span>z2 <span class="op">+</span> x1<span class="op">*</span>y2 <span class="op">-</span> y1<span class="op">*</span>x2 <span class="op">+</span> z1<span class="op">*</span>w2</span> +<span id="cb3-20"><a href="#cb3-20" aria-hidden="true" tabindex="-1"></a> ]</span> +<span id="cb3-21"><a href="#cb3-21" aria-hidden="true" tabindex="-1"></a>}</span> +<span id="cb3-22"><a href="#cb3-22" aria-hidden="true" tabindex="-1"></a></span> +<span id="cb3-23"><a href="#cb3-23" aria-hidden="true" tabindex="-1"></a><span class="kw">function</span> <span class="fu">quatRotateVector</span>(q<span class="op">,</span> v) {</span> +<span id="cb3-24"><a href="#cb3-24" aria-hidden="true" tabindex="-1"></a> <span class="kw">const</span> [w<span class="op">,</span>x<span class="op">,</span>y<span class="op">,</span>z] <span class="op">=</span> q</span> +<span id="cb3-25"><a href="#cb3-25" aria-hidden="true" tabindex="-1"></a> <span class="kw">const</span> [vx<span class="op">,</span>vy<span class="op">,</span>vz] <span class="op">=</span> [v<span class="op">.</span><span class="at">x</span><span class="op">,</span>v<span class="op">.</span><span class="at">y</span><span class="op">,</span>v<span class="op">.</span><span class="at">z</span>]</span> +<span id="cb3-26"><a href="#cb3-26" aria-hidden="true" tabindex="-1"></a></span> +<span id="cb3-27"><a href="#cb3-27" aria-hidden="true" tabindex="-1"></a> <span class="co">// Quaternion multiplication: v&#39; = q * v_quat * q⁻¹</span></span> +<span id="cb3-28"><a href="#cb3-28" aria-hidden="true" tabindex="-1"></a> <span class="kw">const</span> qv <span class="op">=</span> [<span class="dv">0</span><span class="op">,</span> vx<span class="op">,</span> vy<span class="op">,</span> vz]</span> +<span id="cb3-29"><a href="#cb3-29" aria-hidden="true" tabindex="-1"></a> <span class="kw">const</span> qConj <span class="op">=</span> [w<span class="op">,</span> <span class="op">-</span>x<span class="op">,</span> <span class="op">-</span>y<span class="op">,</span> <span class="op">-</span>z]</span> +<span id="cb3-30"><a href="#cb3-30" aria-hidden="true" tabindex="-1"></a></span> +<span id="cb3-31"><a href="#cb3-31" aria-hidden="true" tabindex="-1"></a> <span class="kw">const</span> temp <span class="op">=</span> <span class="fu">quatMultiply</span>(q<span class="op">,</span> qv)</span> +<span id="cb3-32"><a href="#cb3-32" aria-hidden="true" tabindex="-1"></a> <span class="kw">const</span> result <span class="op">=</span> <span class="fu">quatMultiply</span>(temp<span class="op">,</span> qConj)</span> +<span id="cb3-33"><a href="#cb3-33" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> { <span class="dt">x</span><span class="op">:</span> result[<span class="dv">1</span>]<span class="op">,</span> <span class="dt">y</span><span class="op">:</span> result[<span class="dv">2</span>]<span class="op">,</span> <span class="dt">z</span><span class="op">:</span> result[<span class="dv">3</span>] }</span> +<span id="cb3-34"><a href="#cb3-34" aria-hidden="true" tabindex="-1"></a>}</span></code></pre></div> +<hr /> +<h3 id="using-quaternions-in-your-frame">4. <strong>Using Quaternions in +Your Frame</strong></h3> +<p>Instead of your <code>rotate</code> function:</p> +<div class="sourceCode" id="cb4"><pre class="sourceCode js"><code class="sourceCode javascript"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> angleX <span class="op">=</span> angle<span class="op">;</span> </span> +<span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> angleY <span class="op">=</span> angle<span class="op">/</span><span class="dv">2</span><span class="op">;</span> <span class="co">// different speed for demo</span></span> +<span id="cb4-3"><a href="#cb4-3" aria-hidden="true" tabindex="-1"></a></span> +<span id="cb4-4"><a href="#cb4-4" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> qx <span class="op">=</span> <span class="fu">quatFromAxisAngle</span>({<span class="dt">x</span><span class="op">:</span><span class="dv">1</span><span class="op">,</span><span class="dt">y</span><span class="op">:</span><span class="dv">0</span><span class="op">,</span><span class="dt">z</span><span class="op">:</span><span class="dv">0</span>}<span class="op">,</span> angleX)</span> +<span id="cb4-5"><a href="#cb4-5" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> qy <span class="op">=</span> <span class="fu">quatFromAxisAngle</span>({<span class="dt">x</span><span class="op">:</span><span class="dv">0</span><span class="op">,</span><span class="dt">y</span><span class="op">:</span><span class="dv">1</span><span class="op">,</span><span class="dt">z</span><span class="op">:</span><span class="dv">0</span>}<span class="op">,</span> angleY)</span> +<span id="cb4-6"><a href="#cb4-6" aria-hidden="true" tabindex="-1"></a></span> +<span id="cb4-7"><a href="#cb4-7" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> q <span class="op">=</span> <span class="fu">quatMultiply</span>(qy<span class="op">,</span> qx) <span class="co">// combined rotation</span></span> +<span id="cb4-8"><a href="#cb4-8" aria-hidden="true" tabindex="-1"></a></span> +<span id="cb4-9"><a href="#cb4-9" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> (<span class="kw">const</span> f <span class="kw">of</span> fs) {</span> +<span id="cb4-10"><a href="#cb4-10" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span> (<span class="kw">let</span> i <span class="op">=</span> <span class="dv">0</span><span class="op">;</span> i <span class="op">&lt;</span> f<span class="op">.</span><span class="at">length</span><span class="op">;</span> <span class="op">++</span>i) {</span> +<span id="cb4-11"><a href="#cb4-11" aria-hidden="true" tabindex="-1"></a> <span class="kw">const</span> a <span class="op">=</span> vs[f[i]]</span> +<span id="cb4-12"><a href="#cb4-12" aria-hidden="true" tabindex="-1"></a> <span class="kw">const</span> b <span class="op">=</span> vs[f[(i <span class="op">+</span> <span class="dv">1</span>)<span class="op">%</span>f<span class="op">.</span><span class="at">length</span>]]</span> +<span id="cb4-13"><a href="#cb4-13" aria-hidden="true" tabindex="-1"></a></span> +<span id="cb4-14"><a href="#cb4-14" aria-hidden="true" tabindex="-1"></a> <span class="fu">line</span>(</span> +<span id="cb4-15"><a href="#cb4-15" aria-hidden="true" tabindex="-1"></a> <span class="bu">screen</span>(<span class="fu">project</span>(<span class="fu">translateZ</span>(<span class="fu">quatRotateVector</span>(q<span class="op">,</span> a)<span class="op">,</span> dz)))<span class="op">,</span></span> +<span id="cb4-16"><a href="#cb4-16" aria-hidden="true" tabindex="-1"></a> <span class="bu">screen</span>(<span class="fu">project</span>(<span class="fu">translateZ</span>(<span class="fu">quatRotateVector</span>(q<span class="op">,</span> b)<span class="op">,</span> dz)))</span> +<span id="cb4-17"><a href="#cb4-17" aria-hidden="true" tabindex="-1"></a> )</span> +<span id="cb4-18"><a href="#cb4-18" aria-hidden="true" tabindex="-1"></a> }</span> +<span id="cb4-19"><a href="#cb4-19" aria-hidden="true" tabindex="-1"></a>}</span></code></pre></div> +<ul> +<li>Now the cube rotates around <strong>both X and Y axes +simultaneously</strong>.</li> +<li>You can extend this to any combination of axes by multiplying more +quaternions.</li> +</ul> +<hr /> +<h3 id="optional-change-axes-dynamically">5. <strong>Optional: Change +Axes Dynamically</strong></h3> +<p>You could replace your <code>rotationAxis</code> system by letting +the user click to toggle <strong>which axes are active</strong>, then +generate quaternions for each active axis and multiply them +together.</p> +<hr /> +<p>If you want, I can rewrite your <strong>entire code snippet using +quaternions</strong>, so the cube rotates smoothly around multiple axes +and keeps your click-to-change-axis feature.</p> +<p>Do you want me to do that?</p> +<p>Absolutely! Let’s go through your <strong>quaternion-based 3D cube +code</strong> line by line. I’ll break it down into sections so it’s +easier to digest.</p> +<hr /> +<h2 id="canvas-setup"><strong>1. Canvas Setup</strong></h2> +<div class="sourceCode" id="cb5"><pre class="sourceCode js"><code class="sourceCode javascript"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> game <span class="op">=</span> <span class="bu">document</span><span class="op">.</span><span class="fu">getElementById</span>(<span class="st">&quot;game&quot;</span>)<span class="op">;</span></span> +<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a>game<span class="op">.</span><span class="at">width</span> <span class="op">=</span> <span class="dv">400</span><span class="op">;</span></span> +<span id="cb5-3"><a href="#cb5-3" aria-hidden="true" tabindex="-1"></a>game<span class="op">.</span><span class="at">height</span> <span class="op">=</span> <span class="dv">400</span><span class="op">;</span></span></code></pre></div> +<ul> +<li>Gets the <code>&lt;canvas&gt;</code> element with ID +<code>game</code>.</li> +<li>Sets the canvas dimensions to 400×400 pixels.</li> +</ul> +<div class="sourceCode" id="cb6"><pre class="sourceCode js"><code class="sourceCode javascript"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> BACKGROUND <span class="op">=</span> <span class="st">&quot;#101010&quot;</span><span class="op">;</span></span> +<span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> FOREGROUND <span class="op">=</span> <span class="st">&quot;#50FF50&quot;</span><span class="op">;</span></span> +<span id="cb6-3"><a href="#cb6-3" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> TEXT <span class="op">=</span> <span class="st">&quot;#FFFFFF&quot;</span><span class="op">;</span></span></code></pre></div> +<ul> +<li>Defines colors for background, cube lines, and text.</li> +</ul> +<div class="sourceCode" id="cb7"><pre class="sourceCode js"><code class="sourceCode javascript"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> ctx <span class="op">=</span> game<span class="op">.</span><span class="fu">getContext</span>(<span class="st">&quot;2d&quot;</span>)<span class="op">;</span></span></code></pre></div> +<ul> +<li>Gets the 2D drawing context so we can draw shapes on the +canvas.</li> +</ul> +<hr /> +<h2 id="cube-geometry"><strong>2. Cube Geometry</strong></h2> +<div class="sourceCode" id="cb8"><pre class="sourceCode js"><code class="sourceCode javascript"><span id="cb8-1"><a href="#cb8-1" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> vs <span class="op">=</span> [ <span class="op">...</span> ]<span class="op">;</span></span></code></pre></div> +<ul> +<li><code>vs</code> = <strong>vertices</strong> of a cube (8 +points).</li> +<li>Each vertex has <code>x, y, z</code> coordinates in 3D space (values +between -0.25 and 0.25).</li> +</ul> +<div class="sourceCode" id="cb9"><pre class="sourceCode js"><code class="sourceCode javascript"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> fs <span class="op">=</span> [ <span class="op">...</span> ]<span class="op">;</span></span></code></pre></div> +<ul> +<li><code>fs</code> = <strong>faces/edges</strong> to connect +vertices.</li> +<li>Some are quads <code>[0,1,2,3]</code>, some are edges +<code>[0,4]</code>.</li> +<li>This tells the code <strong>which points to draw lines +between</strong>.</li> +</ul> +<div class="sourceCode" id="cb10"><pre +class="sourceCode js"><code class="sourceCode javascript"><span id="cb10-1"><a href="#cb10-1" aria-hidden="true" tabindex="-1"></a><span class="kw">let</span> dz <span class="op">=</span> <span class="dv">1</span><span class="op">;</span></span> +<span id="cb10-2"><a href="#cb10-2" aria-hidden="true" tabindex="-1"></a><span class="kw">let</span> angle <span class="op">=</span> <span class="dv">0</span><span class="op">;</span></span> +<span id="cb10-3"><a href="#cb10-3" aria-hidden="true" tabindex="-1"></a><span class="kw">let</span> activeAxes <span class="op">=</span> [<span class="st">&quot;x&quot;</span>]<span class="op">;</span></span></code></pre></div> +<ul> +<li><code>dz</code>: distance to move the cube along Z (so it’s in front +of the camera).</li> +<li><code>angle</code>: current rotation angle.</li> +<li><code>activeAxes</code>: which axes to rotate around (initially just +<code>x</code>).</li> +</ul> +<hr /> +<h2 id="quaternion-functions"><strong>3. Quaternion +Functions</strong></h2> +<p><strong>a. Create a quaternion from an axis + angle</strong></p> +<div class="sourceCode" id="cb11"><pre +class="sourceCode js"><code class="sourceCode javascript"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a><span class="kw">function</span> <span class="fu">quatFromAxisAngle</span>(axis<span class="op">,</span> angle) {</span> +<span id="cb11-2"><a href="#cb11-2" aria-hidden="true" tabindex="-1"></a> <span class="kw">const</span> half <span class="op">=</span> angle <span class="op">/</span> <span class="dv">2</span><span class="op">;</span></span> +<span id="cb11-3"><a href="#cb11-3" aria-hidden="true" tabindex="-1"></a> <span class="kw">const</span> s <span class="op">=</span> <span class="bu">Math</span><span class="op">.</span><span class="fu">sin</span>(half)<span class="op">;</span></span> +<span id="cb11-4"><a href="#cb11-4" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> [</span> +<span id="cb11-5"><a href="#cb11-5" aria-hidden="true" tabindex="-1"></a> <span class="bu">Math</span><span class="op">.</span><span class="fu">cos</span>(half)<span class="op">,</span> <span class="co">// w</span></span> +<span id="cb11-6"><a href="#cb11-6" aria-hidden="true" tabindex="-1"></a> axis<span class="op">.</span><span class="at">x</span> <span class="op">*</span> s<span class="op">,</span> <span class="co">// x</span></span> +<span id="cb11-7"><a href="#cb11-7" aria-hidden="true" tabindex="-1"></a> axis<span class="op">.</span><span class="at">y</span> <span class="op">*</span> s<span class="op">,</span> <span class="co">// y</span></span> +<span id="cb11-8"><a href="#cb11-8" aria-hidden="true" tabindex="-1"></a> axis<span class="op">.</span><span class="at">z</span> <span class="op">*</span> s <span class="co">// z</span></span> +<span id="cb11-9"><a href="#cb11-9" aria-hidden="true" tabindex="-1"></a> ]<span class="op">;</span></span> +<span id="cb11-10"><a href="#cb11-10" aria-hidden="true" tabindex="-1"></a>}</span></code></pre></div> +<ul> +<li>Converts <strong>axis-angle rotation</strong> into a quaternion +<code>[w, x, y, z]</code>.</li> +<li><code>half</code> = half the rotation angle.</li> +<li><code>s = sin(half)</code> scales the axis.</li> +<li><code>w</code> = cos(half).</li> +</ul> +<p><strong>b. Multiply two quaternions</strong></p> +<div class="sourceCode" id="cb12"><pre +class="sourceCode js"><code class="sourceCode javascript"><span id="cb12-1"><a href="#cb12-1" aria-hidden="true" tabindex="-1"></a><span class="kw">function</span> <span class="fu">quatMultiply</span>(q1<span class="op">,</span> q2) { <span class="op">...</span> }</span></code></pre></div> +<ul> +<li>Standard quaternion multiplication formula.</li> +<li>Combines two rotations into one. <strong>Order matters!</strong> +<code>q1*q2</code> rotates q2 first, then q1.</li> +</ul> +<p><strong>c. Rotate a vector by a quaternion</strong></p> +<div class="sourceCode" id="cb13"><pre +class="sourceCode js"><code class="sourceCode javascript"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a><span class="kw">function</span> <span class="fu">quatRotateVector</span>(q<span class="op">,</span> v) { <span class="op">...</span> }</span></code></pre></div> +<ul> +<li><p>Rotates a 3D vector <code>v</code> using quaternion +<code>q</code>.</p></li> +<li><p>Steps:</p> +<ol type="1"> +<li>Convert vector to quaternion <code>qv = [0, vx, vy, vz]</code>.</li> +<li>Compute conjugate <code>qConj = [w, -x, -y, -z]</code>.</li> +<li>Rotate: <code>v' = q * qv * q⁻¹</code>.</li> +<li>Return new <code>x,y,z</code> of rotated vector.</li> +</ol></li> +</ul> +<hr /> +<h2 id="drawing-functions"><strong>4. Drawing Functions</strong></h2> +<p><strong>Clear canvas and draw text</strong></p> +<div class="sourceCode" id="cb14"><pre +class="sourceCode js"><code class="sourceCode javascript"><span id="cb14-1"><a href="#cb14-1" aria-hidden="true" tabindex="-1"></a><span class="kw">function</span> <span class="fu">clear</span>() {</span> +<span id="cb14-2"><a href="#cb14-2" aria-hidden="true" tabindex="-1"></a> ctx<span class="op">.</span><span class="at">fillStyle</span> <span class="op">=</span> BACKGROUND<span class="op">;</span></span> +<span id="cb14-3"><a href="#cb14-3" aria-hidden="true" tabindex="-1"></a> ctx<span class="op">.</span><span class="fu">fillRect</span>(<span class="dv">0</span><span class="op">,</span> <span class="dv">0</span><span class="op">,</span> game<span class="op">.</span><span class="at">width</span><span class="op">,</span> game<span class="op">.</span><span class="at">height</span>)<span class="op">;</span></span> +<span id="cb14-4"><a href="#cb14-4" aria-hidden="true" tabindex="-1"></a> ctx<span class="op">.</span><span class="at">fillStyle</span> <span class="op">=</span> TEXT<span class="op">;</span></span> +<span id="cb14-5"><a href="#cb14-5" aria-hidden="true" tabindex="-1"></a> ctx<span class="op">.</span><span class="at">font</span> <span class="op">=</span> <span class="st">&quot;20px serif&quot;</span><span class="op">;</span></span> +<span id="cb14-6"><a href="#cb14-6" aria-hidden="true" tabindex="-1"></a> ctx<span class="op">.</span><span class="fu">fillText</span>(<span class="st">&quot; *click me to change rotation axes&quot;</span><span class="op">,</span> <span class="dv">10</span><span class="op">,</span> game<span class="op">.</span><span class="at">height</span> <span class="op">-</span> <span class="dv">15</span>)<span class="op">;</span></span> +<span id="cb14-7"><a href="#cb14-7" aria-hidden="true" tabindex="-1"></a> ctx<span class="op">.</span><span class="fu">fillText</span>(<span class="st">&quot;current axes: &quot;</span> <span class="op">+</span> activeAxes<span class="op">.</span><span class="fu">join</span>(<span class="st">&quot;,&quot;</span>)<span class="op">,</span> <span class="dv">10</span><span class="op">,</span> <span class="dv">20</span>)<span class="op">;</span></span> +<span id="cb14-8"><a href="#cb14-8" aria-hidden="true" tabindex="-1"></a>}</span></code></pre></div> +<ul> +<li>Clears the canvas with the background color.</li> +<li>Draws instructions and current axes at the top/bottom.</li> +</ul> +<p><strong>Draw a point</strong></p> +<div class="sourceCode" id="cb15"><pre +class="sourceCode js"><code class="sourceCode javascript"><span id="cb15-1"><a href="#cb15-1" aria-hidden="true" tabindex="-1"></a><span class="kw">function</span> <span class="fu">point</span>({x<span class="op">,</span> y}) {</span> +<span id="cb15-2"><a href="#cb15-2" aria-hidden="true" tabindex="-1"></a> <span class="kw">const</span> s <span class="op">=</span> <span class="dv">20</span><span class="op">;</span></span> +<span id="cb15-3"><a href="#cb15-3" aria-hidden="true" tabindex="-1"></a> ctx<span class="op">.</span><span class="at">fillStyle</span> <span class="op">=</span> FOREGROUND<span class="op">;</span></span> +<span id="cb15-4"><a href="#cb15-4" aria-hidden="true" tabindex="-1"></a> ctx<span class="op">.</span><span class="fu">fillRect</span>(x <span class="op">-</span> s<span class="op">/</span><span class="dv">2</span><span class="op">,</span> y <span class="op">-</span> s<span class="op">/</span><span class="dv">2</span><span class="op">,</span> s<span class="op">,</span> s)<span class="op">;</span></span> +<span id="cb15-5"><a href="#cb15-5" aria-hidden="true" tabindex="-1"></a>}</span></code></pre></div> +<ul> +<li>Draws a small square at <code>(x, y)</code> on the screen.</li> +<li>Not used in the final version, but useful for debugging +vertices.</li> +</ul> +<p><strong>Draw a line</strong></p> +<div class="sourceCode" id="cb16"><pre +class="sourceCode js"><code class="sourceCode javascript"><span id="cb16-1"><a href="#cb16-1" aria-hidden="true" tabindex="-1"></a><span class="kw">function</span> <span class="fu">line</span>(p1<span class="op">,</span> p2) { <span class="op">...</span> }</span></code></pre></div> +<ul> +<li>Draws a thick line from <code>p1</code> to <code>p2</code> in canvas +coordinates.</li> +</ul> +<p><strong>Convert 3D to screen coordinates</strong></p> +<div class="sourceCode" id="cb17"><pre +class="sourceCode js"><code class="sourceCode javascript"><span id="cb17-1"><a href="#cb17-1" aria-hidden="true" tabindex="-1"></a><span class="kw">function</span> <span class="fu">screen</span>(p) {</span> +<span id="cb17-2"><a href="#cb17-2" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> {</span> +<span id="cb17-3"><a href="#cb17-3" aria-hidden="true" tabindex="-1"></a> <span class="dt">x</span><span class="op">:</span> (p<span class="op">.</span><span class="at">x</span> <span class="op">+</span> <span class="dv">1</span>)<span class="op">/</span><span class="dv">2</span> <span class="op">*</span> game<span class="op">.</span><span class="at">width</span><span class="op">,</span></span> +<span id="cb17-4"><a href="#cb17-4" aria-hidden="true" tabindex="-1"></a> <span class="dt">y</span><span class="op">:</span> (<span class="dv">1</span> <span class="op">-</span>(p<span class="op">.</span><span class="at">y</span> <span class="op">+</span> <span class="dv">1</span>)<span class="op">/</span><span class="dv">2</span>) <span class="op">*</span> game<span class="op">.</span><span class="at">height</span></span> +<span id="cb17-5"><a href="#cb17-5" aria-hidden="true" tabindex="-1"></a> }<span class="op">;</span></span> +<span id="cb17-6"><a href="#cb17-6" aria-hidden="true" tabindex="-1"></a>}</span></code></pre></div> +<ul> +<li>Maps normalized coordinates <code>[-1,1]</code> to screen +coordinates <code>[0, width/height]</code>.</li> +<li>Y-axis is flipped because canvas y=0 is top.</li> +</ul> +<p><strong>Project 3D to 2D</strong></p> +<div class="sourceCode" id="cb18"><pre +class="sourceCode js"><code class="sourceCode javascript"><span id="cb18-1"><a href="#cb18-1" aria-hidden="true" tabindex="-1"></a><span class="kw">function</span> <span class="fu">project</span>({x<span class="op">,</span> y<span class="op">,</span> z}) {</span> +<span id="cb18-2"><a href="#cb18-2" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> { <span class="dt">x</span><span class="op">:</span> x<span class="op">/</span>z<span class="op">,</span> <span class="dt">y</span><span class="op">:</span> y<span class="op">/</span>z }<span class="op">;</span></span> +<span id="cb18-3"><a href="#cb18-3" aria-hidden="true" tabindex="-1"></a>}</span></code></pre></div> +<ul> +<li>Simple <strong>perspective projection</strong>: divides x and y by z +(makes things farther away appear smaller).</li> +</ul> +<p><strong>Translate along Z</strong></p> +<div class="sourceCode" id="cb19"><pre +class="sourceCode js"><code class="sourceCode javascript"><span id="cb19-1"><a href="#cb19-1" aria-hidden="true" tabindex="-1"></a><span class="kw">function</span> <span class="fu">translateZ</span>({x<span class="op">,</span>y<span class="op">,</span>z}<span class="op">,</span> dz) {</span> +<span id="cb19-2"><a href="#cb19-2" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> {x<span class="op">,</span> y<span class="op">,</span> <span class="dt">z</span><span class="op">:</span> z <span class="op">+</span> dz}<span class="op">;</span></span> +<span id="cb19-3"><a href="#cb19-3" aria-hidden="true" tabindex="-1"></a>}</span></code></pre></div> +<ul> +<li>Moves the cube forward/backward so it appears in front of the +camera.</li> +</ul> +<hr /> +<h2 id="animation-loop"><strong>5. Animation Loop</strong></h2> +<div class="sourceCode" id="cb20"><pre +class="sourceCode js"><code class="sourceCode javascript"><span id="cb20-1"><a href="#cb20-1" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> FPS <span class="op">=</span> <span class="dv">60</span><span class="op">;</span></span> +<span id="cb20-2"><a href="#cb20-2" aria-hidden="true" tabindex="-1"></a></span> +<span id="cb20-3"><a href="#cb20-3" aria-hidden="true" tabindex="-1"></a><span class="kw">function</span> <span class="fu">frame</span>() {</span> +<span id="cb20-4"><a href="#cb20-4" aria-hidden="true" tabindex="-1"></a> <span class="kw">const</span> dt <span class="op">=</span> <span class="dv">1</span><span class="op">/</span>FPS<span class="op">;</span></span> +<span id="cb20-5"><a href="#cb20-5" aria-hidden="true" tabindex="-1"></a> angle <span class="op">+=</span> <span class="bu">Math</span><span class="op">.</span><span class="cn">PI</span><span class="op">*</span>dt<span class="op">;</span></span></code></pre></div> +<ul> +<li>Updates rotation angle every frame based on FPS.</li> +</ul> +<p><strong>Create combined quaternion</strong></p> +<div class="sourceCode" id="cb21"><pre +class="sourceCode js"><code class="sourceCode javascript"><span id="cb21-1"><a href="#cb21-1" aria-hidden="true" tabindex="-1"></a> <span class="kw">let</span> q <span class="op">=</span> [<span class="dv">1</span><span class="op">,</span><span class="dv">0</span><span class="op">,</span><span class="dv">0</span><span class="op">,</span><span class="dv">0</span>]<span class="op">;</span> <span class="co">// identity quaternion</span></span> +<span id="cb21-2"><a href="#cb21-2" aria-hidden="true" tabindex="-1"></a> activeAxes<span class="op">.</span><span class="fu">forEach</span>(axis <span class="kw">=&gt;</span> {</span> +<span id="cb21-3"><a href="#cb21-3" aria-hidden="true" tabindex="-1"></a> <span class="kw">let</span> qAxis<span class="op">;</span></span> +<span id="cb21-4"><a href="#cb21-4" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> (axis <span class="op">===</span> <span class="st">&quot;x&quot;</span>) qAxis <span class="op">=</span> <span class="fu">quatFromAxisAngle</span>({<span class="dt">x</span><span class="op">:</span><span class="dv">1</span><span class="op">,</span><span class="dt">y</span><span class="op">:</span><span class="dv">0</span><span class="op">,</span><span class="dt">z</span><span class="op">:</span><span class="dv">0</span>}<span class="op">,</span> angle)<span class="op">;</span></span> +<span id="cb21-5"><a href="#cb21-5" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> (axis <span class="op">===</span> <span class="st">&quot;y&quot;</span>) qAxis <span class="op">=</span> <span class="fu">quatFromAxisAngle</span>({<span class="dt">x</span><span class="op">:</span><span class="dv">0</span><span class="op">,</span><span class="dt">y</span><span class="op">:</span><span class="dv">1</span><span class="op">,</span><span class="dt">z</span><span class="op">:</span><span class="dv">0</span>}<span class="op">,</span> angle)<span class="op">;</span></span> +<span id="cb21-6"><a href="#cb21-6" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> (axis <span class="op">===</span> <span class="st">&quot;z&quot;</span>) qAxis <span class="op">=</span> <span class="fu">quatFromAxisAngle</span>({<span class="dt">x</span><span class="op">:</span><span class="dv">0</span><span class="op">,</span><span class="dt">y</span><span class="op">:</span><span class="dv">0</span><span class="op">,</span><span class="dt">z</span><span class="op">:</span><span class="dv">1</span>}<span class="op">,</span> angle)<span class="op">;</span></span> +<span id="cb21-7"><a href="#cb21-7" aria-hidden="true" tabindex="-1"></a> q <span class="op">=</span> <span class="fu">quatMultiply</span>(qAxis<span class="op">,</span> q)<span class="op">;</span> <span class="co">// multiply in order</span></span> +<span id="cb21-8"><a href="#cb21-8" aria-hidden="true" tabindex="-1"></a> })<span class="op">;</span></span></code></pre></div> +<ul> +<li>Starts with <strong>identity quaternion</strong> (no rotation).</li> +<li>For each active axis, creates a quaternion for rotation around that +axis.</li> +<li>Combines them by multiplying — this lets the cube rotate around +multiple axes <strong>simultaneously</strong>.</li> +</ul> +<p><strong>Draw the cube</strong></p> +<div class="sourceCode" id="cb22"><pre +class="sourceCode js"><code class="sourceCode javascript"><span id="cb22-1"><a href="#cb22-1" aria-hidden="true" tabindex="-1"></a> <span class="fu">clear</span>()<span class="op">;</span></span> +<span id="cb22-2"><a href="#cb22-2" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span> (<span class="kw">const</span> f <span class="kw">of</span> fs) {</span> +<span id="cb22-3"><a href="#cb22-3" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span> (<span class="kw">let</span> i <span class="op">=</span> <span class="dv">0</span><span class="op">;</span> i <span class="op">&lt;</span> f<span class="op">.</span><span class="at">length</span><span class="op">;</span> <span class="op">++</span>i) {</span> +<span id="cb22-4"><a href="#cb22-4" aria-hidden="true" tabindex="-1"></a> <span class="kw">const</span> a <span class="op">=</span> vs[f[i]]<span class="op">;</span></span> +<span id="cb22-5"><a href="#cb22-5" aria-hidden="true" tabindex="-1"></a> <span class="kw">const</span> b <span class="op">=</span> vs[f[(i <span class="op">+</span> <span class="dv">1</span>)<span class="op">%</span>f<span class="op">.</span><span class="at">length</span>]]<span class="op">;</span></span> +<span id="cb22-6"><a href="#cb22-6" aria-hidden="true" tabindex="-1"></a> <span class="fu">line</span>(</span> +<span id="cb22-7"><a href="#cb22-7" aria-hidden="true" tabindex="-1"></a> <span class="bu">screen</span>(<span class="fu">project</span>(<span class="fu">translateZ</span>(<span class="fu">quatRotateVector</span>(q<span class="op">,</span> a)<span class="op">,</span> dz)))<span class="op">,</span></span> +<span id="cb22-8"><a href="#cb22-8" aria-hidden="true" tabindex="-1"></a> <span class="bu">screen</span>(<span class="fu">project</span>(<span class="fu">translateZ</span>(<span class="fu">quatRotateVector</span>(q<span class="op">,</span> b)<span class="op">,</span> dz)))</span> +<span id="cb22-9"><a href="#cb22-9" aria-hidden="true" tabindex="-1"></a> )<span class="op">;</span></span> +<span id="cb22-10"><a href="#cb22-10" aria-hidden="true" tabindex="-1"></a> }</span> +<span id="cb22-11"><a href="#cb22-11" aria-hidden="true" tabindex="-1"></a> }</span></code></pre></div> +<ul> +<li>Clears the canvas.</li> +<li>Loops through each face/edge <code>f</code>.</li> +<li>Rotates each vertex using the combined quaternion.</li> +<li>Translates it forward along Z.</li> +<li>Projects 3D coordinates into 2D screen coordinates.</li> +<li>Draws lines connecting vertices.</li> +</ul> +<p><strong>Schedule next frame</strong></p> +<div class="sourceCode" id="cb23"><pre +class="sourceCode js"><code class="sourceCode javascript"><span id="cb23-1"><a href="#cb23-1" aria-hidden="true" tabindex="-1"></a> <span class="pp">setTimeout</span>(frame<span class="op">,</span> <span class="dv">1000</span><span class="op">/</span>FPS)<span class="op">;</span></span> +<span id="cb23-2"><a href="#cb23-2" aria-hidden="true" tabindex="-1"></a>}</span> +<span id="cb23-3"><a href="#cb23-3" aria-hidden="true" tabindex="-1"></a><span class="pp">setTimeout</span>(frame<span class="op">,</span> <span class="dv">1000</span><span class="op">/</span>FPS)<span class="op">;</span></span></code></pre></div> +<ul> +<li>Calls <code>frame()</code> every 1/60th of a second to animate.</li> +</ul> +<hr /> +<h2 id="handle-clicks"><strong>6. Handle Clicks</strong></h2> +<div class="sourceCode" id="cb24"><pre +class="sourceCode js"><code class="sourceCode javascript"><span id="cb24-1"><a href="#cb24-1" aria-hidden="true" tabindex="-1"></a>game<span class="op">.</span><span class="at">onclick</span> <span class="op">=</span> () <span class="kw">=&gt;</span> {</span> +<span id="cb24-2"><a href="#cb24-2" aria-hidden="true" tabindex="-1"></a> <span class="kw">const</span> allAxes <span class="op">=</span> [[<span class="st">&quot;x&quot;</span>]<span class="op">,</span> [<span class="st">&quot;y&quot;</span>]<span class="op">,</span> [<span class="st">&quot;z&quot;</span>]<span class="op">,</span> [<span class="st">&quot;x&quot;</span><span class="op">,</span><span class="st">&quot;y&quot;</span>]<span class="op">,</span> [<span class="st">&quot;y&quot;</span><span class="op">,</span><span class="st">&quot;z&quot;</span>]<span class="op">,</span> [<span class="st">&quot;x&quot;</span><span class="op">,</span><span class="st">&quot;z&quot;</span>]<span class="op">,</span> [<span class="st">&quot;x&quot;</span><span class="op">,</span><span class="st">&quot;y&quot;</span><span class="op">,</span><span class="st">&quot;z&quot;</span>]]<span class="op">;</span></span> +<span id="cb24-3"><a href="#cb24-3" aria-hidden="true" tabindex="-1"></a> <span class="kw">const</span> index <span class="op">=</span> allAxes<span class="op">.</span><span class="fu">findIndex</span>(a <span class="kw">=&gt;</span> <span class="bu">JSON</span><span class="op">.</span><span class="fu">stringify</span>(a) <span class="op">===</span> <span class="bu">JSON</span><span class="op">.</span><span class="fu">stringify</span>(activeAxes))<span class="op">;</span></span> +<span id="cb24-4"><a href="#cb24-4" aria-hidden="true" tabindex="-1"></a> activeAxes <span class="op">=</span> allAxes[(index <span class="op">+</span> <span class="dv">1</span>) <span class="op">%</span> allAxes<span class="op">.</span><span class="at">length</span>]<span class="op">;</span></span> +<span id="cb24-5"><a href="#cb24-5" aria-hidden="true" tabindex="-1"></a>}<span class="op">;</span></span></code></pre></div> +<ul> +<li>On click, cycles through different <strong>axis +combinations</strong>: x, y, z, xy, yz, xz, xyz.</li> +<li>Updates <code>activeAxes</code> so the cube rotates differently next +frame.</li> +</ul> +<hr /> +<h3 id="summary">✅ <strong>Summary</strong></h3> +<ul> +<li><strong>Vertices + Faces</strong>: Define cube geometry.</li> +<li><strong>Quaternions</strong>: Handle rotation around multiple axes +safely.</li> +<li><strong>Project &amp; Screen Functions</strong>: Convert 3D points +to 2D canvas.</li> +<li><strong>Animation Loop</strong>: Updates rotation, applies +quaternion rotation, and draws cube.</li> +<li><strong>Click Handler</strong>: Change which axes to rotate +around.</li> +</ul> +<hr /> +<p>If you want, I can also <strong>draw a diagram showing the quaternion +rotation concept</strong> visually — it really helps to understand how +it rotates the cube around multiple axes at once.</p> +<p>Do you want me to make that diagram?</p> +<figure> +<img +src="https://nyquistdev.net/git/?p=basic-3d-example.git;a=blob_plain;f=docs/chatgpt-dump/quaternion-rotation.png;hb=HEAD" +alt="diagram" /> +<figcaption aria-hidden="true">diagram</figcaption> +</figure> +<p>Great question — this is exactly the right point to slow down and +understand the <strong>math</strong>, because quaternions feel “magical” +until you see <em>why</em> each line exists.</p> +<p>I’ll go step by step:</p> +<hr /> +<h1 id="what-is-a-quaternion-in-graphics">1. What is a quaternion (in +graphics)?</h1> +<p>A <strong>quaternion</strong> is a 4-component number:</p> +<p>[ q = (w, x, y, z)]</p> +<p>In 3D graphics, we use <strong>unit quaternions</strong> to represent +<strong>rotations</strong>.</p> +<p>A rotation by:</p> +<ul> +<li>angle ( )</li> +<li>around a <strong>unit axis</strong> ( = (u_x, u_y, u_z) )</li> +</ul> +<p>is encoded as:</p> +<p>[ ]</p> +<p>This is the <strong>axis–angle → quaternion</strong> formula.</p> +<p>Why half-angle? Because quaternion rotation comes from complex +numbers and Euler’s formula — doubling happens naturally in the +math.</p> +<hr /> +<h1 id="quatfromaxisangle">2. <code>quatFromAxisAngle</code></h1> +<div class="sourceCode" id="cb25"><pre +class="sourceCode js"><code class="sourceCode javascript"><span id="cb25-1"><a href="#cb25-1" aria-hidden="true" tabindex="-1"></a><span class="kw">function</span> <span class="fu">quatFromAxisAngle</span>(axis<span class="op">,</span> angle) {</span> +<span id="cb25-2"><a href="#cb25-2" aria-hidden="true" tabindex="-1"></a> <span class="kw">const</span> h <span class="op">=</span> angle <span class="op">/</span> <span class="dv">2</span><span class="op">;</span></span> +<span id="cb25-3"><a href="#cb25-3" aria-hidden="true" tabindex="-1"></a> <span class="kw">const</span> s <span class="op">=</span> <span class="bu">Math</span><span class="op">.</span><span class="fu">sin</span>(h)<span class="op">;</span></span> +<span id="cb25-4"><a href="#cb25-4" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> [<span class="bu">Math</span><span class="op">.</span><span class="fu">cos</span>(h)<span class="op">,</span> axis<span class="op">.</span><span class="at">x</span><span class="op">*</span>s<span class="op">,</span> axis<span class="op">.</span><span class="at">y</span><span class="op">*</span>s<span class="op">,</span> axis<span class="op">.</span><span class="at">z</span><span class="op">*</span>s]<span class="op">;</span></span> +<span id="cb25-5"><a href="#cb25-5" aria-hidden="true" tabindex="-1"></a>}</span></code></pre></div> +<h3 id="math-behind-it">Math behind it</h3> +<p>You’re directly implementing:</p> +<p>[ q = ((/2), (/2))]</p> +<ul> +<li><code>h = angle / 2</code> → half-angle</li> +<li><code>cos(h)</code> → scalar part (w)</li> +<li><code>axis * sin(h)</code> → vector part ((x,y,z))</li> +</ul> +<h3 id="important-rule">Important rule</h3> +<p><code>axis</code> <strong>must be normalized</strong> (length = 1), +otherwise rotation will scale or skew.</p> +<hr /> +<h1 id="quaternion-multiplication-rotation-composition">3. Quaternion +multiplication (rotation composition)</h1> +<div class="sourceCode" id="cb26"><pre +class="sourceCode js"><code class="sourceCode javascript"><span id="cb26-1"><a href="#cb26-1" aria-hidden="true" tabindex="-1"></a><span class="kw">function</span> <span class="fu">quatMultiply</span>(a<span class="op">,</span> b) {</span> +<span id="cb26-2"><a href="#cb26-2" aria-hidden="true" tabindex="-1"></a> <span class="kw">const</span> [w1<span class="op">,</span>x1<span class="op">,</span>y1<span class="op">,</span>z1] <span class="op">=</span> a<span class="op">;</span></span> +<span id="cb26-3"><a href="#cb26-3" aria-hidden="true" tabindex="-1"></a> <span class="kw">const</span> [w2<span class="op">,</span>x2<span class="op">,</span>y2<span class="op">,</span>z2] <span class="op">=</span> b<span class="op">;</span></span> +<span id="cb26-4"><a href="#cb26-4" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> [</span> +<span id="cb26-5"><a href="#cb26-5" aria-hidden="true" tabindex="-1"></a> w1<span class="op">*</span>w2 <span class="op">-</span> x1<span class="op">*</span>x2 <span class="op">-</span> y1<span class="op">*</span>y2 <span class="op">-</span> z1<span class="op">*</span>z2<span class="op">,</span></span> +<span id="cb26-6"><a href="#cb26-6" aria-hidden="true" tabindex="-1"></a> w1<span class="op">*</span>x2 <span class="op">+</span> x1<span class="op">*</span>w2 <span class="op">+</span> y1<span class="op">*</span>z2 <span class="op">-</span> z1<span class="op">*</span>y2<span class="op">,</span></span> +<span id="cb26-7"><a href="#cb26-7" aria-hidden="true" tabindex="-1"></a> w1<span class="op">*</span>y2 <span class="op">-</span> x1<span class="op">*</span>z2 <span class="op">+</span> y1<span class="op">*</span>w2 <span class="op">+</span> z1<span class="op">*</span>x2<span class="op">,</span></span> +<span id="cb26-8"><a href="#cb26-8" aria-hidden="true" tabindex="-1"></a> w1<span class="op">*</span>z2 <span class="op">+</span> x1<span class="op">*</span>y2 <span class="op">-</span> y1<span class="op">*</span>x2 <span class="op">+</span> z1<span class="op">*</span>w2</span> +<span id="cb26-9"><a href="#cb26-9" aria-hidden="true" tabindex="-1"></a> ]<span class="op">;</span></span> +<span id="cb26-10"><a href="#cb26-10" aria-hidden="true" tabindex="-1"></a>}</span></code></pre></div> +<h3 id="math-formula">Math formula</h3> +<p>Given: [ q_1 = (w_1, _1), q_2 = (w_2, _2)]</p> +<p>Quaternion multiplication is:</p> +<p>[ ]</p> +<p>Expanded component-wise → exactly your code.</p> +<h3 id="why-this-matters">Why this matters</h3> +<ul> +<li>Quaternion multiplication <strong>combines rotations</strong></li> +<li>Order matters: [ q_{} = q_{} q_{}]</li> +</ul> +<p>This avoids <strong>gimbal lock</strong>, unlike Euler angles.</p> +<hr /> +<h1 id="quaternion-conjugate">4. Quaternion conjugate</h1> +<div class="sourceCode" id="cb27"><pre +class="sourceCode js"><code class="sourceCode javascript"><span id="cb27-1"><a href="#cb27-1" aria-hidden="true" tabindex="-1"></a><span class="kw">function</span> <span class="fu">quatConjugate</span>(q) {</span> +<span id="cb27-2"><a href="#cb27-2" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> [q[<span class="dv">0</span>]<span class="op">,</span> <span class="op">-</span>q[<span class="dv">1</span>]<span class="op">,</span> <span class="op">-</span>q[<span class="dv">2</span>]<span class="op">,</span> <span class="op">-</span>q[<span class="dv">3</span>]]<span class="op">;</span></span> +<span id="cb27-3"><a href="#cb27-3" aria-hidden="true" tabindex="-1"></a>}</span></code></pre></div> +<h3 id="math-meaning">Math meaning</h3> +<p>For: [ q = (w, x, y, z)]</p> +<p>The conjugate is:</p> +<p>[ ]</p> +<h3 id="why-this-works">Why this works</h3> +<p>If (q) is a <strong>unit quaternion</strong>:</p> +<p>[ q^{-1} = q^*]</p> +<p>So conjugate = <strong>inverse rotation</strong>.</p> +<p>This is why cameras use conjugates — moving the world opposite to the +camera rotation.</p> +<hr /> +<h1 id="rotating-a-vector-using-quaternions">5. Rotating a vector using +quaternions</h1> +<div class="sourceCode" id="cb28"><pre +class="sourceCode js"><code class="sourceCode javascript"><span id="cb28-1"><a href="#cb28-1" aria-hidden="true" tabindex="-1"></a><span class="kw">function</span> <span class="fu">quatRotateVector</span>(q<span class="op">,</span> v) {</span> +<span id="cb28-2"><a href="#cb28-2" aria-hidden="true" tabindex="-1"></a> <span class="kw">const</span> qv <span class="op">=</span> [<span class="dv">0</span><span class="op">,</span> v<span class="op">.</span><span class="at">x</span><span class="op">,</span> v<span class="op">.</span><span class="at">y</span><span class="op">,</span> v<span class="op">.</span><span class="at">z</span>]<span class="op">;</span></span> +<span id="cb28-3"><a href="#cb28-3" aria-hidden="true" tabindex="-1"></a> <span class="kw">const</span> r <span class="op">=</span> <span class="fu">quatMultiply</span>(<span class="fu">quatMultiply</span>(q<span class="op">,</span> qv)<span class="op">,</span> <span class="fu">quatConjugate</span>(q))<span class="op">;</span></span> +<span id="cb28-4"><a href="#cb28-4" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> { <span class="dt">x</span><span class="op">:</span> r[<span class="dv">1</span>]<span class="op">,</span> <span class="dt">y</span><span class="op">:</span> r[<span class="dv">2</span>]<span class="op">,</span> <span class="dt">z</span><span class="op">:</span> r[<span class="dv">3</span>] }<span class="op">;</span></span> +<span id="cb28-5"><a href="#cb28-5" aria-hidden="true" tabindex="-1"></a>}</span></code></pre></div> +<h3 id="the-key-formula">The key formula</h3> +<p>To rotate a vector ( ):</p> +<ol type="1"> +<li><p>Convert it to a <strong>pure quaternion</strong> [ v_q = (0, v_x, +v_y, v_z)]</p></li> +<li><p>Apply: [ ]</p></li> +<li><p>Extract the vector part</p></li> +</ol> +<h3 id="why-this-works-1">Why this works</h3> +<p>Quaternion rotation encodes:</p> +<ul> +<li>axis</li> +<li>angle</li> +<li>orientation</li> +</ul> +<p>The sandwich product rotates without:</p> +<ul> +<li>scaling</li> +<li>skewing</li> +<li>drifting</li> +</ul> +<hr /> +<h1 id="why-quaternions-are-better-than-matrices-for-rotation">6. Why +quaternions are better than matrices (for rotation)</h1> +<table> +<thead> +<tr class="header"> +<th>Problem</th> +<th>Euler Angles</th> +<th>Matrices</th> +<th>Quaternions</th> +</tr> +</thead> +<tbody> +<tr class="odd"> +<td>Gimbal lock</td> +<td>❌</td> +<td>❌</td> +<td>✅</td> +</tr> +<tr class="even"> +<td>Smooth interpolation</td> +<td>❌</td> +<td>⚠️</td> +<td>✅ (SLERP)</td> +</tr> +<tr class="odd"> +<td>Compact</td> +<td>✅</td> +<td>❌</td> +<td>✅</td> +</tr> +<tr class="even"> +<td>Numerical stability</td> +<td>⚠️</td> +<td>⚠️</td> +<td>✅</td> +</tr> +</tbody> +</table> +<hr /> +<h1 id="visual-intuition">7. Visual intuition</h1> +<p>A rotation quaternion represents:</p> +<p>[ q = (/2) + (/2)(u_x i + u_y j + u_z k)]</p> +<p>This is the 3D analog of:</p> +<p>[ e^{i} = + i]</p> +<hr /> +<h1 id="best-learning-resources-highly-recommended">8. Best learning +resources (highly recommended)</h1> +<h3 id="beginner-friendly">Beginner-friendly</h3> +<ul> +<li><p><strong>3Blue1Brown – Quaternions</strong></p> +<ul> +<li><a +href="https://www.youtube.com/watch?v=zjMuIxRvygQ">https://www.youtube.com/watch?v=zjMuIxRvygQ</a></li> +</ul></li> +<li><p><strong>Visualizing Quaternions</strong></p> +<ul> +<li><a +href="https://eater.net/quaternions">https://eater.net/quaternions</a></li> +</ul></li> +</ul> +<h3 id="graphics-oriented">Graphics-oriented</h3> +<ul> +<li><p><strong>LearnOpenGL – Camera &amp; Quaternions</strong></p> +<ul> +<li><a +href="https://learnopengl.com/Getting-started/Camera">https://learnopengl.com/Getting-started/Camera</a></li> +</ul></li> +<li><p><strong>Scratchapixel – Quaternions</strong></p> +<ul> +<li><a +href="https://www.scratchapixel.com/lessons/mathematics-physics-for-computer-graphics/quaternions">https://www.scratchapixel.com/lessons/mathematics-physics-for-computer-graphics/quaternions</a></li> +</ul></li> +</ul> +<h3 id="math-reference">Math reference</h3> +<ul> +<li><p>Wikipedia (surprisingly solid)</p> +<ul> +<li><a +href="https://en.wikipedia.org/wiki/Quaternion">https://en.wikipedia.org/wiki/Quaternion</a></li> +<li><a +href="https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation">https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation</a></li> +</ul></li> +</ul> +<hr /> +<h1 id="mental-model-to-keep">9. Mental model to keep</h1> +<blockquote> +<p><strong>Quaternions don’t rotate vectors directly. They rotate +<em>space itself</em> around the vector.</strong></p> +</blockquote> +<p>Once that clicks, camera math suddenly makes sense.</p> +<hr /> +<p>If you want next:</p> +<ul> +<li>🔄 quaternion → rotation matrix</li> +<li>🎥 why camera uses inverse rotation</li> +<li>🎯 SLERP (smooth rotation interpolation)</li> +<li>🧠 geometric intuition with diagrams</li> +</ul> +<p>Just tell me 👍</p> +</body> +</html> diff --git a/docs/chatgpt-dump/quaternions.md b/docs/chatgpt-dump/quaternions.md new file mode 100644 index 0000000..c406140 --- /dev/null +++ b/docs/chatgpt-dump/quaternions.md @@ -0,0 +1,982 @@ +Absolutely! I can show you how to incorporate **quaternions** into your code to +allow smooth rotation around **multiple axes at once**, avoiding the limitations +of your current `rotate` function. + +Let’s go step by step. + +--- + +### 1. **Why Quaternions Here** + +Currently, you rotate around a single axis per frame: + +```js +rotate(a, rotationAxis, angle) +``` + +With quaternions, you can **combine rotations around multiple axes** seamlessly. For example, you could rotate simultaneously around X **and** Y. + +--- + +### 2. **Quaternion Basics** + +A quaternion can be represented as: + +[ +q = w + xi + yj + zk +] + +For 3D rotation: + +```js +q = [w, x, y, z] +``` + +* `w` = cos(angle/2) +* `x, y, z` = axis vector * sin(angle/2) + +To rotate a point `v`: + +1. Convert `v` to a quaternion `[0, vx, vy, vz]`. +2. Apply `v' = q * v * q⁻¹`. + +--- + +### 3. **Quaternion Utilities in JS** + +```js +function quatFromAxisAngle(axis, angle) { + const half = angle / 2 + const s = Math.sin(half) + return [ + Math.cos(half), // w + axis.x * s, // x + axis.y * s, // y + axis.z * s // z + ] +} + +function quatMultiply(q1, q2) { + const [w1,x1,y1,z1] = q1 + const [w2,x2,y2,z2] = q2 + return [ + w1*w2 - x1*x2 - y1*y2 - z1*z2, + w1*x2 + x1*w2 + y1*z2 - z1*y2, + w1*y2 - x1*z2 + y1*w2 + z1*x2, + w1*z2 + x1*y2 - y1*x2 + z1*w2 + ] +} + +function quatRotateVector(q, v) { + const [w,x,y,z] = q + const [vx,vy,vz] = [v.x,v.y,v.z] + + // Quaternion multiplication: v' = q * v_quat * q⁻¹ + const qv = [0, vx, vy, vz] + const qConj = [w, -x, -y, -z] + + const temp = quatMultiply(q, qv) + const result = quatMultiply(temp, qConj) + return { x: result[1], y: result[2], z: result[3] } +} +``` + +--- + +### 4. **Using Quaternions in Your Frame** + +Instead of your `rotate` function: + +```js +const angleX = angle; +const angleY = angle/2; // different speed for demo + +const qx = quatFromAxisAngle({x:1,y:0,z:0}, angleX) +const qy = quatFromAxisAngle({x:0,y:1,z:0}, angleY) + +const q = quatMultiply(qy, qx) // combined rotation + +for (const f of fs) { + for (let i = 0; i < f.length; ++i) { + const a = vs[f[i]] + const b = vs[f[(i + 1)%f.length]] + + line( + screen(project(translateZ(quatRotateVector(q, a), dz))), + screen(project(translateZ(quatRotateVector(q, b), dz))) + ) + } +} +``` + +* Now the cube rotates around **both X and Y axes simultaneously**. +* You can extend this to any combination of axes by multiplying more quaternions. + +--- + +### 5. **Optional: Change Axes Dynamically** + +You could replace your `rotationAxis` system by letting the user click to toggle **which axes are active**, then generate quaternions for each active axis and multiply them together. + +--- + +If you want, I can rewrite your **entire code snippet using quaternions**, so the cube rotates smoothly around multiple axes and keeps your click-to-change-axis feature. + +Do you want me to do that? + +Absolutely! Let’s go through your **quaternion-based 3D cube code** line by line. I’ll break it down into sections so it’s easier to digest. + +--- + +## **1. Canvas Setup** + +```js +const game = document.getElementById("game"); +game.width = 400; +game.height = 400; +``` + +* Gets the `<canvas>` element with ID `game`. +* Sets the canvas dimensions to 400×400 pixels. + +```js +const BACKGROUND = "#101010"; +const FOREGROUND = "#50FF50"; +const TEXT = "#FFFFFF"; +``` + +* Defines colors for background, cube lines, and text. + +```js +const ctx = game.getContext("2d"); +``` + +* Gets the 2D drawing context so we can draw shapes on the canvas. + +--- + +## **2. Cube Geometry** + +```js +const vs = [ ... ]; +``` + +* `vs` = **vertices** of a cube (8 points). +* Each vertex has `x, y, z` coordinates in 3D space (values between -0.25 and 0.25). + +```js +const fs = [ ... ]; +``` + +* `fs` = **faces/edges** to connect vertices. +* Some are quads `[0,1,2,3]`, some are edges `[0,4]`. +* This tells the code **which points to draw lines between**. + +```js +let dz = 1; +let angle = 0; +let activeAxes = ["x"]; +``` + +* `dz`: distance to move the cube along Z (so it’s in front of the camera). +* `angle`: current rotation angle. +* `activeAxes`: which axes to rotate around (initially just `x`). + +--- + +## **3. Quaternion Functions** + +**a. Create a quaternion from an axis + angle** + +```js +function quatFromAxisAngle(axis, angle) { + const half = angle / 2; + const s = Math.sin(half); + return [ + Math.cos(half), // w + axis.x * s, // x + axis.y * s, // y + axis.z * s // z + ]; +} +``` + +* Converts **axis-angle rotation** into a quaternion `[w, x, y, z]`. +* `half` = half the rotation angle. +* `s = sin(half)` scales the axis. +* `w` = cos(half). + +**b. Multiply two quaternions** + +```js +function quatMultiply(q1, q2) { ... } +``` + +* Standard quaternion multiplication formula. +* Combines two rotations into one. **Order matters!** `q1*q2` rotates q2 first, then q1. + +**c. Rotate a vector by a quaternion** + +```js +function quatRotateVector(q, v) { ... } +``` + +* Rotates a 3D vector `v` using quaternion `q`. +* Steps: + + 1. Convert vector to quaternion `qv = [0, vx, vy, vz]`. + 2. Compute conjugate `qConj = [w, -x, -y, -z]`. + 3. Rotate: `v' = q * qv * q⁻¹`. + 4. Return new `x,y,z` of rotated vector. + +--- + +## **4. Drawing Functions** + +**Clear canvas and draw text** + +```js +function clear() { + ctx.fillStyle = BACKGROUND; + ctx.fillRect(0, 0, game.width, game.height); + ctx.fillStyle = TEXT; + ctx.font = "20px serif"; + ctx.fillText(" *click me to change rotation axes", 10, game.height - 15); + ctx.fillText("current axes: " + activeAxes.join(","), 10, 20); +} +``` + +* Clears the canvas with the background color. +* Draws instructions and current axes at the top/bottom. + +**Draw a point** + +```js +function point({x, y}) { + const s = 20; + ctx.fillStyle = FOREGROUND; + ctx.fillRect(x - s/2, y - s/2, s, s); +} +``` + +* Draws a small square at `(x, y)` on the screen. +* Not used in the final version, but useful for debugging vertices. + +**Draw a line** + +```js +function line(p1, p2) { ... } +``` + +* Draws a thick line from `p1` to `p2` in canvas coordinates. + +**Convert 3D to screen coordinates** + +```js +function screen(p) { + return { + x: (p.x + 1)/2 * game.width, + y: (1 -(p.y + 1)/2) * game.height + }; +} +``` + +* Maps normalized coordinates `[-1,1]` to screen coordinates `[0, width/height]`. +* Y-axis is flipped because canvas y=0 is top. + +**Project 3D to 2D** + +```js +function project({x, y, z}) { + return { x: x/z, y: y/z }; +} +``` + +* Simple **perspective projection**: divides x and y by z (makes things farther away appear smaller). + +**Translate along Z** + +```js +function translateZ({x,y,z}, dz) { + return {x, y, z: z + dz}; +} +``` + +* Moves the cube forward/backward so it appears in front of the camera. + +--- + +## **5. Animation Loop** + +```js +const FPS = 60; + +function frame() { + const dt = 1/FPS; + angle += Math.PI*dt; +``` + +* Updates rotation angle every frame based on FPS. + +**Create combined quaternion** + +```js + let q = [1,0,0,0]; // identity quaternion + activeAxes.forEach(axis => { + let qAxis; + if (axis === "x") qAxis = quatFromAxisAngle({x:1,y:0,z:0}, angle); + if (axis === "y") qAxis = quatFromAxisAngle({x:0,y:1,z:0}, angle); + if (axis === "z") qAxis = quatFromAxisAngle({x:0,y:0,z:1}, angle); + q = quatMultiply(qAxis, q); // multiply in order + }); +``` + +* Starts with **identity quaternion** (no rotation). +* For each active axis, creates a quaternion for rotation around that axis. +* Combines them by multiplying — this lets the cube rotate around multiple axes **simultaneously**. + +**Draw the cube** + +```js + clear(); + for (const f of fs) { + for (let i = 0; i < f.length; ++i) { + const a = vs[f[i]]; + const b = vs[f[(i + 1)%f.length]]; + line( + screen(project(translateZ(quatRotateVector(q, a), dz))), + screen(project(translateZ(quatRotateVector(q, b), dz))) + ); + } + } +``` + +* Clears the canvas. +* Loops through each face/edge `f`. +* Rotates each vertex using the combined quaternion. +* Translates it forward along Z. +* Projects 3D coordinates into 2D screen coordinates. +* Draws lines connecting vertices. + +**Schedule next frame** + +```js + setTimeout(frame, 1000/FPS); +} +setTimeout(frame, 1000/FPS); +``` + +* Calls `frame()` every 1/60th of a second to animate. + +--- + +## **6. Handle Clicks** + +```js +game.onclick = () => { + const allAxes = [["x"], ["y"], ["z"], ["x","y"], ["y","z"], ["x","z"], ["x","y","z"]]; + const index = allAxes.findIndex(a => JSON.stringify(a) === JSON.stringify(activeAxes)); + activeAxes = allAxes[(index + 1) % allAxes.length]; +}; +``` + +* On click, cycles through different **axis combinations**: x, y, z, xy, yz, xz, xyz. +* Updates `activeAxes` so the cube rotates differently next frame. + +--- + +### ✅ **Summary** + +* **Vertices + Faces**: Define cube geometry. +* **Quaternions**: Handle rotation around multiple axes safely. +* **Project & Screen Functions**: Convert 3D points to 2D canvas. +* **Animation Loop**: Updates rotation, applies quaternion rotation, and draws cube. +* **Click Handler**: Change which axes to rotate around. + +--- + +If you want, I can also **draw a diagram showing the quaternion rotation concept** visually — it really helps to understand how it rotates the cube around multiple axes at once. + +Do you want me to make that diagram? + +![diagram](https://nyquistdev.net/git/?p=basic-3d-example.git;a=blob_plain;f=docs/chatgpt-dump/quaternion-rotation.png;hb=HEAD) + +Great question — this is exactly the right point to slow down and understand the **math**, because quaternions feel “magical” until you see *why* each line exists. + +I’ll go step by step: + +--- + +# 1. What is a quaternion (in graphics)? + +A **quaternion** is a 4-component number: + +[ +q = (w, x, y, z) +] + +In 3D graphics, we use **unit quaternions** to represent **rotations**. + +A rotation by: + +* angle ( \theta ) +* around a **unit axis** ( \vec{u} = (u_x, u_y, u_z) ) + +is encoded as: + +[ +\boxed{ +q = \left( +\cos\frac{\theta}{2}, +; u_x \sin\frac{\theta}{2}, +; u_y \sin\frac{\theta}{2}, +; u_z \sin\frac{\theta}{2} +\right) +} +] + +This is the **axis–angle → quaternion** formula. + +Why half-angle? +Because quaternion rotation comes from complex numbers and Euler’s formula — doubling happens naturally in the math. + +--- + +# 2. `quatFromAxisAngle` + +```js +function quatFromAxisAngle(axis, angle) { + const h = angle / 2; + const s = Math.sin(h); + return [Math.cos(h), axis.x*s, axis.y*s, axis.z*s]; +} +``` + +### Math behind it + +You’re directly implementing: + +[ +q = (\cos(\theta/2), \vec{u}\sin(\theta/2)) +] + +* `h = angle / 2` → half-angle +* `cos(h)` → scalar part (w) +* `axis * sin(h)` → vector part ((x,y,z)) + +### Important rule + +`axis` **must be normalized** (length = 1), otherwise rotation will scale or skew. + +--- + +# 3. Quaternion multiplication (rotation composition) + +```js +function quatMultiply(a, b) { + const [w1,x1,y1,z1] = a; + const [w2,x2,y2,z2] = b; + return [ + w1*w2 - x1*x2 - y1*y2 - z1*z2, + w1*x2 + x1*w2 + y1*z2 - z1*y2, + w1*y2 - x1*z2 + y1*w2 + z1*x2, + w1*z2 + x1*y2 - y1*x2 + z1*w2 + ]; +} +``` + +### Math formula + +Given: +[ +q_1 = (w_1, \vec{v}_1), \quad q_2 = (w_2, \vec{v}_2) +] + +Quaternion multiplication is: + +[ +\boxed{ +q_1 q_2 = +\left( +w_1 w_2 - \vec{v}_1 \cdot \vec{v}_2,; +w_1 \vec{v}_2 + w_2 \vec{v}_1 + \vec{v}_1 \times \vec{v}_2 +\right) +} +] + +Expanded component-wise → exactly your code. + +### Why this matters + +* Quaternion multiplication **combines rotations** +* Order matters: + [ + q_{\text{final}} = q_{\text{new}} \cdot q_{\text{current}} + ] + +This avoids **gimbal lock**, unlike Euler angles. + +--- + +# 4. Quaternion conjugate + +```js +function quatConjugate(q) { + return [q[0], -q[1], -q[2], -q[3]]; +} +``` + +### Math meaning + +For: +[ +q = (w, x, y, z) +] + +The conjugate is: + +[ +\boxed{ +q^* = (w, -x, -y, -z) +} +] + +### Why this works + +If (q) is a **unit quaternion**: + +[ +q^{-1} = q^* +] + +So conjugate = **inverse rotation**. + +This is why cameras use conjugates — moving the world opposite to the camera rotation. + +--- + +# 5. Rotating a vector using quaternions + +```js +function quatRotateVector(q, v) { + const qv = [0, v.x, v.y, v.z]; + const r = quatMultiply(quatMultiply(q, qv), quatConjugate(q)); + return { x: r[1], y: r[2], z: r[3] }; +} +``` + +### The key formula + +To rotate a vector ( \vec{v} ): + +1. Convert it to a **pure quaternion** + [ + v_q = (0, v_x, v_y, v_z) + ] + +2. Apply: + [ + \boxed{ + v' = q ; v_q ; q^{-1} + } + ] + +3. Extract the vector part + +### Why this works + +Quaternion rotation encodes: + +* axis +* angle +* orientation + +The sandwich product rotates without: + +* scaling +* skewing +* drifting + +--- + +# 6. Why quaternions are better than matrices (for rotation) + +| Problem | Euler Angles | Matrices | Quaternions | +| -------------------- | ------------ | -------- | ----------- | +| Gimbal lock | ❌ | ❌ | ✅ | +| Smooth interpolation | ❌ | ⚠️ | ✅ (SLERP) | +| Compact | ✅ | ❌ | ✅ | +| Numerical stability | ⚠️ | ⚠️ | ✅ | + +--- + +# 7. Visual intuition + +A rotation quaternion represents: + +[ +q = \cos(\theta/2) + \sin(\theta/2)(u_x i + u_y j + u_z k) +] + +This is the 3D analog of: + +[ +e^{i\theta} = \cos\theta + i\sin\theta +] + +--- + +# 8. Best learning resources (highly recommended) + +### Beginner-friendly + +* **3Blue1Brown – Quaternions** + + * [https://www.youtube.com/watch?v=zjMuIxRvygQ](https://www.youtube.com/watch?v=zjMuIxRvygQ) +* **Visualizing Quaternions** + + * [https://eater.net/quaternions](https://eater.net/quaternions) + +### Graphics-oriented + +* **LearnOpenGL – Camera & Quaternions** + + * [https://learnopengl.com/Getting-started/Camera](https://learnopengl.com/Getting-started/Camera) +* **Scratchapixel – Quaternions** + + * [https://www.scratchapixel.com/lessons/mathematics-physics-for-computer-graphics/quaternions](https://www.scratchapixel.com/lessons/mathematics-physics-for-computer-graphics/quaternions) + +### Math reference + +* Wikipedia (surprisingly solid) + + * [https://en.wikipedia.org/wiki/Quaternion](https://en.wikipedia.org/wiki/Quaternion) + * [https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation](https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation) + +--- + +# 9. Mental model to keep + +> **Quaternions don’t rotate vectors directly. +> They rotate *space itself* around the vector.** + +Once that clicks, camera math suddenly makes sense. + +--- + +If you want next: + +* 🔄 quaternion → rotation matrix +* 🎥 why camera uses inverse rotation +* 🎯 SLERP (smooth rotation interpolation) +* 🧠 geometric intuition with diagrams + +Just tell me 👍 + +Absolutely — let’s unpack those sentences slowly and connect the symbols to intuition. + +--- + +## 1. “A quaternion is an expression of the form” + +[ +a + b,\mathbf{i} + c,\mathbf{j} + d,\mathbf{k} +] + +This means that **every quaternion can be written as a sum of four parts**: + +* one **real (scalar)** part +* three **imaginary (vector-like)** parts + +So a quaternion has **four real numbers** in it, not just two like a complex number. + +--- + +## 2. The coefficients (a, b, c, d) + +> “where (a, b, c, d) are real numbers” + +* These are just ordinary real numbers (like (2), (-1.3), (\pi), etc.). +* They tell you **how much** of each basis element the quaternion contains. + +Think of them as coordinates. + +--- + +## 3. The symbols (\mathbf{i}, \mathbf{j}, \mathbf{k}) + +> “and (\mathbf{i}, \mathbf{j}, \mathbf{k}) are symbols that can be interpreted as unit-vectors pointing along the three spatial axes.” + +These symbols are **not numbers** in the usual sense. They are **basis elements** that behave in special ways. + +### Interpretation: + +* (\mathbf{i}) → x-axis direction +* (\mathbf{j}) → y-axis direction +* (\mathbf{k}) → z-axis direction + +So: +[ +b,\mathbf{i} + c,\mathbf{j} + d,\mathbf{k} +] +looks exactly like a 3D vector: +[ +(b, c, d) +] + +--- + +## 4. Scalar part vs vector part + +The quaternion naturally splits into two pieces: + +[ +\underbrace{a}*{\text{scalar part}} +\quad + \quad +\underbrace{(b,\mathbf{i} + c,\mathbf{j} + d,\mathbf{k})}*{\text{vector part}} +] + +This matches the earlier notation: +[ +\mathbf{q} = (r, \vec{v}) +] +where: + +* (r = a) +* (\vec{v} = (b, c, d)) + +--- + +## 5. How this extends complex numbers + +Complex numbers look like: +[ +a + b,\mathbf{i} +] + +Quaternions **extend** this idea: + +* one real axis +* three imaginary axes + +So instead of a 2D number system, quaternions live in **4D**. + +--- + +## 6. Important difference from vectors + +Although (\mathbf{i}, \mathbf{j}, \mathbf{k}) *look* like unit vectors, they are actually **algebraic objects with multiplication rules**: + +[ +\mathbf{i}^2 = \mathbf{j}^2 = \mathbf{k}^2 = \mathbf{i}\mathbf{j}\mathbf{k} = -1 +] + +and multiplication is **not commutative**: +[ +\mathbf{i}\mathbf{j} = \mathbf{k}, \quad +\mathbf{j}\mathbf{i} = -\mathbf{k} +] + +This is why quaternions are so powerful for rotations. + +--- + +## 7. Geometric intuition + +You can think of a quaternion as: + +* a **real number** (a), plus +* a **3D direction and magnitude** ((b, c, d)) + +When used for rotations: + +* (a) relates to the rotation angle +* ((b, c, d)) relates to the rotation axis + +--- + +## 8. Plain-English summary + +> A quaternion is built from one real number and three imaginary components. The imaginary parts behave like directions in 3D space and follow special multiplication rules. Together, they form a 4-dimensional number system that can represent 3D rotations cleanly and efficiently. + +If you’d like, I can next: + +* show how quaternion multiplication works geometrically +* explain why quaternions avoid gimbal lock +* relate quaternions directly to rotation matrices + +Great question — this is where quaternions really become *geometric*, not just algebraic. + +I’ll build this up step by step, from intuition to the final geometric meaning. + +--- + +## 1. Unit quaternions represent rotations + +A **unit quaternion** (length = 1) represents a 3D rotation. + +It is written as: +[ +q = \cos\frac{\theta}{2} + \hat{u},\sin\frac{\theta}{2} +] +where: + +* (\theta) = rotation angle +* (\hat{u} = u_x\mathbf{i} + u_y\mathbf{j} + u_z\mathbf{k}) is a **unit vector** (rotation axis) + +This already encodes geometry: + +* **axis** → direction of the vector part +* **angle** → stored in the sine/cosine terms + +--- + +## 2. Rotating a vector using a quaternion + +A 3D vector (\vec{v}) is treated as a **pure quaternion**: +[ +\vec{v} ;\longrightarrow; 0 + v_x\mathbf{i} + v_y\mathbf{j} + v_z\mathbf{k} +] + +The rotated vector is: +[ +\vec{v},' = q,\vec{v},q^{-1} +] + +This is not arbitrary — this *is* the geometric action of rotation. + +--- + +## 3. What quaternion multiplication does geometrically + +### Key idea: + +> **Quaternion multiplication composes rotations.** + +If: + +* (q_1) rotates by (\theta_1) around axis (\hat{u}_1) +* (q_2) rotates by (\theta_2) around axis (\hat{u}_2) + +Then: +[ +q = q_2 q_1 +] +represents: + +> “First rotate by (q_1), then rotate by (q_2)” + +This is exactly how matrix multiplication works for rotations. + +--- + +## 4. Why order matters (non-commutativity) + +Quaternion multiplication is **not commutative**: +[ +q_1 q_2 \neq q_2 q_1 +] + +Geometrically: + +* Rotating around **x then y** is not the same as **y then x** + +This matches real-world 3D rotations. + +--- + +## 5. Vector–scalar decomposition of multiplication + +Let: +[ +q_1 = (r_1, \vec{v}_1), \quad +q_2 = (r_2, \vec{v}_2) +] + +Their product is: +[ +q_2 q_1 = +\big( +r_2 r_1 - \vec{v}_2 \cdot \vec{v}_1,; +r_2 \vec{v}_1 + r_1 \vec{v}_2 + \vec{v}_2 \times \vec{v}_1 +\big) +] + +### Geometric meaning of each term: + +* **Dot product** → angle alignment +* **Cross product** → axis perpendicular to both rotations +* **Linear terms** → mixing scalar and vector parts + +So quaternion multiplication naturally combines: + +* angle information (dot) +* axis orientation (cross) + +--- + +## 6. Visual intuition + +Think of a unit quaternion as: + +* a point on a **4D unit sphere** +* encoding a rotation in 3D space + +Multiplying two unit quaternions: + +* moves you along that sphere +* in a way that corresponds exactly to doing one rotation after another + +--- + +## 7. Why the half-angle appears + +You might wonder: + +> Why (\theta/2), not (\theta)? + +Geometrically: + +* Rotations form a curved space (SO(3)) +* Unit quaternions live in a simpler space (SU(2)) that *double-covers* rotations +* The half-angle makes multiplication correspond cleanly to rotation composition + +That’s why: +[ +q \text{ and } -q +] +represent the **same rotation**. + +--- + +## 8. Simple concrete example + +Rotate 90° about the z-axis: +[ +q = \cos 45^\circ + \mathbf{k}\sin 45^\circ +] + +Apply it to (\mathbf{i}) (the x-axis): +[ +\mathbf{i}' = q,\mathbf{i},q^{-1} = \mathbf{j} +] + +Geometrically: + +* x-axis rotates into y-axis +* exactly what you expect + +--- + +## 9. Plain-English summary + +> Quaternion multiplication is not “multiplying numbers.” +> It is **combining rotations in 3D space**. +> The algebra mirrors the geometry: dot products handle angles, cross products handle axes, and order encodes rotation order. + +If you want, next I can: + +* derive the rotation formula step-by-step +* show the link to rotation matrices +* explain the double-cover and why (q) and (-q) are the same rotation + diff --git a/docs/chatgpt-dump/rendered.html b/docs/chatgpt-dump/rendered.html new file mode 100644 index 0000000..a9595b6 --- /dev/null +++ b/docs/chatgpt-dump/rendered.html @@ -0,0 +1,468 @@ +<!DOCTYPE html> +<html lang="en"> +<head> +<meta charset="UTF-8"> +<meta http-equiv="X-UA-Compatible" content="IE=edge"> +<meta name="viewport" content="width=device-width, initial-scale=1.0"> +<meta name="generator" content="Asciidoctor 2.0.20"> +<title>Untitled</title> +<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Open+Sans:300,300italic,400,400italic,600,600italic%7CNoto+Serif:400,400italic,700,700italic%7CDroid+Sans+Mono:400,700"> +<style> +/*! 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+ tex2jax: { + inlineMath: [["\\(", "\\)"]], + displayMath: [["\\[", "\\]"]], + ignoreClass: "nostem|nolatexmath" + }, + asciimath2jax: { + delimiters: [["\\$", "\\$"]], + ignoreClass: "nostem|noasciimath" + }, + TeX: { equationNumbers: { autoNumber: "none" } } +}) +MathJax.Hub.Register.StartupHook("AsciiMath Jax Ready", function () { + MathJax.InputJax.AsciiMath.postfilterHooks.Add(function (data, node) { + if ((node = data.script.parentNode) && (node = node.parentNode) && node.classList.contains("stemblock")) { + data.math.root.display = "block" + } + return data + }) +}) +</script> +<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.9/MathJax.js?config=TeX-MML-AM_HTMLorMML"></script> +</body> +</html> \ No newline at end of file diff --git a/example.html b/example.html new file mode 100644 index 0000000..0883092 --- /dev/null +++ b/example.html @@ -0,0 +1,78 @@ +<!DOCTYPE html> +<html lang="en"> +<head> +<meta charset="UTF-8"> +<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/prismjs/themes/prism.css"> +<script src="https://cdn.jsdelivr.net/npm/prismjs/prism.js"></script> +<script src="https://cdn.jsdelivr.net/npm/prismjs/components/prism-javascript.min.js"></script> +<title>Canvas + Code</title> +<style> +/* Container grid: canvas + code side by side */ + +html { + background-image: url(https://media2.giphy.com/media/v1.Y2lkPTc5MGI3NjExNHNmMm03bndxZXlwcW81ZHY4aXc3dXkzMXZhNmx2OTFqYWEwaG13bCZlcD12MV9pbnRlcm5hbF9naWZfYnlfaWQmY3Q9Zw/9ufQTaDtvGX51NkpWL/giphy.gif); + background-size: cover; + background-attachment: fixed; + overflow: hidden; +} + +body { + opacity: 85%; +} + +#container { + display: grid; + grid-template-columns: 1fr 5fr; /* canvas fixed, code fills remaining space */ + height: 100vh; /* full viewport height */ + max-width: 1600px; + gap: 10px; + margin: 0 auto; + align-items: center; +} + +/* Canvas itself */ +#game { + border: 4px solid white; + margin: 0; + padding: 0; + display: block; +} + +/* Code viewer */ +#code-viewer { + margin: 0; + box-sizing: border-box; /* include padding in height */ + overflow: auto; /* scroll if content too long */ + font-family: monospace; + white-space: pre-wrap; + width: 100%; + height: 100%; + max-height: 808px; + max-width: 808px; + margin-left: auto; +} +</style> +</head> +<body> + +<div id="container"> + <canvas id="game"></canvas> + <pre id="code-viewer"> + <code class="language-javascript"> </code> + </pre> + +</div> + +<script src="https://nyquistdev.net/git/?p=basic-3d-example.git;a=blob_plain;f=index.js;hb=HEAD"></script> +<script> +fetch('https://nyquistdev.net/git/?p=basic-3d-example.git;a=blob_plain;f=index.js;hb=HEAD') + .then(res => res.text()) + .then(code => { + const codeEl = document.querySelector('#code-viewer code'); + codeEl.textContent = code; + Prism.highlightElement(codeEl); + }) + .catch(err => console.error('Could not load index.js:', err)); +</script> +</body> +</html> diff --git a/index.html b/index.html new file mode 100644 index 0000000..f2f8a47 --- /dev/null +++ b/index.html @@ -0,0 +1,9 @@ +<style> +#game { + display: block; + margin: 0 auto; +} +</style> + +<canvas id="game"></canvas> +<script src="index.js"></script> diff --git a/index.js b/index.js new file mode 100644 index 0000000..8bfa577 --- /dev/null +++ b/index.js @@ -0,0 +1,186 @@ +const canvas = document.getElementById("game"); +const ctx = canvas.getContext("2d") + +const BACKGROUND = "#000000"; +const FOREGROUND = "#00FF00"; +const FPS = 60; + +// let debug = true; +let debug = false; + +let debugColors = [ + "#FF0000", + "#00FF00", + "#0000FF", +] + +const highContrastColors = [ + "#FFFF00", // 1. Bright Yellow + "#FF00FF", // 2. Magenta + "#00FFFF", // 3. Cyan / Aqua + "#FFA500", // 4. Pure Orange + "#800080", // 5. Deep Purple + "#008080", // 6. Teal + "#A52A2A", // 7. Brown + "#800000", // 8. Maroon + "#008000", // 9. Forest Green + "#000080", // 10. Navy Blue + "#808000", // 11. Olive + "#C71585", // 12. Hot Pink + "#FFC0CB", // 13. Light Pink + "#7FFFD4", // 14. Aquamarine + "#FFE4E1", // 15. Misty Rose + "#F0E68C", // 16. Khaki + "#E6E6FA", // 17. Lavender + "#B0C4DE", // 18. Light Steel Blue + "#FF4500", // 19. Red-Orange + "#1E90FF", // 20. Dodger Blue + "#FFD700", // 21. Gold + "#9370DB", // 22. Medium Purple + "#8B008B", // 23. Dark Magenta + "#556B2F", // 24. Dark Olive Green + "#FF8C00", // 25. Dark Orange + "#48D1CC", // 26. Medium Turquoise + "#32CD32", // 27. Bright Green + "#2F4F4F", // 28. Dark Slate Gray + "#98FF98", // 29. Mint Green + "#FF7F50", // 30. Coral + "#4B0082", // 31. Indigo + "#D2B48C", // 32. Tan / Sand + "#FF7F55", // 30. Coral +]; + +canvas.width = 800; +canvas.height = 800; + +function clear() { + ctx.fillStyle = BACKGROUND; + ctx.fillRect(0, 0, canvas.width, canvas.height); +} + +function line(p1, p2) { + ctx.lineWidth = 8; + if (!debug) ctx.strokeStyle = FOREGROUND; + ctx.beginPath(); + ctx.moveTo(p1.x, p1.y); + ctx.lineTo(p2.x, p2.y); + ctx.stroke(); +} + +function getMagnitude(a, b) { + let x = (b.x - a.x); + let y = (b.y - a.y); + return Math.sqrt(x * x + y * y); +} + +function interpolate(a, b, y) { + let startingPos = a.x; + let incX = (b.x - a.x) / (b.y - a.y); + let numberOfyIncs = y - a.y; + return startingPos + incX * numberOfyIncs; +} + +function drawTriangle(coords, color) { + coords.sort((a,b) => a.y - b.y); + + let blueisSteeperThanRed = + (coords[2].y - coords[0].y) * (coords[1].x - coords[0].x) > + (coords[1].y - coords[0].y) * (coords[2].x - coords[0].x) + + for (let y = coords[0].y; y <= coords[2].y; ++y) { + + let x1 = interpolate(coords[0], coords[2], y); + + let x2 = (y < coords[1].y) ? + interpolate(coords[0], coords[1], y) : + interpolate(coords[1], coords[2], y); + + let fill = (i, y) => { + ctx.fillStyle = color; + ctx.fillRect(i, y, 2, 2); + } + + if (blueisSteeperThanRed) { + for (let i = x1; i < x2; ++i) fill(i, y); + } else { + for (let i = x1; i > x2; --i) fill(i, y); + + } + } + + if (debug) { + for (const i in coords) { + ctx.strokeStyle = debugColors[i]; + let a = coords[i]; + let b = coords[ (i + 1) % coords.length]; + line(a, b); + } + + } + + +} + + +clear(); +let moveX = 0; + +let triangleCoords = [ + // --- CENTER CRADLE (Core focal points) --- + [{x: 400, y: 400}, {x: 300, y: 300}, {x: 500, y: 250}], // High acute top wedge + [{x: 400, y: 400}, {x: 500, y: 250}, {x: 550, y: 420}], // Narrow vertical sliver + [{x: 400, y: 400}, {x: 550, y: 420}, {x: 480, y: 550}], // Rotated downward blade + [{x: 400, y: 400}, {x: 480, y: 550}, {x: 280, y: 480}], // Broad foundational center piece + [{x: 400, y: 400}, {x: 280, y: 480}, {x: 300, y: 300}], // Left-leaning internal wedge + + // --- NORTH FLANK (Radiating to the top edge: y = 0) --- + [{x: 300, y: 300}, {x: 0, y: 0}, {x: 250, y: 0}], + [{x: 300, y: 300}, {x: 250, y: 0}, {x: 500, y: 0}], + [{x: 300, y: 300}, {x: 500, y: 0}, {x: 500, y: 250}], // Massive sweeping North-East shard + [{x: 500, y: 250}, {x: 500, y: 0}, {x: 700, y: 0}], + [{x: 500, y: 250}, {x: 700, y: 0}, {x: 800, y: 150}], // Long, sharp slope + + // --- EAST FLANK (Radiating to the right edge: x = 800) --- + [{x: 500, y: 250}, {x: 800, y: 150}, {x: 550, y: 420}], + [{x: 550, y: 420}, {x: 800, y: 150}, {x: 800, y: 450}], // Tight accent block + [{x: 550, y: 420}, {x: 800, y: 450}, {x: 800, y: 600}], + [{x: 550, y: 420}, {x: 800, y: 600}, {x: 480, y: 550}], // Giant horizontal slice + + // --- SOUTH FLANK (Radiating to the bottom edge: y = 800) --- + [{x: 480, y: 550}, {x: 800, y: 600}, {x: 800, y: 800}], // Clears the bottom right corner + [{x: 480, y: 550}, {x: 800, y: 800}, {x: 550, y: 800}], + [{x: 480, y: 550}, {x: 550, y: 800}, {x: 350, y: 800}], // Long vertical grounding anchor + [{x: 480, y: 550}, {x: 350, y: 800}, {x: 150, y: 800}], + [{x: 480, y: 550}, {x: 150, y: 800}, {x: 280, y: 480}], // Sweeping South-West transitional floor + + // --- WEST FLANK (Radiating to the left edge: x = 0) --- + [{x: 280, y: 480}, {x: 150, y: 800}, {x: 0, y: 800}], // Clears the bottom left corner + [{x: 280, y: 480}, {x: 0, y: 800}, {x: 0, y: 500}], + [{x: 280, y: 480}, {x: 0, y: 500}, {x: 0, y: 300}], // Tall vertical perimeter guard + [{x: 280, y: 480}, {x: 0, y: 300}, {x: 300, y: 300}], // Steep internal left block + + // --- NORTH-WEST FILLER ZONE (Completing upper left frame gaps) --- + [{x: 300, y: 300}, {x: 0, y: 300}, {x: 0, y: 120}], + [{x: 300, y: 300}, {x: 0, y: 120}, {x: 0, y: 0}], + + // --- OUTER SUBDIVISIONS (Adding complexity to the borders for visual contrast) --- + [{x: 700, y: 0}, {x: 800, y: 0}, {x: 800, y: 150}], // Top right micro-corner + [{x: 550, y: 800}, {x: 680, y: 800}, {x: 480, y: 550}], // Tiny baseline shard A + [{x: 680, y: 800}, {x: 800, y: 800}, {x: 480, y: 550}], // Tiny baseline shard B + [{x: 350, y: 800}, {x: 250, y: 800}, {x: 480, y: 550}], // Tiny baseline shard C + [{x: 250, y: 800}, {x: 150, y: 800}, {x: 480, y: 550}], // Tiny baseline shard D + [{x: 0, y: 500}, {x: 0, y: 650}, {x: 280, y: 480}], // Left margin splitter + [{x: 0, y: 650}, {x: 0, y: 800}, {x: 280, y: 480}], // Left floor splitter + [{x: 0, y: 300}, {x: 0, y: 200}, {x: 300, y: 300}], // Left ceiling splitter + [{x: 0, y: 200}, {x: 0, y: 0}, {x: 300, y: 300}] // Left corner lock +]; + +// drawTriangle([{x: moveX + 200, y: 200}, {x: moveX + 600, y: 310},{x: moveX + 450, y: 400}]); +// drawTriangle([{x: 120, y: 200}, {x:200, y: 310},{x: 450, y: 400}], "#FF0000"); +// drawTriangle([{x: 200, y: 200}, {x:600, y: 230},{x: 250, y: 300}]); + + +for (const i in triangleCoords) { + drawTriangle(triangleCoords[i], highContrastColors[i % highContrastColors.length]); + await new 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vertices = [ + {x: 0.25, y: 0.25, z: 0.25}, + {x: -0.25, y: 0.25, z: 0.25}, + {x: -0.25, y: -0.25, z: 0.25}, + {x: 0.25, y: -0.25, z: 0.25}, + {x: 0.25, y: 0.25, z: -0.25}, + {x: -0.25, y: 0.25, z: -0.25}, + {x: -0.25, y: -0.25, z: -0.25}, + {x: 0.25, y: -0.25, z: -0.25}, +]; + +const edges = [ + [0,1,2,3], + [4,5,6,7], + [0,4],[1,5],[2,6],[3,7] +]; + +/* ===================== OBJECT ===================== */ +let objectPos = { x: 0, y: 0, z: 2 }; +let angle = 0; + +const AXIS_MODES = [ + ["x"], ["y"], ["z"], + ["x","y"], ["y","z"], ["x","z"], + ["x","y","z"] +]; +let axisModeIndex = AXIS_MODES.length-1; +let activeAxes = AXIS_MODES[axisModeIndex]; + +/* ===================== CAMERA ===================== */ +let camera = { + position: { x: 0, y: 0, z: 0 }, + rotation: [1,0,0,0], + speed: 0.05, + sensitivity: 0.002 +}; + +/* ===================== INPUT ===================== */ +const keys = {}; +window.addEventListener("keydown", e => keys[e.key.toLowerCase()] = true); +window.addEventListener("keyup", e => keys[e.key.toLowerCase()] = false); + +/* ===================== QUATERNIONS ===================== */ +function quatFromAxisAngle(axis, angle) { + const h = angle / 2; + const s = Math.sin(h); + return [Math.cos(h), axis.x*s, axis.y*s, axis.z*s]; +} + +function quatMultiply(a, b) { + const [w1,x1,y1,z1] = a; + const [w2,x2,y2,z2] = b; + return [ + w1*w2 - x1*x2 - y1*y2 - z1*z2, + w1*x2 + x1*w2 + y1*z2 - z1*y2, + w1*y2 - x1*z2 + y1*w2 + z1*x2, + w1*z2 + x1*y2 - y1*x2 + z1*w2 + ]; +} + +function quatConjugate(q) { + return [q[0], -q[1], -q[2], -q[3]]; +} + +function quatRotateVector(q, v) { + // whenever w is zero then we are on the unit sphere + // unit quanternion + const qv = [0, v.x, v.y, v.z]; + + // the conjugate is equal to the inverse quanternion if q is a unit-norm + // quanternion. in this case it is a unit-norm quanternion + const r = quatMultiply(quatMultiply(q, qv), quatConjugate(q)); + return { x: r[1], y: r[2], z: r[3] }; +} + +/* ===================== CAMERA VECTORS ===================== */ +function cameraForward() { + return quatRotateVector(camera.rotation, {x:0,y:0,z:1}); +} +function cameraRight() { + return quatRotateVector(camera.rotation, {x:1,y:0,z:0}); +} +function cameraUp() { + return quatRotateVector(camera.rotation, {x:0,y:1,z:0}); +} + +function applyCamera(p) { + const v = { + x: p.x - camera.position.x, + y: p.y - camera.position.y, + z: p.z - camera.position.z + }; + // we use the conjugate here because the camera should move everything in the + // opposite direction that we are moving + return quatRotateVector(quatConjugate(camera.rotation), v); +} + +/* ===================== PROJECTION ===================== */ +function project(p) { + return { x: p.x / p.z, y: p.y / p.z }; +} +function screen(p) { + return { + x: (p.x + 1) * 0.5 * game.width, + y: (1 - (p.y + 1) * 0.5) * game.height + }; +} + +/* ===================== DRAW ===================== */ +function clear() { + ctx.fillStyle = BACKGROUND; + ctx.fillRect(0,0,game.width,game.height); + + ctx.fillStyle = TEXT; + ctx.font = "14px monospace"; + ctx.fillText("WASD: move | QE: up/down", 10, 18); + ctx.fillText("Mouse: look | Click canvas to lock", 10, 36); + ctx.fillText("Click canvas again: change cube axes", 10, 54); + ctx.fillText("Cube axes: " + activeAxes.join(","), 10, 72); +} + +function line(a,b) { + ctx.strokeStyle = FOREGROUND; + ctx.lineWidth = 2; + ctx.beginPath(); + ctx.moveTo(a.x,a.y); + ctx.lineTo(b.x,b.y); + ctx.stroke(); +} + +/* ===================== LOOP ===================== */ +function frame() { + angle += Math.PI / FPS; + + // --- cube rotation quaternion --- + + // This is the center of the cube. we are going to rotate around the center + // of the cube. + let q = [1,0,0,0]; + + // First, have the correct unit quanternion + for (const ax of activeAxes) { + // quatFromAxixAngle are unit quanternions + // as we keep pulling the quanternion in the axis direction further and + // further, you keep rotating the quanternion + + // we will use the right hand rule to figure out what is going on. + + // the left side of the multiplication is the function of the right + + // REMEMBER + // These axis angles are part of the fourth demension and won't be + // visible + + // x is i. so we are going in the positive i direction + // --------------------------------------------------- + // the jk circle (The part we see) rotates from j to kk + if (ax==="x") q = quatMultiply(quatFromAxisAngle({x:1,y:0,z:0}, angle), q); + + // y is j. So we are going in the positive j direction + // --------------------------------------------------- + // the ij circle (part we see) rotates from -i to k + if (ax==="y") q = quatMultiply(quatFromAxisAngle({x:0,y:1,z:0}, angle), q); + + // z is k. So we are going in the positve k direction + // --------------------------------------------------- + // The ij circle rotates from j to -i + if (ax==="z") q = quatMultiply(quatFromAxisAngle({x:0,y:0,z:1}, angle), q); + } + + // --- camera movement --- + if (keys["w"]) { + const f = cameraForward(); + camera.position.x += f.x * camera.speed; + camera.position.y += f.y * camera.speed; + camera.position.z += f.z * camera.speed; + } + if (keys["s"]) { + const f = cameraForward(); + camera.position.x -= f.x * camera.speed; + camera.position.y -= f.y * camera.speed; + camera.position.z -= f.z * camera.speed; + } + if (keys["a"]) { + const r = cameraRight(); + camera.position.x -= r.x * camera.speed; + camera.position.y -= r.y * camera.speed; + camera.position.z -= r.z * camera.speed; + } + if (keys["d"]) { + const r = cameraRight(); + camera.position.x += r.x * camera.speed; + camera.position.y += r.y * camera.speed; + camera.position.z += r.z * camera.speed; + } + if (keys["q"]) { + const u = cameraUp(); + camera.position.x -= u.x * camera.speed; + camera.position.y -= u.y * camera.speed; + camera.position.z -= u.z * camera.speed; + } + if (keys["e"]) { + const u = cameraUp(); + camera.position.x += u.x * camera.speed; + camera.position.y += u.y * camera.speed; + camera.position.z += u.z * camera.speed; + } + + clear(); + + for (const e of edges) { + for (let i=0;i<e.length;i++) { + let a = vertices[e[i]]; + let b = vertices[e[(i+1)%e.length]]; + + a = quatRotateVector(q, a); + b = quatRotateVector(q, b); + + a = { x:a.x+objectPos.x, y:a.y+objectPos.y, z:a.z+objectPos.z }; + b = { x:b.x+objectPos.x, y:b.y+objectPos.y, z:b.z+objectPos.z }; + + a = applyCamera(a); + b = applyCamera(b); + + if (a.z<=NEAR || b.z<=NEAR) continue; + line(screen(project(a)), screen(project(b))); + } + } + + setTimeout(frame, 1000/FPS); +} +frame(); + +/* ===================== INTERACTION ===================== */ +game.onclick = () => { + game.requestPointerLock(); + axisModeIndex = (axisModeIndex + 1) % AXIS_MODES.length; + activeAxes = AXIS_MODES[axisModeIndex]; +}; + +window.addEventListener("mousemove", e => { + if (document.pointerLockElement !== game) return; + + const yaw = quatFromAxisAngle({x:0,y:1,z:0}, -e.movementX * camera.sensitivity); + const pitch = quatFromAxisAngle({x:1,y:0,z:0}, -e.movementY * camera.sensitivity); + + camera.rotation = quatMultiply(yaw, camera.rotation); + camera.rotation = quatMultiply(camera.rotation, pitch); +}); diff --git a/rendered.html b/rendered.html new file mode 100644 index 0000000..45d3678 --- /dev/null +++ b/rendered.html @@ -0,0 +1,1538 @@ +<!DOCTYPE html> +<html lang="en"> +<head> +<meta charset="UTF-8"> +<meta http-equiv="X-UA-Compatible" content="IE=edge"> +<meta name="viewport" content="width=device-width, initial-scale=1.0"> +<meta name="generator" content="Asciidoctor 2.0.20"> +<title>Basic 3D Example</title> +<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Open+Sans:300,300italic,400,400italic,600,600italic%7CNoto+Serif:400,400italic,700,700italic%7CDroid+Sans+Mono:400,700"> +<style> +/*! 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} +} +h3 ~ *:not(h2):not(h3) { + margin-left: 1.5em; +} + +h4 ~ *:not(h2):not(h3):not(.h3):not(h4) { + margin-left: 2em; +} +h5 ~ *:not(h2):not(h3):not(.h3):not(h4):not(.h4):not(h5) { + margin-left: 2.5em; +} + + +.sect1 h2 { + + /* + background: linear-gradient( + to bottom, + rgba(251, 189, 47, 0.2), /* yellow */ + rgba(0, 0, 0, 0.4) /* black */ + ); + */ + /*opacity: .9;*/ + } + +#preamble p{ + padding-left: 10px; + color: var(--gruvbox-white) !important; +} + +#footer { + max-width: 62.5em !important; + padding-left: 10px !important; + text-align: center; +} + +.logo img { + display: block !important; + margin-left: auto !important; + margin-right: auto !important; +} + +#header > h1:first-child { + margin-top: .3em !important; + padding-top: .2em !important; + text-align: center; +} + +footer { + max-width: 62.5em; + margin-left: auto; + margin-right: auto; + display: block; + +} + +details[open] > summary::before { + /* display: none;*/ + border-top: none; + content: "▼"; + margin-top: -12px; +} + +.example.stemblock { + border: solid; + font-size: 2vw; + max-width: 40%; + margin: auto; +} + +.center { + max-width: 500px; + display: flex; + justify-content: center; +} + +.highlight { + color: var(--gruvbox-yellow) +} + +.border { + padding: 4px; + border: 2px solid rgba(255, 0, 0, .3); +} + +.dull { + opacity: 40%; +} + +.inline { + margin-right: 5%; + display: inline-block; +} + +img { + margin: auto; + width: 100%; +} + + +</style> + +</head> + +<script> +function sendHeight() { + const height = document.documentElement.scrollHeight; + parent.postMessage({ type: "HEIGHT", height }, "*"); +} + +window.addEventListener("load", sendHeight); +window.addEventListener("resize", sendHeight); +</script> +</head> +<body class="article"> +<div id="header"> +<h1>Basic 3D Example</h1> +</div> +<div id="content"> +<div id="preamble"> +<div class="sectionbody"> +<div class="paragraph"> +<p>click <a href="https://nyquistdev.net/git/?p=basic-3d-example.git;a=blob_plain;f=example.html;hb=HEAD"">here</a> for an interactive example</p> +</div> +<div class="paragraph"> +<p>A good explaination of how 3D works.</p> +</div> +<div class="admonitionblock note"> +<table> +<tr> +<td class="icon"> +<i class="fa icon-note" title="Note"></i> +</td> +<td class="content"> +This started off as code from Tsoding, I used chatgpt to change + specific things, and also made some of my own changes. This is a learning + tool! not something I wrote off the top of my head +</td> +</tr> +</table> +</div> +</div> +</div> +<div class="sect1"> +<h2 id="_graphics_pipeline">Graphics Pipeline</h2> +<div class="sectionbody"> +<div class="listingblock"> +<div class="content"> +<pre>vertex + -&gt; rotate cube + -&gt; translate cube (object position) + -&gt; apply camera (inverse transform) + -&gt; project + -&gt; screen</pre> +</div> +</div> +<div class="admonitionblock note"> +<table> +<tr> +<td class="icon"> +<i class="fa icon-note" title="Note"></i> +</td> +<td class="content"> +Some parts of these pipeline may be left out if that part of the pipeline + is not needed +</td> +</tr> +</table> +</div> +</div> +</div> +<div class="sect1"> +<h2 id="_basic_projection_formula">Basic Projection Formula</h2> +<div class="sectionbody"> +<div class="paragraph"> +<p>Projection formula for a vector \([x,y,z\)]:</p> +</div> +<div class="stemblock"> +<div class="content"> +\[ +\begin{aligned} +x' &amp;= x/z \\ +y' &amp;= y/z +\end{aligned} +\] +</div> +</div> +</div> +</div> +<div class="sect1"> +<h2 id="_vertex_rotation_formulas">Vertex Rotation Formulas</h2> +<div class="sectionbody"> +<div class="paragraph"> +<p>click <a href="https://nyquistdev.net/git/?p=basic-3d-example.git;a=blob_plain;f=example.html;hb=refs/heads/simple-rotation">here</a> for an interactive example</p> +</div> +<div class="paragraph"> +<p>Rotate vector formula for x-axis (y-z plane):</p> +</div> +<div class="stemblock"> +<div class="content"> +\[ +\begin{aligned} +x' &amp;= x \\ +y' &amp;= y\cos\theta - z\sin\theta \\ +z' &amp;= y\sin\theta + z\cos\theta +\end{aligned} +\] +</div> +</div> +<div class="paragraph"> +<p>Rotate vector formula for y-axis (x-z plane):</p> +</div> +<div class="stemblock"> +<div class="content"> +\[ +\begin{aligned} +x' &amp;= x\cos\theta - z\sin\theta \\ +y' &amp;= y \\ +z' &amp;= x\sin\theta + z\cos\theta +\end{aligned} \] +</div> +</div> +<div class="paragraph"> +<p>Rotate vector formula for z-axis (x-y plane):</p> +</div> +<div class="stemblock"> +<div class="content"> +\[ +\begin{aligned} +x' &amp;= x\cos\theta - y\sin\theta \\ +y' &amp;= x\sin\theta + y\cos\theta \\ +z' &amp;= z +\end{aligned} +\] +</div> +</div> +</div> +</div> +<div class="sect1"> +<h2 id="_2_single_axis_rotation_matrices_euler_angles">2. Single-axis rotation matrices (Euler Angles)</h2> +<div class="sectionbody"> +<div class="sect2"> +<h3 id="_rotation_about_x_axis_roll">Rotation about X-axis (roll)</h3> +<div class="stemblock"> +<div class="content"> +\[\boxed{ +R_x(\alpha) = +\begin{bmatrix} +1 &amp; 0 &amp; 0 \\ +0 &amp; \cos\alpha &amp; -\sin\alpha \\ +0 &amp; \sin\alpha &amp; \cos\alpha +\end{bmatrix} +}\] +</div> +</div> +</div> +<div class="sect2"> +<h3 id="_rotation_about_y_axis_pitch">Rotation about Y-axis (pitch)</h3> +<div class="stemblock"> +<div class="content"> +\[\boxed{ +R_y(\beta) = +\begin{bmatrix} +\cos\beta &amp; 0 &amp; \sin\beta \\ +0 &amp; 1 &amp; 0 \\ +-\sin\beta &amp; 0 &amp; \cos\beta +\end{bmatrix} +}\] +</div> +</div> +</div> +<div class="sect2"> +<h3 id="_rotation_about_z_axis_yaw">Rotation about Z-axis (yaw)</h3> +<div class="stemblock"> +<div class="content"> +\[\boxed{ +R_z(\gamma) = +\begin{bmatrix} +\cos\gamma &amp; -\sin\gamma &amp; 0 \\ +\sin\gamma &amp; \cos\gamma &amp; 0 \\ +0 &amp; 0 &amp; 1 +\end{bmatrix} +}\] +</div> +</div> +<div class="paragraph"> +<p>Euler angles will always have gimbal lock. You can switch which axis has gimbal +lock by switching the order. Most application use quaternions instead of eular +angles.</p> +</div> +</div> +</div> +</div> +<div class="sect1"> +<h2 id="_quaternions">Quaternions</h2> +<div class="sectionbody"> +<div class="paragraph"> +<p>click <a href="https://nyquistdev.net/git/?p=basic-3d-example.git;a=blob_plain;f=example.html;hb=refs/heads/quaternions">here</a> for an interactive example</p> +</div> +<div class="paragraph"> +<p>Quaternions allow you to rotate smoothly around multiple axes without gimbal +lock. Chat-gpt rewrote the code to use quaternions. Here are some of the +replys about <a href="https://nyquistdev.net/git/?p=basic-3d-example.git;a=blob_plain;f=docs/chatgpt-dump/quaternions.html;hb=HEAD">quaternions</a>.</p> +</div> +<div class="sect2"> +<h3 id="_1_what_is_a_quaternion_in_graphics">1. What is a quaternion (in graphics)?</h3> +<div class="paragraph"> +<p>A quaternion is a 4-component number:</p> +</div> +<div class="stemblock"> +<div class="content"> +\[q = (w, x, y, z)\] +</div> +</div> +<div class="paragraph"> +<p>In 3D graphics, we use <strong>unit quaternions</strong> to represent <strong>rotations</strong>.</p> +</div> +<div class="paragraph"> +<p>A rotation by:</p> +</div> +<div class="ulist"> +<ul> +<li> +<p>angle \(\theta\)</p> +</li> +<li> +<p>around a <strong>unit axis</strong> \(\vec{u} = (u_x, u_y, u_z)\)</p> +</li> +</ul> +</div> +<div class="paragraph"> +<p>is encoded as:</p> +</div> +<div class="stemblock"> +<div class="content"> +\[\boxed{ +q = \left( +\cos\frac{\theta}{2}, +\; u_x \sin\frac{\theta}{2}, +\; u_y \sin\frac{\theta}{2}, +\; u_z \sin\frac{\theta}{2} +\right) +}\] +</div> +</div> +<div class="paragraph"> +<p>This is the <strong>axis–angle &#8594; quaternion</strong> formula.</p> +</div> +<div class="sect3"> +<h4 id="_why_half_angle">Why half-angle?</h4> +<div class="paragraph"> +<p>Because quaternion rotation comes from complex numbers and Euler’s formula — +doubling happens naturally in the math.</p> +</div> +</div> +</div> +<div class="sect2"> +<h3 id="_2_quatfromaxisangle">2. quatFromAxisAngle</h3> +<div class="listingblock"> +<div class="content"> +<pre class="pygments highlight"><code data-lang="js"><span></span><span class="tok-kd">function</span><span class="tok-w"> </span><span class="tok-nx">quatFromAxisAngle</span><span class="tok-p">(</span><span class="tok-nx">axis</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-nx">angle</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-p">{</span> +<span class="tok-w"> </span><span class="tok-kd">const</span><span class="tok-w"> </span><span class="tok-nx">h</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-nx">angle</span><span class="tok-w"> </span><span class="tok-o">/</span><span class="tok-w"> </span><span class="tok-mf">2</span><span class="tok-p">;</span> +<span class="tok-w"> </span><span class="tok-kd">const</span><span class="tok-w"> </span><span class="tok-nx">s</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-nb">Math</span><span class="tok-p">.</span><span class="tok-nx">sin</span><span class="tok-p">(</span><span class="tok-nx">h</span><span class="tok-p">);</span> +<span class="tok-w"> </span><span class="tok-k">return</span><span class="tok-w"> </span><span class="tok-p">[</span><span class="tok-nb">Math</span><span class="tok-p">.</span><span class="tok-nx">cos</span><span class="tok-p">(</span><span class="tok-nx">h</span><span class="tok-p">),</span><span class="tok-w"> </span><span class="tok-nx">axis</span><span class="tok-p">.</span><span class="tok-nx">x</span><span class="tok-o">*</span><span class="tok-nx">s</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-nx">axis</span><span class="tok-p">.</span><span class="tok-nx">y</span><span class="tok-o">*</span><span class="tok-nx">s</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-nx">axis</span><span class="tok-p">.</span><span class="tok-nx">z</span><span class="tok-o">*</span><span class="tok-nx">s</span><span class="tok-p">];</span> +<span class="tok-p">}</span></code></pre> +</div> +</div> +<div class="paragraph"> +<p>You’re directly implementing:</p> +</div> +<div class="stemblock"> +<div class="content"> +\[q = (\cos(\theta/2), \vec{u}\sin(\theta/2))\] +</div> +</div> +<div class="ulist"> +<ul> +<li> +<p>\(h = angle / 2\) &#8594; half-angle</p> +</li> +<li> +<p>\(cos(h)\) → scalar part (w)</p> +</li> +<li> +<p>\(axis * sin(h)\) → vector part \((x,y,z)\)</p> +</li> +</ul> +</div> +<div class="admonitionblock important"> +<table> +<tr> +<td class="icon"> +<i class="fa icon-important" title="Important"></i> +</td> +<td class="content"> +<code>axis</code> <strong>must be normalized</strong> (length = 1), otherwise rotation will + scale or skew. +</td> +</tr> +</table> +</div> +</div> +<div class="sect2"> +<h3 id="_3_quaternion_multiplication_rotation_composition">3. Quaternion multiplication (rotation composition)</h3> +<div class="listingblock"> +<div class="content"> +<pre class="pygments highlight"><code data-lang="js"><span></span><span class="tok-kd">function</span><span class="tok-w"> </span><span class="tok-nx">quatMultiply</span><span class="tok-p">(</span><span class="tok-nx">a</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-nx">b</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-p">{</span> +<span class="tok-w"> </span><span class="tok-kd">const</span><span class="tok-w"> </span><span class="tok-p">[</span><span class="tok-nx">w1</span><span class="tok-p">,</span><span class="tok-nx">x1</span><span class="tok-p">,</span><span class="tok-nx">y1</span><span class="tok-p">,</span><span class="tok-nx">z1</span><span class="tok-p">]</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-nx">a</span><span class="tok-p">;</span> +<span class="tok-w"> </span><span class="tok-kd">const</span><span class="tok-w"> </span><span class="tok-p">[</span><span class="tok-nx">w2</span><span class="tok-p">,</span><span class="tok-nx">x2</span><span class="tok-p">,</span><span class="tok-nx">y2</span><span class="tok-p">,</span><span class="tok-nx">z2</span><span class="tok-p">]</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-nx">b</span><span class="tok-p">;</span> +<span class="tok-w"> </span><span class="tok-k">return</span><span class="tok-w"> </span><span class="tok-p">[</span> +<span class="tok-w"> </span><span class="tok-nx">w1</span><span class="tok-o">*</span><span class="tok-nx">w2</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-nx">x1</span><span class="tok-o">*</span><span class="tok-nx">x2</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-nx">y1</span><span class="tok-o">*</span><span class="tok-nx">y2</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-nx">z1</span><span class="tok-o">*</span><span class="tok-nx">z2</span><span class="tok-p">,</span> +<span class="tok-w"> </span><span class="tok-nx">w1</span><span class="tok-o">*</span><span class="tok-nx">x2</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-nx">x1</span><span class="tok-o">*</span><span class="tok-nx">w2</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-nx">y1</span><span class="tok-o">*</span><span class="tok-nx">z2</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-nx">z1</span><span class="tok-o">*</span><span class="tok-nx">y2</span><span class="tok-p">,</span> +<span class="tok-w"> </span><span class="tok-nx">w1</span><span class="tok-o">*</span><span class="tok-nx">y2</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-nx">x1</span><span class="tok-o">*</span><span class="tok-nx">z2</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-nx">y1</span><span class="tok-o">*</span><span class="tok-nx">w2</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-nx">z1</span><span class="tok-o">*</span><span class="tok-nx">x2</span><span class="tok-p">,</span> +<span class="tok-w"> </span><span class="tok-nx">w1</span><span class="tok-o">*</span><span class="tok-nx">z2</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-nx">x1</span><span class="tok-o">*</span><span class="tok-nx">y2</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-w"> </span><span class="tok-nx">y1</span><span class="tok-o">*</span><span class="tok-nx">x2</span><span class="tok-w"> </span><span class="tok-o">+</span><span class="tok-w"> </span><span class="tok-nx">z1</span><span class="tok-o">*</span><span class="tok-nx">w2</span> +<span class="tok-w"> </span><span class="tok-p">];</span> +<span class="tok-p">}</span></code></pre> +</div> +</div> +<div class="paragraph"> +<p>formula:</p> +</div> +<div class="stemblock"> +<div class="content"> +\[q_1 = (w_1, \vec{v}_1), \quad q_2 = (w_2, \vec{v}_2)\] +</div> +</div> +<div class="paragraph"> +<p>Quaternion multiplication is:</p> +</div> +<div class="stemblock"> +<div class="content"> +\[\boxed{ +q_1 q_2 = +\left( +w_1 w_2 - \vec{v}_1 \cdot \vec{v}_2,\; +w_1 \vec{v}_2 + w_2 \vec{v}_1 + \vec{v}_1 \times \vec{v}_2 +\right) +}\] +</div> +</div> +<div class="paragraph"> +<p>Expanded component-wise &#8594; exactly your code.</p> +</div> +<div class="sect3"> +<h4 id="_why_this_matters">Why this matters</h4> +<div class="ulist"> +<ul> +<li> +<p>Quaternion multiplication <strong>combines rotations</strong></p> +</li> +<li> +<p>Order matters:</p> +</li> +</ul> +</div> +<div class="stemblock"> +<div class="content"> +\[q_{\text{final}} = q_{\text{new}} \cdot q_{\text{current}}\] +</div> +</div> +<div class="paragraph"> +<p>This avoids <strong>gimbal lock</strong>, unlike Euler angles.</p> +</div> +</div> +</div> +<div class="sect2"> +<h3 id="_4_quaternion_conjugate">4. Quaternion conjugate</h3> +<div class="listingblock"> +<div class="content"> +<pre class="pygments highlight"><code data-lang="js"><span></span><span class="tok-kd">function</span><span class="tok-w"> </span><span class="tok-nx">quatConjugate</span><span class="tok-p">(</span><span class="tok-nx">q</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-p">{</span> +<span class="tok-w"> </span><span class="tok-k">return</span><span class="tok-w"> </span><span class="tok-p">[</span><span class="tok-nx">q</span><span class="tok-p">[</span><span class="tok-mf">0</span><span class="tok-p">],</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-nx">q</span><span class="tok-p">[</span><span class="tok-mf">1</span><span class="tok-p">],</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-nx">q</span><span class="tok-p">[</span><span class="tok-mf">2</span><span class="tok-p">],</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-nx">q</span><span class="tok-p">[</span><span class="tok-mf">3</span><span class="tok-p">]];</span> +<span class="tok-p">}</span></code></pre> +</div> +</div> +<div class="paragraph"> +<p>For:</p> +</div> +<div class="stemblock"> +<div class="content"> +\[q = (w, x, y, z)\] +</div> +</div> +<div class="paragraph"> +<p>The conjugate is:</p> +</div> +<div class="stemblock"> +<div class="content"> +\[\boxed{ +q^* = (w, -x, -y, -z) +}\] +</div> +</div> +</div> +<div class="sect2"> +<h3 id="_why_this_works">Why this works</h3> +<div class="paragraph"> +<p>If (q) is a <strong>unit quaternion</strong>:</p> +</div> +<div class="stemblock"> +<div class="content"> +\[q^{-1} = q^*\] +</div> +</div> +<div class="paragraph"> +<p>So conjugate = <strong>inverse rotation</strong>.</p> +</div> +<div class="paragraph"> +<p>This is why cameras use conjugates — moving the world opposite to the camera +rotation.</p> +</div> +</div> +<div class="sect2"> +<h3 id="_5_rotating_a_vector_using_quaternions">5. Rotating a vector using quaternions</h3> +<div class="listingblock"> +<div class="content"> +<pre class="pygments highlight"><code data-lang="js"><span></span><span class="tok-kd">function</span><span class="tok-w"> </span><span class="tok-nx">quatRotateVector</span><span class="tok-p">(</span><span class="tok-nx">q</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-nx">v</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-p">{</span> +<span class="tok-w"> </span><span class="tok-kd">const</span><span class="tok-w"> </span><span class="tok-nx">qv</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-p">[</span><span class="tok-mf">0</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-nx">v</span><span class="tok-p">.</span><span class="tok-nx">x</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-nx">v</span><span class="tok-p">.</span><span class="tok-nx">y</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-nx">v</span><span class="tok-p">.</span><span class="tok-nx">z</span><span class="tok-p">];</span> +<span class="tok-w"> </span><span class="tok-kd">const</span><span class="tok-w"> </span><span class="tok-nx">r</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-nx">quatMultiply</span><span class="tok-p">(</span><span class="tok-nx">quatMultiply</span><span class="tok-p">(</span><span class="tok-nx">q</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-nx">qv</span><span class="tok-p">),</span><span class="tok-w"> </span><span class="tok-nx">quatConjugate</span><span class="tok-p">(</span><span class="tok-nx">q</span><span class="tok-p">));</span> +<span class="tok-w"> </span><span class="tok-k">return</span><span class="tok-w"> </span><span class="tok-p">{</span><span class="tok-w"> </span><span class="tok-nx">x</span><span class="tok-o">:</span><span class="tok-w"> </span><span class="tok-nx">r</span><span class="tok-p">[</span><span class="tok-mf">1</span><span class="tok-p">],</span><span class="tok-w"> </span><span class="tok-nx">y</span><span class="tok-o">:</span><span class="tok-w"> </span><span class="tok-nx">r</span><span class="tok-p">[</span><span class="tok-mf">2</span><span class="tok-p">],</span><span class="tok-w"> </span><span class="tok-nx">z</span><span class="tok-o">:</span><span class="tok-w"> </span><span class="tok-nx">r</span><span class="tok-p">[</span><span class="tok-mf">3</span><span class="tok-p">]</span><span class="tok-w"> </span><span class="tok-p">};</span> +<span class="tok-p">}</span></code></pre> +</div> +</div> +<div class="paragraph"> +<p>The key formula</p> +</div> +<div class="paragraph"> +<p>To rotate a vector \(\vec{v}\):</p> +</div> +<div class="olist arabic"> +<ol class="arabic"> +<li> +<p>Convert it to a <strong>pure quaternion</strong></p> +</li> +</ol> +</div> +<div class="stemblock"> +<div class="content"> +\[v_q = (0, v_x, v_y, v_z)\] +</div> +</div> +<div class="olist arabic"> +<ol class="arabic"> +<li> +<p>Apply:</p> +</li> +</ol> +</div> +<div class="stemblock"> +<div class="content"> +\[\boxed{ +v' = q \; v_q \; q^{-1} +}\] +</div> +</div> +<div class="olist arabic"> +<ol class="arabic"> +<li> +<p>Extract the vector part</p> +</li> +</ol> +</div> +<div class="sect3"> +<h4 id="_why_this_works_2">Why this works</h4> +<div class="paragraph"> +<p>Quaternion rotation encodes:</p> +</div> +<div class="ulist"> +<ul> +<li> +<p>axis</p> +</li> +<li> +<p>angle</p> +</li> +<li> +<p>orientation</p> +</li> +</ul> +</div> +<div class="paragraph"> +<p>The sandwich product rotates without:</p> +</div> +<div class="ulist"> +<ul> +<li> +<p>scaling</p> +</li> +<li> +<p>skewing</p> +</li> +<li> +<p>drifting</p> +</li> +</ul> +</div> +</div> +</div> +<div class="sect2"> +<h3 id="_visual_intuition">Visual intuition</h3> +<div class="paragraph"> +<p>A rotation quaternion represents:</p> +</div> +<div class="stemblock"> +<div class="content"> +\[q = \cos(\theta/2) + \sin(\theta/2)(u_x i + u_y j + u_z k)\] +</div> +</div> +<div class="paragraph"> +<p>This is the 3D analog of:</p> +</div> +<div class="stemblock"> +<div class="content"> +\[e^{i\theta} = \cos\theta + i\sin\theta\] +</div> +</div> +</div> +</div> +</div> +<div class="sect1"> +<h2 id="_camera">Camera</h2> +<div class="sectionbody"> +<div class="paragraph"> +<p>chat-gpt replys about <a href="https://nyquistdev.net/git/?p=basic-3d-example.git;a=blob_plain;f=docs/chatgpt-dump/camera.html;hb=HEAD">camera</a>.</p> +</div> +</div> +</div> +<div class="sect1"> +<h2 id="_simple_clickable_object">Simple clickable object</h2> +<div class="sectionbody"> +<div class="paragraph"> +<p>click <a href="https://nyquistdev.net/git/?p=basic-3d-example.git;a=blob_plain;f=example.html;hb=refs/heads/clickable">here</a> for an interactive example</p> +</div> +<div class="paragraph"> +<p>This does not account for the rotation of the object but it does check to see if +your mouse is within the rectangular bounds of the mesh.</p> +</div> +<div class="olist arabic"> +<ol class="arabic"> +<li> +<p>finding the min and max values for x and y of the mesh:</p> +</li> +</ol> +</div> +<div class="listingblock"> +<div class="content"> +<pre class="pygments highlight"><code data-lang="js"><span></span><span class="tok-kd">function</span><span class="tok-w"> </span><span class="tok-nx">computeBounds</span><span class="tok-p">(</span><span class="tok-nx">mesh</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-p">{</span> +<span class="tok-w"> </span><span class="tok-kd">let</span><span class="tok-w"> </span><span class="tok-nx">min</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-p">{</span><span class="tok-nx">x</span><span class="tok-o">:</span><span class="tok-w"> </span><span class="tok-kc">Infinity</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-nx">y</span><span class="tok-o">:</span><span class="tok-w"> </span><span class="tok-kc">Infinity</span><span class="tok-p">}</span> +<span class="tok-w"> </span><span class="tok-kd">let</span><span class="tok-w"> </span><span class="tok-nx">max</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-p">{</span><span class="tok-nx">x</span><span class="tok-o">:</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-kc">Infinity</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-nx">y</span><span class="tok-o">:</span><span class="tok-w"> </span><span class="tok-o">-</span><span class="tok-kc">Infinity</span><span class="tok-p">}</span> + +<span class="tok-w"> </span><span class="tok-k">for</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-kd">const</span><span class="tok-w"> </span><span class="tok-nx">v</span><span class="tok-w"> </span><span class="tok-k">of</span><span class="tok-w"> </span><span class="tok-nx">mesh</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-p">{</span> +<span class="tok-w"> </span><span class="tok-nx">min</span><span class="tok-p">.</span><span class="tok-nx">x</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-nb">Math</span><span class="tok-p">.</span><span class="tok-nx">min</span><span class="tok-p">(</span><span class="tok-nx">min</span><span class="tok-p">.</span><span class="tok-nx">x</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-nx">v</span><span class="tok-p">.</span><span class="tok-nx">x</span><span class="tok-p">)</span> +<span class="tok-w"> </span><span class="tok-nx">min</span><span class="tok-p">.</span><span class="tok-nx">y</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-nb">Math</span><span class="tok-p">.</span><span class="tok-nx">min</span><span class="tok-p">(</span><span class="tok-nx">min</span><span class="tok-p">.</span><span class="tok-nx">y</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-nx">v</span><span class="tok-p">.</span><span class="tok-nx">y</span><span class="tok-p">)</span> + +<span class="tok-w"> </span><span class="tok-nx">max</span><span class="tok-p">.</span><span class="tok-nx">x</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-nb">Math</span><span class="tok-p">.</span><span class="tok-nx">max</span><span class="tok-p">(</span><span class="tok-nx">max</span><span class="tok-p">.</span><span class="tok-nx">x</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-nx">v</span><span class="tok-p">.</span><span class="tok-nx">x</span><span class="tok-p">)</span> +<span class="tok-w"> </span><span class="tok-nx">max</span><span class="tok-p">.</span><span class="tok-nx">y</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-nb">Math</span><span class="tok-p">.</span><span class="tok-nx">max</span><span class="tok-p">(</span><span class="tok-nx">max</span><span class="tok-p">.</span><span class="tok-nx">y</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-nx">v</span><span class="tok-p">.</span><span class="tok-nx">y</span><span class="tok-p">)</span> +<span class="tok-w"> </span><span class="tok-p">}</span> +<span class="tok-w"> </span><span class="tok-k">return</span><span class="tok-w"> </span><span class="tok-p">{</span><span class="tok-nx">min</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-nx">max</span><span class="tok-p">}</span> +<span class="tok-p">}</span></code></pre> +</div> +</div> +<div class="olist arabic"> +<ol class="arabic"> +<li> +<p>Define the box corners of the rect for the mesh</p> +</li> +</ol> +</div> +<div class="listingblock"> +<div class="content"> +<pre class="pygments highlight"><code data-lang="js"><span></span><span class="tok-kd">function</span><span class="tok-w"> </span><span class="tok-nx">boundingBoxCorners</span><span class="tok-p">({</span><span class="tok-nx">min</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-nx">max</span><span class="tok-p">})</span><span class="tok-w"> </span><span class="tok-p">{</span> +<span class="tok-w"> </span><span class="tok-k">return</span><span class="tok-w"> </span><span class="tok-p">{</span> +<span class="tok-w"> </span><span class="tok-nx">bottom</span><span class="tok-o">:</span><span class="tok-w"> </span><span class="tok-p">{</span><span class="tok-nx">left</span><span class="tok-o">:</span><span class="tok-w"> </span><span class="tok-p">{</span><span class="tok-nx">x</span><span class="tok-o">:</span><span class="tok-w"> </span><span class="tok-nx">min</span><span class="tok-p">.</span><span class="tok-nx">x</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-nx">y</span><span class="tok-o">:</span><span class="tok-w"> </span><span class="tok-nx">max</span><span class="tok-p">.</span><span class="tok-nx">y</span><span class="tok-p">},</span><span class="tok-w"> </span><span class="tok-nx">right</span><span class="tok-o">:</span><span class="tok-w"> </span><span class="tok-p">{</span><span class="tok-nx">x</span><span class="tok-o">:</span><span class="tok-w"> </span><span class="tok-nx">max</span><span class="tok-p">.</span><span class="tok-nx">x</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-nx">y</span><span class="tok-o">:</span><span class="tok-w"> </span><span class="tok-nx">max</span><span class="tok-p">.</span><span class="tok-nx">y</span><span class="tok-p">}},</span> +<span class="tok-w"> </span><span class="tok-nx">top</span><span class="tok-o">:</span><span class="tok-w"> </span><span class="tok-p">{</span><span class="tok-nx">left</span><span class="tok-o">:</span><span class="tok-w"> </span><span class="tok-p">{</span><span class="tok-nx">x</span><span class="tok-o">:</span><span class="tok-w"> </span><span class="tok-nx">min</span><span class="tok-p">.</span><span class="tok-nx">x</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-nx">y</span><span class="tok-o">:</span><span class="tok-w"> </span><span class="tok-nx">min</span><span class="tok-p">.</span><span class="tok-nx">y</span><span class="tok-p">},</span><span class="tok-w"> </span><span class="tok-nx">right</span><span class="tok-o">:</span><span class="tok-w"> </span><span class="tok-p">{</span><span class="tok-nx">x</span><span class="tok-o">:</span><span class="tok-w"> </span><span class="tok-nx">max</span><span class="tok-p">.</span><span class="tok-nx">x</span><span class="tok-p">,</span><span class="tok-w"> </span><span class="tok-nx">y</span><span class="tok-o">:</span><span class="tok-w"> </span><span class="tok-nx">min</span><span class="tok-p">.</span><span class="tok-nx">y</span><span class="tok-p">}},</span> + +<span class="tok-w"> </span><span class="tok-p">}</span> +<span class="tok-p">}</span></code></pre> +</div> +</div> +<div class="olist arabic"> +<ol class="arabic"> +<li> +<p>Check to see if the mouse is in the rect in the game loop. If it is, and the +mouse button is down, then you can move the object</p> +</li> +</ol> +</div> +<div class="listingblock"> +<div class="content"> +<pre class="pygments highlight"><code data-lang="js"><span></span><span class="tok-nx">inObject</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-p">(</span> +<span class="tok-w"> </span><span class="tok-nx">mousePos</span><span class="tok-p">.</span><span class="tok-nx">x</span><span class="tok-w"> </span><span class="tok-o">&gt;</span><span class="tok-w"> </span><span class="tok-nx">rect</span><span class="tok-p">.</span><span class="tok-nx">top</span><span class="tok-p">.</span><span class="tok-nx">left</span><span class="tok-p">.</span><span class="tok-nx">x</span> +<span class="tok-w"> </span><span class="tok-o">&amp;&amp;</span> +<span class="tok-w"> </span><span class="tok-nx">mousePos</span><span class="tok-p">.</span><span class="tok-nx">x</span><span class="tok-w"> </span><span class="tok-o">&lt;</span><span class="tok-w"> </span><span class="tok-nx">rect</span><span class="tok-p">.</span><span class="tok-nx">top</span><span class="tok-p">.</span><span class="tok-nx">right</span><span class="tok-p">.</span><span class="tok-nx">x</span> +<span class="tok-w"> </span><span class="tok-o">&amp;&amp;</span> +<span class="tok-w"> </span><span class="tok-nx">mousePos</span><span class="tok-p">.</span><span class="tok-nx">y</span><span class="tok-w"> </span><span class="tok-o">&gt;</span><span class="tok-w"> </span><span class="tok-nx">rect</span><span class="tok-p">.</span><span class="tok-nx">top</span><span class="tok-p">.</span><span class="tok-nx">left</span><span class="tok-p">.</span><span class="tok-nx">y</span> +<span class="tok-w"> </span><span class="tok-o">&amp;&amp;</span> +<span class="tok-w"> </span><span class="tok-nx">mousePos</span><span class="tok-p">.</span><span class="tok-nx">y</span><span class="tok-w"> </span><span class="tok-o">&lt;</span><span class="tok-w"> </span><span class="tok-nx">rect</span><span class="tok-p">.</span><span class="tok-nx">bottom</span><span class="tok-p">.</span><span class="tok-nx">right</span><span class="tok-p">.</span><span class="tok-nx">y</span> +<span class="tok-p">)</span> + +<span class="tok-k">if</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-nx">mouseDown</span><span class="tok-w"> </span><span class="tok-o">&amp;&amp;</span><span class="tok-w"> </span><span class="tok-nx">inObject</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-nx">canMoveObject</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-kc">true</span> + +<span class="tok-k">if</span><span class="tok-w"> </span><span class="tok-p">(</span><span class="tok-o">!</span><span class="tok-nx">mouseDown</span><span class="tok-p">)</span><span class="tok-w"> </span><span class="tok-nx">canMoveObject</span><span class="tok-w"> </span><span class="tok-o">=</span><span class="tok-w"> </span><span class="tok-kc">false</span></code></pre> +</div> +</div> +</div> +</div> +<div class="sect1"> +<h2 id="_links">Links</h2> +<div class="sectionbody"> +<div class="ulist"> +<ul> +<li> +<p><a href="https://www.youtube.com/watch?v=qjWkNZ0SXfo">One Formula That demystifies 3D Graphics</a></p> +</li> +<li> +<p><a href="https://www.youtube.com/watch?v=EZufiIwwqFA">Rotation matrix derivation (step-by-step prove)</a></p> +</li> +<li> +<p><a href="https://matthew-brett.github.io/teaching/rotation_2d.html">rotation 2D by mathew</a></p> +</li> +<li> +<p><a href="https://www.youtube.com/watch?v=zc8b2Jo7mno">gimbal lock visual representation</a></p> +</li> +<li> +<p><a href="https://www.youtube.com/watch?v=zjMuIxRvygQ">3Blue1Brown | Quaternions and 3d rotation, explained interactively</a></p> +</li> +<li> +<p><a href="https://www.youtube.com/watch?v=d4EgbgTm0Bg&amp;t=17s">3Blue1Brown | Visualizing the 4d numbers Quanternions</a></p> +</li> +<li> +<p><a href="https://phas.ubc.ca/~berciu/TEACHING/PHYS206/LECTURES/FILES/euler.pdf">euler.pdf</a></p> +</li> +</ul> +</div> +</div> +</div> +</div> +<script type="text/x-mathjax-config"> +MathJax.Hub.Config({ + messageStyle: "none", + tex2jax: { + inlineMath: [["\\(", "\\)"]], + displayMath: [["\\[", "\\]"]], + ignoreClass: "nostem|nolatexmath" + }, + asciimath2jax: { + delimiters: [["\\$", "\\$"]], + ignoreClass: "nostem|noasciimath" + }, + TeX: { equationNumbers: { autoNumber: "none" } } +}) +MathJax.Hub.Register.StartupHook("AsciiMath Jax Ready", function () { + MathJax.InputJax.AsciiMath.postfilterHooks.Add(function (data, node) { + if ((node = data.script.parentNode) && (node = node.parentNode) && node.classList.contains("stemblock")) { + data.math.root.display = "block" + } + return data + }) +}) +</script> +<script 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