From: A-Larsen Date: Thu, 15 Oct 2026 01:22:41 +0000 (-0400) Subject: first commit X-Git-Url: https://nyquistdev.net/git/?a=commitdiff_plain;h=refs%2Fheads%2Fsimple-rotation;p=basic-3d-example.git first commit --- diff --git a/README.adoc b/README.adoc index ac1ab78..856ccd4 100644 --- a/README.adoc +++ b/README.adoc @@ -1,28 +1,23 @@ = 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 ----- +A good explaination how 3D works. If you click the cube you can change the axis +it will rotate on. -NOTE: Some parts of these pipeline may be left out if that part of the pipeline - is not needed +NOTE: This started off as code from Tsoding -== Basic Projection Formula +== Formulas Projection formula for a vector latexmath:[[x,y,z]]: @@ -36,9 +31,6 @@ y' &= y/z \] ++++ -== 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] @@ -64,6 +56,7 @@ z' &= x\sin\theta + z\cos\theta \end{aligned} \] ++++ + Rotate vector formula for z-axis (x-y plane): [latexmath] @@ -77,344 +70,15 @@ z' &= z \] ++++ -== 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 **axisâangle -> quaternion** formula. - -==== Why half-angle? -Because quaternion rotation comes from complex numbers and Eulerâs formula â -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]; -} ----- - -Youâre directly implementing: - -[latexmath] -++++ -q = (\cos(\theta/2), \vec{u}\sin(\theta/2)) -++++ - -* latexmath:[h = angle / 2] -> half-angle -* latexmath:[cos(h)] â scalar part (w) -* latexmath:[axis * sin(h)] â 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 â 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 +lock. -if (!mouseDown) canMoveObject = false ----- +== ChatGpt +for my chatGpt Responses checkout my https://nyquistdev.net/git/?p=basic-3d-example.git;a=tree;f=docs/chatgpt-dump/docs;h=60b7a07cb968290e6068fc7e5f52dfd9e16a06e0;hb=HEAD[chatgpt-dumps]. == 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/docs/chatgpt-dump/Makehtml b/docs/chatgpt-dump/Makehtml index 615bffe..f65394f 100755 --- a/docs/chatgpt-dump/Makehtml +++ b/docs/chatgpt-dump/Makehtml @@ -1,15 +1,2 @@ #!/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 +pandoc -s quaternions.md -o quaternions.html diff --git a/docs/chatgpt-dump/quaternions.html b/docs/chatgpt-dump/quaternions.html index 35f63d2..4b54d57 100644 --- a/docs/chatgpt-dump/quaternions.html +++ b/docs/chatgpt-dump/quaternions.html @@ -161,6 +161,7 @@ margin: 0 0.8em 0.2em -1.6em; vertical-align: middle; } + .display.math{display: block; text-align: center; margin: 0.5rem auto;} /* CSS for syntax highlighting */ pre > code.sourceCode { white-space: pre; position: relative; } pre > code.sourceCode > span { line-height: 1.25; } @@ -335,475 +336,5 @@ together.

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! Letâs go through your quaternion-based 3D cube -code line by line. Iâll break it down into sections so itâs -easier to digest.

1. Canvas Setup

const game = document.getElementById("game");
-game.width = 400;
-game.height = 400;

Gets the <canvas> element with ID -game.
Sets the canvas dimensions to 400Ã400 pixels.

const BACKGROUND = "#101010";
-const FOREGROUND = "#50FF50";
-const TEXT = "#FFFFFF";

Defines colors for background, cube lines, and text.

const ctx = game.getContext("2d");

Gets the 2D drawing context so we can draw shapes on the -canvas.

2. Cube Geometry

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).

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.

let dz = 1;
-let angle = 0;
-let activeAxes = ["x"];

dz: distance to move the cube along Z (so itâs 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

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

function quatMultiply(q1, q2) { ... }

Standard quaternion multiplication formula.
Combines two rotations into one. Order matters! -q1*q2 rotates q2 first, then q1.

c.Â Rotate a vector by a quaternion

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 * qâ»Â¹.
4. Return new x,y,z of rotated vector.

4. Drawing Functions

Clear canvas and draw text

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

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

function line(p1, p2) { ... }

Draws a thick line from p1 to p2 in canvas -coordinates.

Convert 3D to screen coordinates

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

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

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

const FPS = 60;
-
-function frame() {
-    const dt = 1/FPS;
-    angle += Math.PI*dt;

Updates rotation angle every frame based on FPS.

Create combined quaternion

    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 â this lets the cube rotate around -multiple axes simultaneously.

Draw the cube

    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

    setTimeout(frame, 1000/FPS);
-}
-setTimeout(frame, 1000/FPS);

Calls frame() every 1/60th of a second to animate.

6. Handle Clicks

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.

â 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 â it really helps to understand how -it rotates the cube around multiple axes at once.

Do you want me to make that diagram?

Great question â this is exactly the right point to slow down and -understand the math, because quaternions feel âmagicalâ -until you see why each line exists.

Iâll 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 ( )
around a unit axis ( = (u_x, u_y, u_z) )

is encoded as:

[ ]

This is the axisâangle â quaternion formula.

Why half-angle? Because quaternion rotation comes from complex -numbers and Eulerâs formula â doubling happens naturally in the -math.

2. `quatFromAxisAngle`

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

Youâre directly implementing:

[ q = ((/2), (/2))]

h = angle / 2 â half-angle
cos(h) â scalar part (w)
axis * sin(h) â vector part ((x,y,z))

Important rule

axis must be normalized (length = 1), -otherwise rotation will scale or skew.

3. Quaternion -multiplication (rotation composition)

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, _1), q_2 = (w_2, _2)]

Quaternion multiplication is:

[ ]

Expanded component-wise â exactly your code.

Why this matters

Quaternion multiplication combines rotations
Order matters: [ q_{} = q_{} q_{}]

This avoids gimbal lock, unlike Euler angles.

4. Quaternion conjugate

function quatConjugate(q) {
-    return [q[0], -q[1], -q[2], -q[3]];
-}

Math meaning

For: [ q = (w, x, y, z)]

The conjugate is:

[ ]

Why this works

If (q) is a unit quaternion:

[ q^{-1} = q^*]

So conjugate = inverse rotation.

This is why cameras use conjugates â moving the world opposite to the -camera rotation.

5. Rotating a vector using -quaternions

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 ( ):

Convert it to a pure quaternion [ v_q = (0, v_x, -v_y, v_z)]
Apply: [ ]
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	â	â	â
Smooth interpolation	â	â ï¸	â (SLERP)
Compact	â	â	â
Numerical stability	â ï¸	â ï¸	â

7. Visual intuition

A rotation quaternion represents:

[ q = (/2) + (/2)(u_x i + u_y j + u_z k)]

This is the 3D analog of:

[ e^{i} = + i]

8. Best learning -resources (highly recommended)

Beginner-friendly

3Blue1Brown â Quaternions
-
- https://www.youtube.com/watch?v=zjMuIxRvygQ
Visualizing Quaternions
-
- https://eater.net/quaternions

Graphics-oriented

LearnOpenGL â Camera & Quaternions
-
- https://learnopengl.com/Getting-started/Camera
Scratchapixel â 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/Quaternions_and_spatial_rotation

9. Mental model to keep

-
Quaternions donât rotate vectors directly. They rotate -space itself around the vector.
-

Once that clicks, camera math suddenly makes sense.

If you want next:

ð quaternion â rotation matrix
ð¥ why camera uses inverse rotation
ð¯ SLERP (smooth rotation interpolation)
ð§ geometric intuition with diagrams

Just tell me ð

diff --git a/docs/chatgpt-dump/quaternions.md b/docs/chatgpt-dump/quaternions.md index c406140..6c5ad5b 100644 --- a/docs/chatgpt-dump/quaternions.md +++ b/docs/chatgpt-dump/quaternions.md @@ -1,6 +1,4 @@ -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. +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. Letâs go step by step. @@ -124,859 +122,3 @@ If you want, I can rewrite your **entire code snippet using quaternions**, so th Do you want me to do that? -Absolutely! Letâs go through your **quaternion-based 3D cube code** line by line. Iâll break it down into sections so itâs easier to digest. - ---- - -## **1. Canvas Setup** - -```js -const game = document.getElementById("game"); -game.width = 400; -game.height = 400; -``` - -* Gets the `

1. Canvas Setup

2. Cube Geometry

3. Quaternion -Functions

4. Drawing Functions

5. Animation Loop

6. Handle Clicks

â Summary

1. What is a quaternion (in -graphics)?

2. quatFromAxisAngle

Math behind it

Important rule

3. Quaternion -multiplication (rotation composition)

Math formula

Why this matters

4. Quaternion conjugate

Math meaning

Why this works

5. Rotating a vector using -quaternions

The key formula

Why this works

6. Why -quaternions are better than matrices (for rotation)

7. Visual intuition

8. Best learning -resources (highly recommended)

Beginner-friendly

Graphics-oriented

Math reference

9. Mental model to keep

Graphics Pipeline

Basic Projection Formula

Formulas

Vertex Rotation Formulas

2. Single-axis rotation matrices (Euler Angles)

Rotation about X-axis (roll)

Rotation about Y-axis (pitch)

Rotation about Z-axis (yaw)

Quaternions