Coding guide maths

Table of Contents Maths conventions Coordinate spaces Matrix layout Multiplication order Quaternions Our matrices look like... DO NOT...

Maths conventions

Coordinate spaces

	Handedness	X	Y	Z
World	Right	Right	Forward	Up
Camera	Left	Right	Up	Forward
NDC	Left	Right [-1; 1]	Up [-1; 1]	Forward [0; 1]

Matrix layout

SSE: row-major
Packed: row-major
Shader: row-major
Never explicitly specify layout in shaders!
Do not index matrices in a way that depends on layout (i.e. vector of first 4 elements in memory, m[0].x is forbidden)!

Multiplication order

The vector is followed by the matrix:

vector' = vector*MATRIX1*MATRIX2*MATRIX3

This way, the order in which transforms are applied aligns with left-to-right reading. Matrix functions are automatically configured to generate such matrices.

Quaternions

Mathematical quaternion multiplication is used, thus q = q2*q1 denotes the rotation q1 followed by the rotation q2.

Our matrices look like...

Scale:
$\begin{bmatrix} s_x & 0 & 0 & 0\\ 0 & s_y & 0 & 0\\ 0 & 0 & s_z & 0\\ 0 & 0 & 0 & 1 \end{bmatrix}$

Rotation X, Y and Z:
$\begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & cos(\phi) & sin(\phi) & 0\\ 0 & -sin(\phi) & cos(\phi) & 0\\ 0 & 0 & 0 & 1 \end{bmatrix}$

$\begin{bmatrix} cos(\phi) & 0 & -sin(\phi) & 0\\ 0 & 1 & 0 & 0\\ sin(\phi) & 0 & cos(\phi) & 0\\ 0 & 0 & 0 & 1 \end{bmatrix}$

$\begin{bmatrix} cos(\phi) & sin(\phi) & 0 & 0\\ -sin(\phi) & cos(\phi) & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{bmatrix}$

Translation:
$\begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ T_x & T_y & T_z & 1 \end{bmatrix}$

View:
$\begin{bmatrix} \mathbf{col_2}\times \mathbf{up} & \mathbf{col_0}\times \mathbf{col_2} & \mathbf{norm(target-eye)} & 0\\ \mathbf{eye \cdot col_0} & \mathbf{eye \cdot col_1} & \mathbf{eye \cdot col_2} & 1 \end{bmatrix}$

Projection:
$\newline w = \tan(FoV_x/2) \newline h = w\cdot AspectRatio \newline A = \frac{z_f}{z_f-z_n} \newline B = \frac{-z_fz_n}{z_f-z_n} \newline \begin{bmatrix} w & 0 & 0 & 0\\ 0 & h & 0 & 0\\ 0 & 0 & A & 1\\ 0 & 0 & B & 0 \end{bmatrix}$

DO NOT...

Do not explicitly assign matrix elements if you want to create a matrix that rotates or translates or similar. All these have handy functions in the math library. Something is missing? -> Open an issue on https://github.com/petiaccja/Mathter and contact Péter.

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