Transformation matrix

2D

Rotation

${\begin{pmatrix}\cos \theta &-\sin \theta &0\\\sin \theta &\cos \theta &0\\0&0&1\end{pmatrix}}$

Source: https://en.wikipedia.org/wiki/Rotation_matrix

3D

Translation

${\begin{pmatrix}1&0&0&x\\0&1&0&y\\0&0&1&z\\0&0&0&1\end{pmatrix}}$

Source: https://www.khronos.org/registry/OpenGL-Refpages/gl2.1/xhtml/glTranslate.xml

Scaling

${\begin{pmatrix}x&0&0&0\\0&y&0&0\\0&0&z&0\\0&0&0&1\end{pmatrix}}$

Source: https://www.khronos.org/registry/OpenGL-Refpages/gl2.1/xhtml/glScale.xml

Rotation

${\begin{pmatrix}x^{2}c'+c&xyc'-zs&xzc'+ys&0\\yxc'+zs&y^{2}c'+c&yzc'-xs&0\\xzc'-ys&yzc'+xs&z^{2}c'+c&0\\0&0&0&1\end{pmatrix}}$

with

${\begin{aligned}c&=\cos({\mathit {angle}})\\c'&=1-c\\s&=\sin({\mathit {angle}})\end{aligned}}$

Source: https://www.khronos.org/registry/OpenGL-Refpages/gl2.1/xhtml/glRotate.xml

See also

• Homogeneous coordinates
• https://math.stackexchange.com/questions/237369/given-this-transformation-matrix-how-do-i-decompose-it-into-translation-rotati/417813