Transformations

$\textbf{M}\textbf{v} = v_x \textbf{a} + v_y \textbf{b} + v_z \textbf{c}$	Where a,b,c are the columns of M
$\textbf{M}^T \textbf{M} = \begin{bmatrix} a^2 & \textbf{a} \cdot \textbf{b} & ... ...textbf{c} \cdot \textbf{a} & \textbf{c} \cdot \textbf{b} & c^2 \end{bmatrix}$	Orthogonal transform; a,b,c are cols of M. Unit cols and all perpindicular. $M^T = M^-1$
$\textbf{B} = \textbf{M} \textbf{A} \textbf{M}^{-1}$	Transform from coord space A applied in coord space B. M = coord space transform.
$M_{rot_x}(\theta) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & cos \theta & -sin \theta \\ 0 & sin \theta & cos \theta \end{bmatrix}$	Rotation matrix about X axis
$M_{rot_y}(\theta) = \begin{bmatrix} cos \theta & 0 & sin \theta \\ 0 & 1 & 0 \\ sin \theta & 0 & cos \theta \end{bmatrix}$	Rotation matrix about Y axis
$M_{rot_z}(\theta) = \begin{bmatrix} cos \theta & -sin \theta & 0 \\ sin \theta & cos \theta & 0 \\ 0 & 0 & 1 \end{bmatrix}$	Rotation matrix about Z axis
$M_{rot}(\theta, \textbf{a}) = \begin{bmatrix} c + (1-c)a_x^2 & (1-c)a_x a_y - ... ... (1-c)a_x a_z - s a_y & (1-c)a_y a_z + s a_x & c + (1-c) a_z^2 \end{bmatrix}$	Rotation matrix about arbitrary axis a. c = $cos \theta$, s = $sin \theta$
$M_{reflect}(\textbf{a}) = \begin{bmatrix} 1 - 2a_x^2 & -2a_x a_y & -2a_x a_z \... ... 1 -2a_y^2 & -2a_y a_z \\ -2a_x a_z & -2a_y a_z & 1 - 2a_z^2 \end{bmatrix}$	Reflection matrix about axis a, assumes a unit length.
$M_{invol}(\textbf{a}) = \begin{bmatrix} 2a_x^2 - 1 & 2a_x a_y & 2a_x a_z \\ ... ... & 2a_y^2 - 1 & 2a_y a_z \\ 2a_x a_z & 2a_y a_z & 2a_z^2 - 1 \end{bmatrix}$	Involution matrix about axis a, assumes a unit length. Negation of reflect mat.
$M_{scale}(s_x,s_y,s_z) = \begin{bmatrix} s_x & 0 & 0 \\ 0 & s_y & 0 \\ 0 & 0 & s_z \end{bmatrix}$	Scale matrix about x,y,z axes
$M_{scale}(s,\textbf{a}) = \begin{bmatrix} (s-1)a_x^2 + 1 & (s-1)a_x a_y & (s-1... ...s-1)a_y a_z \\ (s-1)a_x a_z & (s-1) a_y a_z & (s-1)a_z^2 + 1 \end{bmatrix}$	Scale matrix along direction a

$M_{skew}(\theta, \textbf{a}, \textbf{b}) = \begin{bmatrix} a_x b_x tan \theta ... ...z b_x tan \theta & a_z b_y tan \theta & a_z b_z tan \theta + 1 \end{bmatrix}$	Skew matrix along direction a based on length projection b
$M_{skew}(\theta, \textbf{a}, \textbf{b}) = \textbf{I} + tan \theta (\textbf{a} \otimes \textbf{b})$	Alternative skew matrix formation
$\textbf{H} = \begin{bmatrix}\textbf{M} & \textbf{t} \\ \textbf{0} & 1 \end{bmatrix}$	Homogenous transformation matrix; M = 3x3 transf mat, t = translate vec
$\textbf{H}^{-1} = \begin{bmatrix} \textbf{M}^{-1} & -\textbf{M}^{-1}\textbf{t} \\ \textbf{0} & 1 \end{bmatrix}$	Homogenous transformation matrix inverse