Geometry

$\textbf{n} = (\textbf{p}_1 - \textbf{p}_0) \times (\textbf{p}_2 - \textbf{p}_0)$	Outward facing normal vector from points p0,p1,p2
$\textbf{n}^B = \textbf{n}^A \textbf{M}^{-1}$	transform normal from space A to B with transform matrix M
$\textbf{n}^B = \textbf{n}^A adj(\textbf{M}) = \textbf{n}^A det(\textbf{M})\textbf{M}^{-1}$	transform a normal formed from cross prodect from space A to B
$d = \sqrt{u^2 - \frac{(\textbf{u} \cdot \textbf{v})^2}{v^2}}$	Distance from point q to line v, u = q-p, p = point on line
$d = \sqrt{\frac{(\textbf{u} \times \textbf{v})^2}{v^2}}$	Alternate distance formula
$\begin{bmatrix}t_1 \\ t_2 \end{bmatrix} = \frac{1}{(\textbf{v}_1 \cdot \textb... ...\textbf{v}_1 \\ (\textbf{p}_2 - \textbf{p}_1) \cdot \textbf{v}_2\end{bmatrix}$	Time parameters for distance between to parameteric lines at points $p_1, p_2 (L1(t) = p_1 + t_1 v_1)$
$d = \vert\vert L_2(t_2) - L_1(t_1)\vert\vert$	Distance obtained from the above time parameters between two parametric lines
$d = \sqrt{\frac{[(\textbf{p}_2 - \textbf{p}_1) \times \textbf{v}_1]^2}{v_1^2}}$	Distance between two lines if they are parallel (determinant = 0)
$\textbf{f} \cdot \textbf{p} = 0, \textbf{f} = [n_x n_y n_z d] = [\textbf{n}\vert d]$	implicit plane representation, n is the normal to the plane, d = distance from plane to origin
$d = \textbf{f} \cdot \textbf{p}$	distance d from point p to plane f
$\textbf{p}' = \textbf{p} - 2(\textbf{f} \cdot \textbf{p}) \textbf{n}$	Reflection of point p through normalized plane f
$\textbf{H}_{reflect}(\textbf{f}) = \begin{bmatrix} 1 - 2n_x^2 & -2n_xn_y & -2n... ... -2n_xn_z & -2n_yn_z & 1 - 2n_z^2 & -2n_zd \\ 0 & 0 & 0 & 1 \end{bmatrix}$	Reflection matrix through plane f
$\textbf{q} = \textbf{p} - \frac{\textbf{f} \cdot \textbf{p}}{\textbf{f} \cdot \textbf{v}} \textbf{v}$	Intersection point of a line L(t) = p + tv with plane f
$\textbf{p} = \frac{d_1(\textbf{n}_3 \times \textbf{n}_2) + d_2(\textbf{n}_1 \ti... ...textbf{n}_2 \times \textbf{n}_1)} {[\textbf{n}_1, \textbf{n}_2, \textbf{n}_3]}$	Intersection point of 3 planes (divisor is scalar triple product)
$\textbf{p} = \frac{d_1(\textbf{v} \times \textbf{n}_2) + d_2(\textbf{n}_1 \times \textbf{v})} {v^2}$	Intersection point of two planes, $\textbf{v} = \textbf{n}_1 \times \textbf{n}_2$
$\textbf{f}^B = \textbf{f}^A det(\textbf{H})\textbf{H}^{-1} = \textbf{f}^A adj(\textbf{H})$	Transformation of plane f in space A to space B with transform mat H
$\{\textbf{v}\vert\textbf{m}\}, \textbf{m} = \textbf{p}_1 \times \textbf{p}_2$	Plucker coords rep of a line, v = direction, and p1,p2 are any points on the line (called 'moment')
$(\textbf{p} \vert w)$	Plucker coords rep of a 4D point vector with w component
$d = \frac{\vert\textbf{v}_1 \cdot \textbf{m}_2 + \textbf{v}_2 \cdot \textbf{m}_1\vert} {\vert\vert\textbf{v}_1 \times \textbf{v}_2\vert\vert}$	Distance between two lines in Plucker rep

$\{\textbf{v}^B \vert \textbf{m}^B\} = \{\textbf{M}\textbf{v}^A \vert \textbf{m}^A adj(\textbf{M}) + \textbf{t} \times (\textbf{M}\textbf{v}^A)\}$

Line transform from mat H, M = upper 3x3 matrix from H, t = last column (translation) of H