$\textbf{n} = (\textbf{p}_1 - \textbf{p}_0) \times (\textbf{p}_2 - \textbf{p}_0)$ 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
$ 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{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