Reflection

The operation of exchanging all points of a mathematical object with their Mirror Images (i.e., reflections in a mirror). Objects which do not change Handedness under reflection are said to be Amphichiral; those that do are said to be Chiral.

If the Plane of reflection is taken as the -Plane, the reflection in 2- or 3-D Space consists of making the transformation $x\to -x$ for each point. Consider an arbitrary point ${\bf x}_0$ and a Plane specified by the equation

$\begin{displaymath} ax+by+xz+d=0. \end{displaymath}$

(1)

This Plane has Normal Vector

$\begin{displaymath} {\bf n}=\left[{\matrix{a\cr b\cr c\cr}}\right], \end{displaymath}$

(2)

and the Point-Plane Distance is

$\begin{displaymath} D={\vert ax_0+by_0+cz_0+d\vert\over\sqrt{a^2+b^2+c^2}}. \end{displaymath}$

(3)

The position of the point reflected in the given plane is therefore given by

$\displaystyle {\bf x}_0'$	$\textstyle =$	$\displaystyle {\bf x}_0-2D\hat{\bf n}$
	$\textstyle =$	$\displaystyle \left[\begin{array}{c}x_0\\ y_0\\ z_0\end{array}\right]-2\vert ax_0+by_0+cz_0+d\vert\left[\begin{array}{c}a\\ b\\ c\end{array}\right].$	(4)