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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 $yz$-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

\end{displaymath} (1)

This Plane has Normal Vector
{\bf n}=\left[{\matrix{a\cr b\cr c\cr}}\right],
\end{displaymath} (2)

and the Point-Plane Distance is
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)

See also Amphichiral, Chiral, Dilation, Enantiomer, Expansion, Glide, Handedness, Improper Rotation, Inversion Operation, Mirror Image, Projection, Reflection Property, Reflection Relation, Reflexible, Rotation, Rotoinversion, Translation

© 1996-9 Eric W. Weisstein