TOPICS

# Reflection

The operation of exchanging all points of a mathematical object with their mirror images (i.e., reflections in a mirror). Objects that do not change handedness under reflection are said to be amphichiral; those that do are said to be chiral.

Consider the geometry of the left figure in which a point is reflected in a mirror (blue line). Then

 (1)

so the reflection of is given by

 (2)

The term reflection can also refer to the reflection of a ball, ray of light, etc. off a flat surface. As shown in the right diagram above, the reflection of a points off a wall with normal vector satisfies

 (3)

If the plane of reflection is taken as the -plane, the reflection in two- or three-dimensional space consists of making the transformation for each point. Consider an arbitrary point and a plane specified by the equation

 (4)

This plane has normal vector

 (5)

and the signed point-plane distance is

 (6)

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

 (7) (8)

The reflection of a point with trilinear coordinates in a point is given by , where

 (9) (10) (11)

Affine Transformation, Amphichiral, Chiral, Dilation, Enantiomer, Expansion, Glide, Handedness, Improper Rotation, Inversion Operation, Mirror Image, Projection, Reflection Triangle, Reflection Property, Reflection Relation, Reflexible, Rotation, Translation Explore this topic in the MathWorld classroom

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## References

Addington, S. "The Four Types of Symmetry in the Plane." http://mathforum.org/sum95/suzanne/symsusan.html.Coxeter, H. S. M. and Greitzer, S. L. "Reflection." §4.4 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 86-87, 1967.Voisin, C. Mirror Symmetry. Providence, RI: Amer. Math. Soc., 1999.Yaglom, I. M. Geometric Transformations I. New York: Random House, 1962.

Reflection

## Cite this as:

Weisstein, Eric W. "Reflection." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Reflection.html