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Reflection

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A reflection is the operation of exchanging all points of a mathematical object with their mirror images. Objects that do not change handedness under reflection are said to be amphichiral; those that do are said to be chiral.

Reflection1

Consider the geometry of the left figure in which a point x_1 is reflected in a mirror (blue line). Then

x_r=x_0+n^^[(x_1-x_0)·n^^],
(1)

so the reflection of x_1 is given by

x_1^'=-x_1+2x_0+2n^^[(x_1-x_0)·n^^].
(2)
Reflection2

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 x_1 off a wall with normal vector n satisfies

x_1^'-x_0=v-2(v·n^^)n^^.
(3)

If the plane of reflection is taken as the yz-plane, the reflection in two- or three-dimensional space consists of making the transformation x->-x for each point. Consider an arbitrary point x_0 and a plane specified by the equation

ax+by+cz+d=0.
(4)

This plane has normal vector

n=[a; b; c],
(5)

and the signed point-plane distance is

D=(ax_0+by_0+cz_0+d)/(sqrt(a^2+b^2+c^2)).
(6)

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

x_0^'=x_0-2Dn^^
(7)
=[x_0; y_0; z_0]-(2(ax_0+by_0+cz_0+d))/(a^2+b^2+c^2)[a; b; c].
(8)

The reflection of a point with trilinear coordinates alpha_0:beta_0:gamma_0 in a point alpha_1:beta_1:gamma_1 is given by alpha:beta:gamma, where

alpha=2alpha_1(bbeta_0+cgamma_0)+alpha_0(aalpha_1-bbeta_1-cgamma_1)
(9)
beta=2beta_1(aalpha_0+cgamma_0)+beta_0(-aalpha_1+bbeta_1-cgamma_1)
(10)
gamma=2gamma_1(aalpha_0+bbeta_0)+gamma_0(-aalpha_1-bbeta_1+cgamma_1).
(11)

See also

Affine Transformation, Amphichiral, Chiral, Dilation, Enantiomer, Expansion, Glide, Handedness, Improper Rotation, Inversion Operation, Mirror Image, Projection, Reflection Property, Reflection Relation, Reflection Triangle, 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." https://web.archive.org/web/20020223025136/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.

Referenced on Wolfram|Alpha

Reflection

Cite this as:

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

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