TOPICS

Reflection

The operation of exchanging all points of a mathematical object with theirmirror images (i.e., reflections in a mirror). Objects that do not changehandedness under reflection are said to beamphichiral; those that do are said to bechiral.

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

(1)

so the reflection of x_1 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 x_1 off a wall withnormal vector satisfies

(3)

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

(4)

Thisplane hasnormal vector

(5)

and the signedpoint-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

			(7)
			(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

			(9)
			(10)
			(11)

Explore with Wolfram|Alpha

More things to try:

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 inGeometry 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." FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/Reflection.html

Movatterモバイル変換

Reflection

See also

Explore with Wolfram|Alpha

References

Referenced on Wolfram|Alpha

Cite this as:

Subject classifications