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Reflection Property


In the plane, the reflection property can be stated as three theorems (Ogilvy 1990, pp. 73-77):

1. The locus of the center of a variable circle, tangent to a fixed circle and passing through a fixed point inside that circle, is an ellipse.

2. If a variable circle is tangent to a fixed circle and also passes through a fixed point outside the circle, then the locus of its moving center is a hyperbola.

3. If a variable circle is tangent to a fixed straight line and also passes through a fixed point not on the line, then the locus of its moving center is a parabola.

Let alpha:I->R^2 be a smooth regular parameterized curve in R^2 defined on an open interval I, and let F_1 and F_2 be points in P^2\alpha(I), where P^n is an n-dimensional projective space. Then alpha has a reflection property with foci F_1 and F_2 if, for each point P in alpha(I),

1. Any vector normal to the curve alpha at P lies in the vector space span of the vectors F_1P^-> and F_2P^->.

2. The line normal to alpha at P bisects one of the pairs of opposite angles formed by the intersection of the lines joining F_1 and F_2 to P.

A smooth connected plane curve has a reflection property iff it is part of an ellipse, hyperbola, parabola, circle, or straight line.

focisignboth foci finiteone focus finiteboth foci infinite
distinctpositiveconfocal ellipsesconfocal parabolasparallel lines
distinctnegativeconfocal hyperbola and perpendicularconfocal parabolasparallel lines
bisector of interfoci line segment
equalconcentric circlesparallel lines

Let S in R^3 be a smooth connected surface, and let F_1 and F_2 be points in P^3\S, where P^n is an n-dimensional projective space. Then S has a reflection property with foci F_1 and F_2 if, for each point P in S,

1. Any vector normal to S at P lies in the vector space span of the vectors F_1P^-> and F_2P^->.

2. The line normal to S at P bisects one of the pairs of opposite angles formed by the intersection of the lines joining F_1 and F_2 to P.

A smooth connected surface has a reflection property iff it is part of an ellipsoid of revolution, a hyperboloid of revolution, a paraboloid of revolution, a sphere, or a plane.

focisignboth foci finiteone focus finiteboth foci infinite
distinctpositiveconfocal ellipsoidsconfocal paraboloidsparallel planes
distinctnegativeconfocal hyperboloids and plane perpendicularconfocal paraboloidsparallel planes
bisector of interfoci line segment
equalconcentric spheresparallel planes

See also

Billiards

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References

Drucker, D. "Euclidean Hypersurfaces with Reflective Properties." Geometrica Dedicata 33, 325-329, 1990.Drucker, D. "Reflective Euclidean Hypersurfaces." Geometrica Dedicata 39, 361-362, 1991.Drucker, D. "Reflection Properties of Curves and Surfaces." Math. Mag. 65, 147-157, 1992.Drucker, D. and Locke, P. "A Natural Classification of Curves and Surfaces with Reflection Properties." Math. Mag. 69, 249-256, 1996.Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 73-77, 1990.Wegner, B. "Comment on 'Euclidean Hypersurfaces with Reflective Properties.' " Geometrica Dedicata 39, 357-359, 1991.

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Reflection Property

Cite this as:

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

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