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 $\alpha: I\to\Bbb{R}^2$ be a smooth regular parameterized curve in $\Bbb{R}^2$ defined on an Open Interval $I$, and let $F_1$ and $F_2$ be points in $\Bbb{P}^2\backslash \alpha(I)$, where $\Bbb{P}^n$ is an $n$-D 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 Span of the vectors $\overrightarrow{F_1 P}$ and $\overrightarrow{F_2P}$.

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.

Foci	Sign	Both foci finite	One focus finite	Both foci infinite
distinct	Positive	confocal ellipses	confocal parabolas	parallel lines
distinct	Negative	confocal hyperbola and perpendicular	confocal parabolas	parallel lines
		bisector of interfoci line segment
equal		concentric circles		parallel lines

Let $S\in\Bbb{R}^3$ be a smooth connected surface, and let $F_1$ and $F_2$ be points in $\Bbb{P}^3\backslash S$, where $\Bbb{P}^n$ is an $n$-D 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 Span of the vectors $\overrightarrow{F_1 P}$ and $\overrightarrow{F_2P}$.

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.

Foci	Sign	Both foci finite	One focus finite	Both foci infinite
distinct	Positive	confocal ellipsoids	confocal paraboloids	parallel planes
distinct	Negative	confocal hyperboloids and plane perpendicular	confocal paraboloids	parallel planes
		bisector of interfoci line segment
equal		concentric spheres		parallel planes

See also Billiards

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.