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 be a smooth regular parameterized curve in defined on an Open Interval , and let and be points in , where is an -D Projective Space. Then has a reflection property with Foci and if, for each point ,

- 1. Any vector normal to the curve at lies in the Span of the vectors and .
- 2. The line normal to at bisects one of the pairs of opposite Angles formed by the intersection of the lines joining and to .

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 be a smooth connected surface, and let and be points in , where is an -D Projective Space. Then has a reflection property with Foci and if, for each point ,

- 1. Any vector normal to at lies in the Span of the vectors and .
- 2. The line normal to at bisects one of the pairs of opposite angles formed by the intersection of the lines joining and to .

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 |

**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.

© 1996-9

1999-05-25