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 -dimensional 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 vector space 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 .
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 be a smooth connected surface, and let and be points in , where is an -dimensional projective space. Then has a reflection property with foci and if, for each point ,
1. Any vector normal to at lies in the vector space 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 .
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.