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 .

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 .

Drucker, D. "Euclidean Hypersurfaces with Reflective Properties." Geometrica Dedicata33, 325-329, 1990.Drucker,
D. "Reflective Euclidean Hypersurfaces." Geometrica Dedicata39,
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
Dedicata39, 357-359, 1991.