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.
