TOPICS
Reflection Property
In the plane, the reflection property can be stated as three theorems (Ogilvy 1990, pp. 73-77):
1. Thelocus of the center of a variablecircle, tangent to a fixedcircle and passing through a fixed point inside thatcircle, is anellipse.
2. If a variablecircle is tangent to a fixedcircle and also passes through a fixed point outside thecircle, then thelocus of its moving center is ahyperbola.
3. If a variablecircle is tangent to a fixed straight line and also passes through a fixed point not on the line, then thelocus of its moving center is aparabola.
Let
be a smooth regular parameterized curve in
defined on anopen interval
, and let
and
be points in
, where
is an
-dimensionalprojective space. Then
has a reflection property withfoci
and
if, for each point
,
1. Any vector normal to the curve
at
lies in thevector space span of the vectors
and
.
2. The line normal to
at
bisects one of the pairs of oppositeangles formed by the intersection of the lines joining
and
to
.
A smooth connected plane curve has a reflection propertyiff it is part of anellipse,hyperbola,parabola,circle, or straightline.
| 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
-dimensionalprojective space. Then
has a reflection property withfoci
and
if, for each point
,
1. Any vector normal to
at
lies in thevector 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 propertyiff it is part of anellipsoid of revolution, ahyperboloid of revolution, aparaboloid of revolution, asphere, or aplane.
See also
BilliardsExplore with Wolfram|Alpha
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References
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.Referenced on Wolfram|Alpha
Reflection PropertyCite this as:
Weisstein, Eric W. "Reflection Property."FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/ReflectionProperty.html