Reflection Property

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 alpha:I->R^2 be a smooth regular parameterized curve in R^2 defined on an open interval I, and let F_1 and F_2 be points in P^2\alpha(I), where P^n is an n-dimensional projective space. Then alpha has a reflection property with foci F_1 and F_2 if, for each point P in alpha(I),

1. Any vector normal to the curve alpha at P lies in the vector space span of the vectors F_1P^-> and F_2P^->.

2. The line normal to alpha at P bisects one of the pairs of opposite angles formed by the intersection of the lines joining F_1 and F_2 to P.

A smooth connected plane curve has a reflection property iff it is part of an ellipse, hyperbola, parabola, circle, or straight line.

Let S in R^3 be a smooth connected surface, and let F_1 and F_2 be points in P^3\S, where P^n is an n-dimensional projective space. Then S has a reflection property with foci F_1 and F_2 if, for each point P in S,

1. Any vector normal to S at P lies in the vector space span of the vectors F_1P^-> and F_2P^->.

2. The line normal to S at P bisects one of the pairs of opposite angles formed by the intersection of the lines joining F_1 and F_2 to P.

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.

See also

Explore with Wolfram|Alpha

References

Referenced on Wolfram|Alpha

Cite this as:

Weisstein, Eric W. "Reflection Property." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ReflectionProperty.html

Subject classifications