“Reflection Property”的概念、定义、习题解题方法及翻译-数学辞典

单词

Reflection Property

释义

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 througha 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 outsidethe 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 onthe line, then the Locus of its moving center is a Parabola.

Let be a smooth regular parameterized curve in defined on an Open Interval , andlet and be points in , where is an -D 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 Span of the vectors and .
: 2. The line normal to at bisects one of the pairs of opposite Angles formed by theintersection 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 -D Projective Space. Then has a reflection property with Foci and if, for each point ,

: 1. Any vector normal to at lies in the Span of the vectors and .
: 2. The line normal to at bisects one of the pairs of opposite angles formed by the intersection ofthe lines joining and to .

A smooth connected surface has a reflection property Iff it is part of an Ellipsoid of revolution, aHyperboloid of revolution, a Paraboloid of revolution, a Sphere, or a Plane.

Foci	Sign	Both foci finite	One focus finite	Both foci infinite
distinct	Positive	confocal ellipsoids	confocal paraboloids	parallel planes
distinct	Negative	confocal hyperboloids and plane perpendicular	confocal paraboloids	parallel planes
		bisector of interfoci line segment
equal		concentric spheres		parallel planes

See also Billiards

References

Drucker, D. ``Euclidean Hypersurfaces with Reflective Properties.'' Geometrica Dedicata 33, 325-329, 1990.

Drucker, D. ``Reflective Euclidean Hypersurfaces.'' Geometrica Dedicata 39, 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 Dedicata 39, 357-359, 1991.

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