用反射论焦散的顶点:关于四顶点定理和雅可比最后几何陈述的台球变型

IF 0.4 4区数学 Q4 MATHEMATICS American Mathematical Monthly Pub Date : 2023-03-17 DOI:10.1080/00029890.2023.2179842

Gil Bor, S. Tabachnikov

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引用次数: 0

摘要

摘要一个点光源被放置在一个椭圆内。反射的第n个焦散是在从曲线反射n次之后从光源发出的光线的包络。我们表明，对于通用点光源，这些焦散中的每一个都至少有4个尖端。这是关于凸表面上点的共轭轨迹的尖端数的Jacobi最后一个几何陈述的台球变体。我们用不同的思想提出了各种各样的证明，包括曲线缩短流和勒让德结理论。

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On Cusps of Caustics by Reflection: Billiard Variations on the Four Vertex Theorem and on Jacobi’s Last Geometric Statement

Abstract A point source of light is placed inside an oval. The nth caustic by reflection is the envelope of the light rays emanating from the light source after n reflections off the curve. We show that, for a generic point light source, each of these caustics has at least 4 cusps. This is a billiard variation on Jacobi’s Last Geometric Statement concerning the number of cusps of the conjugate locus of a point on a convex surface. We present various proofs, using different ideas, including the curve shortening flow and Legendrian knot theory.

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来源期刊

American Mathematical Monthly Mathematics-General Mathematics

CiteScore

0.80

自引率

20.00%

发文量

127

审稿时长

6-12 weeks

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