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On Cusps of Caustics by Reflection: Billiard Variations on the Four Vertex Theorem and on Jacobi's Last Geometric Statement.
- Source :
-
American Mathematical Monthly . May2023, Vol. 130 Issue 5, p454-467. 14p. - Publication Year :
- 2023
-
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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00029890
- Volume :
- 130
- Issue :
- 5
- Database :
- Academic Search Index
- Journal :
- American Mathematical Monthly
- Publication Type :
- Academic Journal
- Accession number :
- 163387731
- Full Text :
- https://doi.org/10.1080/00029890.2023.2179842