TY - JOUR

T1 - On Cusps of Caustics by Reflection

T2 - Billiard Variations on the Four Vertex Theorem and on Jacobi’s Last Geometric Statement

AU - Bor, Gil

AU - Tabachnikov, Serge

N1 - Funding Information:
GBacknowledges support from the Shapiro visiting program in Penn State and CONACYT Grant A1-S-4588. ST was supported by NSF grant DMS-2005444.
Publisher Copyright:
© 2023 The Mathematical Association of America.

PY - 2023

Y1 - 2023

N2 - 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.

AB - 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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U2 - 10.1080/00029890.2023.2179842

DO - 10.1080/00029890.2023.2179842

M3 - Article

AN - SCOPUS:85150776006

SN - 0002-9890

VL - 130

SP - 454

EP - 467

JO - American Mathematical Monthly

JF - American Mathematical Monthly

IS - 5

ER -