二维射影 Finsler 度量中反射焦散线的尖点
On cusps of caustics by reflection in two dimensional projective Finsler metrics
AI总结：
本文研究射影 Finsler 度量下凸平面区域中的 Finsler 台球，证明经历固定反射次数的反射焦散线至少具有四个尖点，推广了欧氏台球的相关结果。
摘要：
平面或其凸子集上的 Finsler 度量（不必对称）若其测地线为直线段，则称为射影的。我们考虑在赋予射影 Finsler 度量的凸平面区域中的 Finsler 台球。反射焦散线是有向直线（即台球轨迹）的包络，这些轨迹从台球内部一点出发，并经历固定次数的反射。我们证明这种焦散线至少有四个尖点。该问题的动机来自“Jacobi 最后的几何陈述”：三轴椭球面上非脐点的共轭轨迹恰有四个尖点。本札记推广了此方向上关于欧氏台球的近期结果。
英文摘要：
A Finsler, not necessarily symmetric, metric in the plane or its convex subset is called projective if its geodesics are straight segments. We consider Finsler billiards in a convex planar domain endowed with a projective Finsler metric. A caustic by reflection is the envelope of the oriented lines, the billiard trajectories, that start at a point inside the billiard and undergo a fixed number of reflections. We show that such a caustic has at least four cusps. This problem is motivated by the "Last Geometric Statement of Jacobi" that the conjugate locus of a non-umbilic point of a triaxial ellipsoid has exactly four cusps. The present note extends the recent results in this direction concerning Euclidean billiards.
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