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<p>This papers studies properties of caustics of different billiard models in dimension <i>d</i>, with <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(d\ge 3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. Namely, we consider the cases where the law of reflection is defined by 1) a Riemannian metric projectively equivalent to the Euclidean one; 2) a constant non-degenerate quadratic form (<i>pseudo-Euclidean billiards</i>); 3) a smooth field of transverse lines to the boundary defining a law of reflection (<i>projective billiards</i>). Cases 1) and 2) are particular cases of 3). The paper gives a necessary and sufficient condition so that if such billiards have a caustic then the latter is a quadric. In the case of pseudo-Euclidean billiards, we even show that the only billiards having a caustic are the quadrics, for which the caustics are pseudo-confocal quadrics.</p>

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Only quadrics have pseudo-caustics – on caustics of Riemannian, pseudo-Euclidean and projective billiards in higher dimensions

  • 期刊论文

  • 发表日期：

    2026年2月6日

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  • Corentin Fierobe

摘要

This papers studies properties of caustics of different billiard models in dimension d, with \(d\ge 3\) d ≥ 3 . Namely, we consider the cases where the law of reflection is defined by 1) a Riemannian metric projectively equivalent to the Euclidean one; 2) a constant non-degenerate quadratic form (pseudo-Euclidean billiards); 3) a smooth field of transverse lines to the boundary defining a law of reflection (projective billiards). Cases 1) and 2) are particular cases of 3). The paper gives a necessary and sufficient condition so that if such billiards have a caustic then the latter is a quadric. In the case of pseudo-Euclidean billiards, we even show that the only billiards having a caustic are the quadrics, for which the caustics are pseudo-confocal quadrics.