Is every triangle a trajectory of an elliptical billiard?

TL;DR

The paper proves that every triangle is a $3$-periodic billiard trajectory inside a unique ellipse by combining Marden's theorem from polynomial geometry with classical triangle results (Ceva and Menelaus); it further shows that a convex $4$-periodic elliptical trajectory is exactly a parallelogram, with a unique ellipse for each parallelogram, and provides both synthetic and analytic proofs. It extends the nonconvex case to Darboux butterflies, showing each butterfly is a $4$-periodic trajectory within a unique ellipse and that such butterflies are unions of two $3$-periodic trajectories in congruent acute triangles, with explicit foci computations. The work thus bridges Marden's theorem and the Poncelet porism, illuminating deep connections between polynomial geometry, conic configurations, and elliptic billiards, and supplies constructive methods for determining the boundary ellipses and focal data. Overall, it offers a unified framework for realizing any triangle, parallelogram, or Darboux butterfly as a billiard trajectory inside a uniquely determined ellipse, enriching the geometry of billiards with explicit analytic and synthetic tools.

Abstract

Using Marden's Theorem from geometric theory of polynomials, we show that for every triangle there is a unique ellipse such that the triangle is a billiard trajectory within that ellipse. Since $3$-periodic trajectories of billiards within ellipses are examples of the Poncelet polygons, our considerations provide a new insight into the relationship between Marden's Theorem and the Poncelet Porism, two gems of exceptional classical beauty. We also show that every parallelogram is a billiard trajectory within a unique ellipse. We prove a similar result for the self-intersecting polygonal lines consisting of two pairs of congruent sides, named "Darboux butterflies". In each of three considered cases, we effectively calculate the foci of the boundary ellipses.

Figures (32)

• Figure 1: Billiard reflection law: the angle of incidence with the tangent line at the bouncing point on the boundary equals the angle of reflection.
• Figure 2: Billiard motion in a triangle.
• Figure 3: The minimization property of billiard reflection: the shortest path connecting points $X$ and $Y$ which visits line $\ell$ is the billiard path $XLY$. Its length equals the segment $XY'$, where $Y'$ is symmetric to $Y$ with respect to $\ell$. For any other point $L'\in\ell$, we have $XL'+L'Y=XL'+L'Y'>XY'$.
• Figure 4: For a fixed point $M'$ on $AB$, we construct points $K'$, $L'$ on the remaining two sides of the triangle, such that $\triangle K'L'M'$ has smallest possible perimeter.
• Figure 5: The vertices of the triangle with smallest perimeter inscribed in $\triangle ABC$ are the feet $K$, $L$, $M$ of the altitudes.

Theorems & Definitions (42)

• Remark 2.1
• Theorem 2.2: see e.g. DragRadn2011book*Theorem 2.2
• Theorem 2.3: Ceva's Theorem
• Remark 2.4
• Remark 2.5
• Lemma 2.6
• Theorem 2.7: Theorem of Menelaus
• Corollary 2.8
• Theorem 2.9: Simson's line
• Proposition 3.1: First focal property of ellipses