Circum- and Inconic Invariants of 3-Periodics in the Elliptic Billiard (2004.02680v4)

Abstract: A Circumconic passes through a triangle's vertices; an Inconic is tangent to the sidelines. We study the variable geometry of certain conics derived from the 1d family of 3-periodics in the Elliptic Billiard. Some display intriguing invariances such as aspect ratio and pairwise ratio of focal lengths.

• The paper identifies invariant properties of circumconics (circumellipses, circumhyperbolas) and inconics associated with 3-periodic trajectories in an elliptic billiard.
• Key findings include fixed axis ratios for specific circumellipses and inconics, and constant focal length ratios for certain circumhyperbolas across the family of 3-periodics.
• These invariant properties contribute to the understanding of geometric interactions in elliptic billiards and have potential implications for areas like computer graphics and dynamical systems.

Invariants of Circumconics and Inconics in Elliptic Billiards

This paper presents a comprehensive examination of specific conics derived from 3-periodic trajectories within an Elliptic Billiard (EB). The main focus is on identifying invariants in the geometrical properties of these conics, and exploring their implications within the context of elliptic billiards and broader applications in geometry. The authors introduce the concept of circumconics and inconics, specifically their circumellipse and circumhyperbola variants, which exhibit fascinating invariances such as fixed aspect ratios and consistent relationships in their focal lengths across varying configurations of 3-periodic trajectories in the EB.

Key Findings

Circumellipses:

1. Axis Ratio of $E_1$: The $X_1$-centered circumellipse of the 3-periodic family holds a fixed ratio of its semiaxes even as the semiaxes themselves vary. This invariant aspect ratio is determined by intrinsic properties of the triangle's geometry.
2. Moses Pencil of Circumellipses: A pencil of circumellipses, whose centers lie on the Feuerbach Circumhyperbola of the Medial Triangle, share mutually parallel axes and also maintain invariant axis ratio among them. This finding suggests an underlying harmonization between these conic sections, contributing to the paper of circumconics associated with classic triangle centers such as the Incenter and Circumcenter.

Circumhyperbolas:

1. Focal Lengths in Hyperbolas: For the Feuerbach hyperbola ($F$) and the Excentral Jerabek hyperbola ($J_{exc}$), the focal lengths maintain a constant ratio over the 3-periodic family. This ratio, given by $\sqrt{2/\rho}$, presents a novel link between these focal properties and the overall geometry of the conic sections with respect to periodic trajectories inside the EB.

Inconics:

1. Invariance in Inconic Aspect Ratios:
   • Excentral $X_3$-Centered Inconic ($I_3'$): This inconic, a 90-degree rotated copy of the traditional $X_1$-centered circumellipse, maintains an invariant aspect ratio.
   • Excentral MacBeath Inconic ($I_5'$): The axis ratio of this inconic is invariant over 3-periodics, providing insightful connections between inelliptical properties and circumconic configurations in the EB setup.

The exploration of these properties enhances our understanding of the geometric and algebraic interactions that occur within elliptic billiards. The results provide a rigorous set of tools and principles that can potentially extend to other types of periodic billiard systems, offering a deeper understanding of dynamical systems governed by periodic reflections.

Implications and Future Directions

The invariant properties of circumconics and inconics associated with 3-periodics in an elliptic billiard contribute to the broader paper of geometric invariants in mathematical physics and dynamical systems. These findings may influence the development of computational models in arenas such as computer graphics, where curve fitting and trajectory modeling rely on precise geometric underpinnings. Further research could focus on generalizing these results to $N$-periodics, potentially unearthing new classes of geometric and algebraic invariants applicable to multidimensional and higher-complexity systems.

Future challenges lie in formulating expressions for the focal lengths of circumhyperbolas and proving the invariance of circumconic axis ratios using algebraic or projective geometric methods. Moreover, investigating the loci of foci, parabola configurations, and cubic relationships could open avenues for discovering even more intricate connections linking topology, symmetry, and integrable systems. As the understanding of these invariants progresses, it can potentially lead to novel insights into the mechanics of elliptic motion and its numerous applications within theoretical and applied mathematics.