The Ballet of Triangle Centers on the Elliptic Billiard (2002.00001v2)

Abstract: The dynamic geometry of the family of 3-periodics in the Elliptic Billiard is mystifying. Besides conserving perimeter and the ratio of inradius-to-circumradius, it has a stationary point. Furthermore, its triangle centers sweep out mesmerizing loci including ellipses, quartics, circles, and a slew of other more complex curves. Here we explore a bevy of new phenomena relating to (i) the shape of 3-periodics and (ii) the kinematics of certain Triangle Centers constrained to the Billiard boundary, specifically the non-monotonic motion some can display with respect to 3-periodics. Hypnotizing is the joint motion of two such non-monotonic Centers, whose many stops-and-gos are akin to a Ballet.

• The paper explores the geometric loci and kinematic behaviors of various triangle centers associated with 3-periodic trajectories in an elliptic billiard.
• It identifies critical aspect ratios of the ellipse where triangles change shape (obtuse/right) and centers' loci form complex algebraic curves, including ellipses and self-intersecting paths.
• A key finding describes the 'non-monotonic' motion or 'ballet' of certain triangle centers along their paths, exhibiting backward and forward movement depending on the billiard's aspect ratio.

Overview of "The Ballet of the Triangle Centers on the Elliptic Billiard"

The paper presented in the paper explores the intricate behaviors of triangle centers associated with the family of 3-periodic trajectories in an elliptic billiard table. An elliptic billiard is a specific type of Poncelet's Porism, allowing for a continuous family of polygons—triangles in this case—that remain periodic and maintain certain invariant properties.

Key Abstract Concepts and Properties

In the context of elliptic billiards, the 3-periodic triangles are particularly notable due to their conservation properties, such as perimeter constancy and the invariance of the inradius-to-circumradius ratio. This unique integrability offers a rich field for geometric exploration.

Properties and Phenomena

1. Shape Properties: The paper explores the conditions under which the 3-periodic triangles become obtuse or even right triangles. It is noted that such transitions in the geometrical shape correspond to specific aspect ratios of the billiard table, discoverable through derived mathematical expressions.
2. Locus Complexity: A major focus of the work is on the geometric loci traced by various classical triangle centers like the Incenter, Orthocenter, and Circumcenter. Noteworthy phenomena include the identification of loci that form ellipses and other higher-degree algebraic curves, even exhibiting features such as non-compactness and self-intersection.
3. Non-Monotonic Kinematics: Perhaps one of the most intriguing insights is the non-monotonic motion nature of some triangle centers when constrained to the billiard boundary. This aspect infamously termed as a "ballet," sees triangle centers engaging in backward and forward dances along their paths, influenced by specific aspect ratios.

Strong Results and Conclusions

The numerical and theoretical results provide insights into a variety of geometrical behaviors:

• The critical values of aspect ratios (e.g., specific thresholds) are articulated, where triangle episodic motions transition from monotonic to non-monotonic or where specific trilinear conditions are satisfied.
• Examples include the motion analysis of the Mittenpunkt ($X_9$), which intriguingly stays fixed at the center of the billiard, as well as the Bevan Point ($X_{40}$), whose locus turns into a rotated copy of the billiard at the golden ratio aspect.

Implications and Future Directions

The paper suggests several implications in theoretical geometry, particularly in understanding the properties of triangle centers in dynamic and constrained settings. Moreover, these insights might find practical relevance in computational geometry, computer graphics, and perhaps enlightening connections to integrable systems.

Future inquiries are directed towards more profound analyses of loci degree determination, further classifications of kinematic behaviors among other triangle centers, and broader implications of these surprising symmetric properties uncovered in planar billiards.

In summary, the paper provides an articulate and detailed exploration of the geometric ballet performed by triangle centers within an elliptic billiard table, opening avenues for further mathematical discovery both analytically and computationally.