Related by Similiarity: Poristic Triangles and 3-Periodics in the Elliptic Billiard

Abstract: Discovered by William Chapple in 1746, the Poristic family is a set of variable-perimeter triangles with common Incircle and Circumcircle. By definition, the family has constant Inradius-to-Circumradius ratio. Interestingly, this invariance also holds for the family of 3-periodics in the Elliptic Billiard, though here Inradius and Circumradius are variable and perimeters are constant. Indeed, we show one family is mapped onto the other via a varying similarity transform. This implies that any scale-free quantities and invariants observed in one family must hold on the other.

• The paper establishes a mapping between Poristic triangles and 3-periodic billiards using a similarity transform, preserving scale-free geometric invariants.
• Key theoretical results include delineation of invariant properties for circum- and inconics and the identification of semi-axes invariance properties across configurations.
• This research provides a rigorous framework for geometric invariants, potentially improving numerical algorithms in computational geometry and aiding future applications in simulations and machine learning.

Analyzing Poristic Triangles and 3-Periodics in the Elliptic Billiard

This paper explores the intriguing mathematical relationship between the Poristic family of triangles and 3-periodic families within the context of the Elliptic Billiard. Focusing on the geometric properties and scale-free invariants of these triangles, the authors provide a formal foundation for mapping one family onto the other through a similarity transform. This approach ensures that invariants in one family are preserved in the other, highlighting the deeper interconnectedness between these distinct geometric configurations.

Background on Poristic Families

The Poristic family of triangles, historically rooted in the work of William Chapple and expanded by Euler and Poncelet, consists of triangles that share a fixed Incircle and Circumcircle while exhibiting a constant Inradius-to-Circumradius ratio, $r/R$. This family presents a range of configurations including acute and obtuse triangles, contingent upon the relative alignment of the triangle's center and circle radii. The inherent invariant $r/R$ serves as a pivotal metric for analyzing these configurations, forming a basis for comparisons with elliptic systems.

Theoretical Contributions

The primary contribution of this study lies in establishing a direct correspondence between Poristic families and 3-periodic billiards through a similarity transform. This transform is meticulously derived, considering parameters such as semi-axes and focal configurations, allowing for the preservation of scale-free invariants. The insights presented in this work are supported by several key theoretical results:

1. Conic Invariant Properties: The study delineates the invariant properties of circum- and inconics within Poristic triangles, demonstrating that semi-axis ratios remain constant across configurations. Theorems addressing the Excentral Caustic and Excentral $X_3$-Centered Inconic elucidate these invariances.
2. Axes and Aspect Ratios: The paper highlights specific semi-axes invariance properties for certain conics which are crucial for maintaining the comparative framework between Poristic and 3-periodic families. These properties emphasize the consistent geometric relationships that underpin the broader family mappings.
3. Elliptic Transform Equivalence: Perhaps the most compelling result is the mapping through a similarity transform linking Poristic triangles to 3-periodics. This conceptual bridge allows the authors to show that invariant quantities such as perimeter ratios and axis parallels remain unaffected, underscoring the analytical robustness of their approach.

Implications and Future Directions

From a theoretical perspective, the work offers a profound understanding of geometric invariants in classical systems, which could extend to modern computational methods in geometry. By establishing a rigorous framework for these invariants, the research aids in enhancing numerical algorithms that rely on geometric properties, potentially facilitating more efficient computational geometry applications.

In practical terms, extending these findings to broader classes of geometric configurations or higher-dimensional counterparts could yield new insights, particularly in numerical simulations and model validations where geometric constancy is paramount.

Looking ahead, further research might explore the integration of these geometric insights within machine learning frameworks, where invariance principles could enhance the fidelity and interpretability of models dealing with complex spatial data.

Conclusion

The presented work deftly navigates the complex landscape of Poristic triangles and 3-periodic paths, offering robust theoretical insights that further our understanding of geometric invariance. By bridging classical geometry with contemporary analytical techniques, this study not only enriches the existing body of knowledge but also sets the stage for future exploration into more profound geometric relationships in both theorem and application.