New Invariants of Poncelet-Jacobi Bicentric Polygons (2103.11260v3)

Abstract: The 1d family of Poncelet polygons interscribed between two circles is known as the Bicentric family. Using elliptic functions and Liouville's theorem, we show (i) that this family has invariant sum of internal angle cosines and (ii) that the pedal polygons with respect to the family's limiting points have invariant perimeter. Interestingly, both (i) and (ii) are also properties of elliptic billiard N-periodics. Furthermore, since the pedal polygons in (ii) are identical to inversions of elliptic billiard N-periodics with respect to a focus-centered circle, an important corollary is that (iii) elliptic billiard focus-inversive N-gons have constant perimeter. Interestingly, these also conserve their sum of cosines (except for the N=4 case).

• The paper demonstrates that bicentric polygons maintain constant cosine sum invariants analogous to properties in elliptic billiard trajectories.
• It establishes invariant perimeters of pedal polygons by applying elliptic functions and Liouville’s theorem.
• The findings reveal that focus-inversive pedal polygons remain invariant under inversion, extending classical insights into Poncelet polygons.

An Insightful Overview of "New Invariants of Poncelet-Jacobi Bicentric Polygons"

The paper "New Invariants of Poncelet-Jacobi Bicentric Polygons" by Roitman, Garcia, and Reznik investigates geometric properties of bicentric N-gons, specifically looking at invariant sums of internal angle cosines and invariant perimeters of pedal polygons. Utilizing elliptic functions and Liouville's theorem, the authors present new findings pertaining to Poncelet polygons, which are a subset of polygons interscribed between two circles.

Main Results and Contributions

The paper presents several key results:

1. Invariant Cosine Sums: It is shown that the sum of cosines of bicentric polygons remains constant across the family. This characteristic mirrors properties observed in elliptic billiard N-periodics, a class of trajectories studied in the context of billiard dynamics inside ellipses.
2. Invariant Perimeters: The perimeter of pedal polygons, derived from the bicentric polygons with respect to the limiting points (specific geometric loci related to the circles in question), is also invariant. This is similar to a known feature of elliptic billiard N-periodics, which conserve their perimeter.
3. Focus-Inversive N-gons: The paper reveals that bicentric pedal polygons, considered with respect to a limiting point, are equivalent to the inversion of elliptic billiard N-gons around a circle centered at a focus, confirming constant perimeter in these configurations.

The paper explores the geometric transformations and symmetries within bicentric families, expanding on 200 years of mathematical discourse since Poncelet's time. These well-defined invariants connect the paper to broad implications in classical geometry and extend to insights on related mathematical fields, such as dynamic systems and algebraic geometry.

Theoretical and Practical Implications

The research enriches theoretical understanding by highlighting intrinsic invariants of polygons in confocal and bicentric arrangements. From a practical perspective, these invariants may facilitate new computational techniques in computer graphics and CAD systems, where precise geometric constructs are crucial.

Speculatively, these findings could pave the way for further exploration into the integrable systems theory, enhancing the algorithms modeling complex dynamical systems, translating into real-world applications in physics, engineering, and beyond.

Future Directions

The results presented incite several avenues for further investigation:

• Extension to Higher Dimensions: Could the invariants found in planar configurations extend to solider geometries involving spheres and other higher-dimensional analogs?
• Applications to Computational Algorithms: How might these invariants be applied or extend computational algorithms that rely on geometric precision and configurations?
• Further Exploration of Elliptic Billiards: While this paper complements the extensive research on elliptic billiards and N-periodics, probing the linked dynamic properties through algebraic and numeric simulations could develop a richer theoretical framework.

In conclusion, this paper contributes substantially to the field of geometric properties of bicentric N-gons, revealing invariants that align with previous findings in elliptic systems and opening promising pathways for further research in advanced geometric analysis and application. The methodologies employed in dealing with these elliptic constructs can provide novel insights into the pursuit of universal geometric truths.