Poncelet Triangles: a Theory for Locus Ellipticity (2106.00715v4)

Abstract: We present a theory which predicts if the locus of a triangle center over certain Poncelet triangle families is a conic or not. We consider families interscribed in (i) the confocal pair and (ii) an outer ellipse and an inner concentric circular caustic. Previously, determining if a locus was a conic was done on a case-by-case basis. In the confocal case, we also derive conditions under which a locus degenerates to a segment or a circle. We show the locus' turning number is +/- 3, while predicting its monotonicity with respect to the motion of a vertex of the triangle family.

• The paper establishes predictive criteria showing that triangle center loci form conics through specific affine combinations.
• The methodology leverages Blaschke's parametrization to analyze turning numbers and degeneracies in Poncelet triangle families.
• The findings impact computational geometry and set a foundation for further exploration of generalized Poncelet configurations.

Poncelet Triangles: A Theory for Locus Ellipticity

This paper introduces a comprehensive theory aiming to predict the ellipticity of loci traced by the centers of triangles within specific Poncelet triangle families. The focus lies on two main types of interscribed geometries: triangles inscribed in (i) an ellipse-circular caustic pair, and (ii) confocal ellipses, often referred to as elliptic billiards. Determination of locus ellipticity has traditionally required a case-by-case analysis, but this research advances a systematic approach, particularly addressing families described by the Poncelet closure theorem.

The work is grounded in solid mathematical foundations, leveraging Blaschke's parametrization—a technique where triangle parameters are connected through complex roots of symmetric polynomials. The paper postulates conditions under which the locus of a triangle center degenerates into a simpler geometric structure like a segment or a circle.

Main Contributions

1. Ellipticity Prediction: The authors establish criteria for when the locus of a triangle center is a conic, extending from the specific case of the incenter to a broader array of triangle centers. Predictive conditions are linked to whether the center can be expressed as a fixed affine combination of known geometrical points, namely the barycenter, circumcenter, and any stationary center within the triangle family, such as the mittenpunkt in confocal cases.
2. Locus Characteristics: The research delineates parameters affecting the shape and nature of loci. It demonstrates that the loci of certain centers, under specified conditions, trace out segments or circles—providing explicit ratios of affine combinations resulting in degenerate (segment-like or circular) loci.
3. Turning Number: The paper calculates that the winding number of such loci is consistently ±3, aligning with the triangle’s full sweep motion over its containing ellipse—brought forth by the monotonic nature of Blaschke parameterization, drawing on Lemma 3.4 from the literature.
4. Stationary Centers: Beyond the geometric assessment of loci, the paper identifies certain triangle centers that remain stationary across various families of Poncelet triangles, facilitating a predictable framework for locus behavior.

Implications and Future Directions

The implications of this research are significant in the domain of theoretical geometry, offering a unifying approach to understanding and predicting the behavior of triangle center loci. The theoretical model exhibits robustness across variations of the Poncelet problem, expanding previous findings constrained to specific geometric configurations.

From a practical standpoint, the insights can potentially influence computational geometry, particularly in algorithms where the determination of locus shape plays a role. The proposed conjecture concerning the locus of the incenter being a non-degenerate conic solely when the pair is confocal invites further mathematical exploration and verification, possibly leading to refined geometric classification theories.

Furthermore, the research lays the groundwork for exploring similar properties in non-concentric or non-axis-aligned configurations, as hinted by unsolved questions and speculative areas for research extension. The applicability of the theory to related geometric constructs such as bicentric families or the Brocard porism stands as an avenue for probing the generalizability and limits of these geometric principles.

In conclusion, this paper provides a rich theoretical toolkit for researchers interested in the intersection of dynamic geometry and classical theorems, presenting new avenues for exploration and computational verification within the field of mathematical sciences.