Ana C. Chavez-Caliz
We study the locus of the Circumcenter of Mass of Poncelet polygons, and the limit of the Center of Mass (when we consider the polygon as a "homogeneous lamina") for degenerate Poncelet polygons. We also provide a proof for one of Dan Reznik invariants for billiard trajectories. Plus, we take a look at how the scene looks like when we shift to spherical geometry.
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We study loci and properties of a Parabola-inscribed family of Poncelet polygons whose caustic is a focus-centered circle. This family is the polar image of a special case of the bicentric family with respect to its circumcircle. We describe closure conditions, curious loci, and new conserved quantities.
The locus of the centers of mass of the family of Poncelet polygons, inscribed into a conic $Γ$ and circumscribed about a conic $γ$, is a conic homothetic to $Γ$.
Explicit Constructions for Poncelet Polygons
本文从射影角度研究彭赛列n边形，给出特定n下的显式构造与括号多项式代数刻画，利用其与(N₄)-构型的关联，构造出此前被认为刚性的可动平凡天体4-构型。
An Affine Approach to Metric Invariants of Poncelet Polygons
本文推广Dan Reznik的几何观察结果，基于Marden定理的三角形对称参数化给出更强形式的证明，还推广了其奇数边形相关结论，并提出圆内接n边形外切于中心二次曲线时的正方形总面积不变性新猜想。
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)…
We study center power with respect to circles derived from Poncelet 3-periodics (triangles) in a generic pair of ellipses as well as loci of their triangle centers. We show that (i) for any concentric pair, the power of the center with respect to either circumcircle or Euler's circle is invariant, and (ii) if a triangle center of a 3-periodic in a generic nested pair is a fixed affine combination of barycenter and circumcenter, its locus over the family is an ellipse.
We explore properties and loci of a Poncelet family of polygons -- called here Steiner-Soddy -- whose vertices are centers of circles in the Steiner porism, including conserved quantities, loci, and its relationship to other Poncelet families.
Via simulation, we discover and prove curious new Euclidean properties and invariants of the Poncelet family of harmonic polygons.
Four special Poncelet triangle families about the incircle
本文研究了四个关于内切圆的特殊彭赛列三角形家族，探讨了其保持三角中心不变的性质及守恒量。
We analyze loci of triangle centers over variants of two-well known triangle porisms: the bicentric and confocal families. Specifically, we evoke the general version of Poncelet's closure theorem whereby individual sides can be made tangent to separate in-pencil caustics. We show that despite the more complicated dynamic geometry, the locus of certain triangle centers and associated points remain conics and/or circles.
Given a triangle, a trio of circumellipses can be defined, each centered on an excenter. Over the family of Poncelet 3-periodics (triangles) in a concentric ellipse pair (axis-aligned or not), the trio resembles a rotating propeller, where each "blade" has variable area. Amazingly, their total area is invariant, even when the ellipse pair is not axis-aligned. We also prove a closely-related invariant involving the sum of blade-to-excircle area ratios.
When Grünbaum meets Poncelet -- Infinite Classes of Movable $n_4$ Configurations
本文将Poncelet Porism的运动特性推广到$(n_4)$构型，证明所有平凡天体构型均可动，还给出了关联离散可积系统的替代研究路径。
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.
In this article we introduce a general definition of the concept of center of an $n$-gon, for $n\geq 3$, generalizing the idea of C. Kimberling for triangle. We define centers associated to functions instead of to geometrical properties. We discuss the definition of those functions depending on both, the vertices of the polygons or the lengths of sides and diagonals. We explore the problem of characterization of regular polygons in terms of these $n$-gon center functions and we study the relation between our general definition of center of a polygon and other approaches arising from Applied Ma…
There are several remarkable points, defined for polygons and multisets of points in the plane, called centers (such as the centroid). To make possible their study, there exists a formal definition for the concept of center in both cases. In this paper, the relation between symmetries of polygons and multisets of points in the plane and the coincidence and collinearity of their centers is studied. First, a precise statement for the problem is given. Then, it is proved that, given a polygon or a multiset of points in the plane, a given point in the plane is a center for this object if and only …
In this paper, an efficient algorithm to find the center of the biggest circle inscribed in a given polygon is described. This work was inspired by the publication of Daniel Garcia-Castellanos & Umberto Lombardo and their algorithm used to find a landmass' poles of inaccessibility. Two more efficient algorithms were found, one of them only applicable when the problem can be described as a linear problem, like in the case of a convex polygon. Keywords: distance geometry, euclidean distance, inscribed circle, irregular polygon, algorithm, mathematical optimization, Monte Carlo, linear programmin…
The excenters of bicentric polygons are concyclic
本文证明双心多边形的所有旁心共圆，并给出该圆圆心与外接圆、内切圆圆心之间的中点关系，以及用内外圆半径和圆心距表示的旁心圆半径公式。
We define and study a variant of the center of mass of a polygon and, more generally, of a simplicial polytope which we call the Circumcenter of Mass (CCM). The Circumcenter of Mass is an affine combination of the circumcenters of the simplices in a triangulation of a polytope, weighted by their volumes. For an inscribed polytope, CCM coincides with the circumcenter. Our motivation comes from the study of completely integrable discrete dynamical systems, where the CCM is an invariant of the discrete bicycle (Darboux) transformation and of recuttings of polygons. We show that the CCM satisfies …
Maximum-Area Small Polygons of Even Order
该研究解决了直径不超过1的最大面积小多边形的缺失偶数阶问题，证明n≥8时最大面积小n边形的构型，结合已有结果得到n≥3时所有情况的精确解。
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