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've built a web-based tool for the real-time interaction with loci of Poncelet triangle families. Our initial goals were to facilitate exploratory detection of geometric properties of such families. During frequent walks in my neighborhood, it appeared to me Poncelet loci shared a palette of motifs with those found in wrought iron gates at the entrance of many a residential building. As a result, I started to look at Poncelet loci aesthetically, a kind of generative art. Features were gradually added to the tool with the sole purpose of beautifying the output. Hundreds of interesting loci we…
Math.&Stats. Inst., Federal Univ. of Goiás（戈亚斯联邦大学数学与统计研究所）；Geom. Dept., Univ. Applied Arts, Vienna（维也纳应用艺术大学几何系）；Data Science Consulting Ltd.（数据科学咨询有限公司）
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.
ENS-Lyon（里昂高等师范学院）；Inst. Mat. e Estatística, Univ. Fed. de Goiás（戈亚斯联邦大学数学与统计研究所）；Data Science Consulting（数据科学咨询公司）
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.
We present a criterion when six points chosen on the sides of a triangle belong to the same conic. Using this tool we show how the two geometrical gems - celebrated Poncelet's theorem of projective geometry and incredible Morley's theorem of Euclidean geometry - can meet in a one construction.
Rice University（莱斯大学）；Enseeiht（国立高等电力技术、电子学、计算机、水力学与电信学院）；Federal Univ. of Goiás（戈亚斯联邦大学）；Data Science Consulting Ltd.（数据科学咨询有限公司）
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.
We study the loci of the Brocard points over two selected families of triangles: (i) 2 vertices fixed on a circumference and a third one which sweeps it, (ii) Poncelet 3-periodics in the homothetic ellipse pair. Loci obtained include circles, ellipses, and teardrop-like curves. We derive expressions for both curves and their areas. We also study the locus of the vertices of Brocard triangles over the homothetic and Brocard-poristic Poncelet families.
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.
We compare loci types and invariants across Poncelet families interscribed in three distinct concentric Ellipse pairs: (i) ellipse-incircle, (ii) circumcircle-inellipse, and (iii) homothetic. Their metric properties are mostly identical to those of 3 well-studied families: elliptic billiard (confocal pair), Chapple's poristic triangles, and the Brocard porism. We therefore organized them in three related groups.
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.
We describe all triangles that shares the same circumcircle and Euler circle. Although this two circles do not form a poristic pair of circles, we find a poristic circle "in-between" that enable to solve this problem using Poncelet porism.
Previously we showed the family of 3-periodics in the elliptic billiard (confocal pair) is the image under a variable similarity transform of poristic triangles (those with non-concentric, fixed incircle and circumcircle). Both families conserve the ratio of inradius to circumradius and therefore also the sum of cosines. This is consisten with the fact that a similarity preserves angles. Here we study two new Poncelet 3-periodic families also tied to each other via a variable similarity: (i) a first one interscribed in a pair of concentric, homothetic ellipses, and (ii) a second non-concentric…
We describe invariants of centers of ellipse-inscribed triangle families with two vertices fixed to the ellipse boundary and a third one which sweeps it. We prove that: (i) if a triangle center is a fixed affine combination of barycenter and orthocenter, its locus is an ellipse; (ii) and that over the family of said affine combinations, the centers of said loci sweep a line; (iii) over the family of parallel fixed vertices, said loci rigidly translate along a second line. Additionally, we study invariants of the envelope of elliptic loci over combinations of two fixed vertices on the ellipse.
Inverting the vertices of elliptic billiard N-periodics with respect to a circle centered on one focus yields a new "focus-inversive" family inscribed in Pascal's Limaçon. The following are some of its surprising invariants: (i) perimeter, (ii) sum of cosines, and (iii) sum of distances from inversion center (the focus) to vertices. We prove these for the N=3 case, showing that this family (a) has a stationary Gergonne point, (b) is a 3-periodic family of a second, rigidly moving elliptic billiard, and (c) the loci of incenter, barycenter, circumcenter, orthocenter, nine-point center, and a gr…
