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Dan Reznik

Publications and source records attributed to Dan Reznik.

At least 37 records · Page 2Linked to original sources
arXiv2021

New Properties of Triangular Orbits in Elliptic Billiards

New invariants in the one-dimensional family of 3-periodic orbits in the elliptic billiard were introduced by the authors in "Can the Elliptic Billiard Still Surprise Us?" (2020), Math. Intelligencer, 42(1): 6--17, some of which were generalized to $N>3$. Invariants mentioned there included ratios of radii and/or areas, sum of angle cosines, and a special stationary circle. Here we present some of the proofs omitted there as well as a few new related facts.

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arXiv2021

Poncelet Triangles: a Theory for Locus Ellipticity

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.

math.MG↗
arXiv2021

Poncelet Plectra: Harmonious Curves in Cosine Space

It has been shown that the family of Poncelet N-gons in the confocal pair (elliptic billiard) conserves the sum of cosines of its internal angles. Curiously, this quantity is equal to the sum of cosines conserved by its affine image where the caustic is a circle. We show that furthermore, (i) when N=3, the cosine triples of both families sweep the same planar curve: an equilateral cubic resembling a plectrum (guitar pick). We also show that (ii) the family of triangles excentral to the confocal family conserves the same product of cosines as the one conserved by its affine image inscribed in a circle; and that (iii) cosine triples of both families sweep the same spherical curve. When the triple of log-cosines is considered, this curve becomes a planar, plectrum-shaped curve, rounder than the one swept by its parent confocal family.

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arXiv2021

Family Ties: Relating Poncelet 3-Periodics by their Properties

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.

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arXiv2021

New Invariants of Poncelet-Jacobi Bicentric Polygons

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).

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arXiv2021

The Talented Mr. Inversive Triangle in the Elliptic Billiard

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 great many other triangle centers are circles.

math.MG↗
arXiv2021

Invariant Center Power and Elliptic Loci of Poncelet Triangles

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.

math.MG↗
arXiv2021

Intriguing Invariants of Centers of Ellipse-Inscribed Triangles

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.

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arXiv2021

Average Elliptic Billiard Invariants with Spatial Integrals

We compare invariants of N-periodic trajectories in the elliptic billiard, classic and new, to their aperiodic counterparts via a spatial integrals evaluated over the boundary of the elliptic billiard. The integrand is weighed by a universal measure equal to the density of rays hitting a given boundary point. We find that aperiodic averages are smooth and monotonic on caustic eccentricity, and perfectly match N-periodic average invariants at the discrete caustic parameters which admit a given N-periodic family.

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arXiv2021

Poncelet Propellers: Invariant Total Blade Area

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.

math.MG↗
arXiv2021

Invariants of Self-Intersected N-Periodics in the Elliptic Billiard

We study self-intersected N-periodics in the elliptic billiard, describing new facts about their geometry (e.g., self-intersected 4-periodics have vertices concyclic with the foci). We also check if some invariants listed in "Eighty New Invariants of N-Periodics in the Elliptic Billiard" (2020), arXiv:2004.12497, remain invariant in the self-intersected case. Toward that end, we derive explicit expressions for many low-N simple and self-intersected cases. We identify two special cases (one simple, one self-intersected) where a quantity prescribed to be invariant is actually variable.

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arXiv2020

Eighty New Invariants of N-Periodics in the Elliptic Billiard

We introduce several-dozen experimentally-found invariants of Poncelet N-periodics in the confocal ellipse pair (Elliptic Billiard). Recall this family is fully defined by two integrals of motion (linear and angular momentum), so any "new" invariants are dependent upon them. Nevertheless, proving them may require sophisticated methods. We reference some two-dozen proofs already contributed. We hope this article will motivate contributions for those still lacking proof.

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arXiv2020

A Special Conic Associated with the Reuleaux Negative Pedal Curve

The Negative Pedal Curve of the Reuleaux Triangle w.r. to a point $M$ on its boundary consists of two elliptic arcs and a point $P_0$. Interestingly, the conic passing through the four arc endpoints and by $P_0$ has a remarkable property: one of its foci is $M$. We provide a synthetic proof based on Poncelet's polar duality and inversive techniques. Additional intriguing properties of Reuleaux negative pedal are proved using straightforward techniques.

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arXiv2020

An Infinite, Converging, Sequence of Brocard Porisms

The Brocard porism is a known 1d family of triangles inscribed in a circle and circumscribed about an ellipse. Remarkably, the Brocard angle is invariant and the Brocard points are stationary at the foci of the ellipse. In this paper we show that a certain derived triangle spawns off a second, smaller, Brocard porism so that repeating this calculation produces an infinite, converging sequence of porisms. We also show that this sequence is embedded in a continuous family of porisms.

math.MG↗
arXiv2020

Loci of the Brocard Points over Selected Triangle Families

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.

math.MG↗
arXiv2020

Area-Invariant Pedal-Like Curves Derived from the Ellipse

We study six pedal-like curves associated with the ellipse which are area-invariant for pedal points lying on one of two shapes: (i) a circle concentric with the ellipse, or (ii) the ellipse boundary itself. Case (i) is a corollary to properties of the Curvature Centroid (Krümmungs-Schwerpunkt) of a curve, proved by Steiner in 1825. For case (ii) we prove area invariance algebraically. Explicit expressions for all invariant areas are also provided.

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arXiv2020

Related by Similarity II: Poncelet 3-Periodics in the Homothetic Pair and the Brocard 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 one known as the Brocard porism: fixed circumcircle and Brocard inellipse. The Brocard points of this family are stationary at the foci of the inellipse. A key common invariant is the Brocard angle, and therefore the sum of cotangents. This raises an interesting question: given a non-concentric Poncelet family (limited or not to the outer conic being a circle), can a similar doppelgänger always be found interscribed in a concentric, axis-aligned ellipse and/or conic pair?

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