Dan Reznik

Data Science Consulting

A triangle center such as the incenter, barycenter, etc., is specified by a function thrice- and cyclically applied on sidelengths and/or angles. Consider the 1d family of 3-periodics in the elliptic billiard, and the loci of its triangle centers. Some will sweep ellipses, and others higher-degree algebraic curves. We propose two rigorous methods t...

Loci of Poncelet triangle families are explored with a bespoke web-based, interactive tool. Hundreds of beautiful loci have been compiled into a web “gallery”. Some loci aesthetically resemble, or seem to have a common motif with, wrought iron gate designs of the early 20th century. Are Poncelet loci attractive and surprising enough to inspire new...

We describe some three-dozen curious phenomena manifested by parabolas inscribed or circumscribed about certain Poncelet triangle families. Despite their pirouetting motion, parabolas' focus, vertex, directrix, etc., will often sweep or envelop rather elementary loci such as lines, circles, or points. Most phenomena are unproven though supported by...

We give closed-form expressions for recently discovered invariants of Poncelet N-periodic trajectories in the elliptic billiard with spatial integrals evaluated over the boundary of the elliptic billiard. The integrand is weighed by a measure equal to the density of rays hitting a given boundary point. We find that aperiodic averages are smooth and...

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. KeywordsTriangleInversivePedal polygonPorismPonceletInvariant

We study the two-parameter manifold of parabolas circumscribed to triangles in a Poncelet porism between two circles (Chapple’s porism). It turns out that the focal points of the parabolas in a certain one-parameter subfamily trace a straight line. The vertices of these parabolas move on rational cubic curves whose acnodes trace an ellipse centered...

The two-parameter family of circum parabolas of triangles moving through the poristic family of Chapple's porism are described

We study the two-parameter manifold of parabolas circum-scribed to triangles in a Poncelet porism between two circles (Chapple's porism). It turns out that the focal points of the parabolas in a certain one-parameter subfamily trace a straight line. The vertices of these parabolas move on rational cubic curves whose acnodes trace an ellipse centere...

The Brocard porism is a 1d family of Poncelet 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 triangle derived from the family spawns off a second, smaller, Brocard porism so that repe...

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 cert...

Using experimental techniques, we study properties of the "circumcenter map'', which, upon $n$ iterations sends an $n$-gon to a scaled and rotated copy of itself. We also explore the topology of area-expanding and area-contracting regions induced by this map.

We explore geometric and topological properties of 3d surfaces swept by Poncelet triangles, as well as tangles formed by associated points.

We've built a web-based tool for the real-time interaction with loci of Poncelet triangle families which has been enhanced with tools to beautify and share the output. We've been inspired by the fact that some loci have features found in wrought iron gate designs. Hundreds of interesting loci have been collected into an online ``gallery'', with som...

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 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 confoca...

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 the blades of a rotating propeller. Though the area of each blade is variable, their total area is invariant, even when the ellipse pair i...

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.

Podsjećamo na konstrukcije temeljene na trijadama konika sa žarištima u parovima vrhova referetnog trokuta. Nalazimo da njihovih 6 vrhova leži na dobro poznatim konikama čiji tip analiziramo. Za ove konike dajemo uvjete da budu kružnice i/ili degenerirane konike. U slučaju degeneriranih konika proučavamo geometrijsko mjesto njihovog središta.

We revisit constructions based on triads of conics with foci at pairs of vertices of a reference triangle. We find that their 6 vertices lie on well-known conics, whose type we analyze. We give conditions for these to be circles and/or degenerate. In the latter case, we study the locus of their center.

The usual Poncelet porisms deal with polygons which are inscribed into one conic and circumscribed to another conic. A more general form of Poncelet porisms considers polygons whose sides are tangent to more than one conic of a pencil of conics. We shall study the case of poristic triangles inscribed into a circle c_1 with sides tangent to two furt...

Via simulation, we discover and prove curious new Euclidean properties and invariants of the Poncelet family of harmonic polygons.

If one erects regular hexagons upon the sides of a triangle $T$, several surprising properties emerge, including: (i) the triangles which flank said hexagons have an isodynamic point common with $T$, (ii) the construction can be extended iteratively, forming an infinite grid of regular hexagons and flank triangles, (iii) a web of confocal parabolas...

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...

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.

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 spe...

The Cramer-Castillon problem (CCP) consists in finding one or more polygons inscribed in a circle such that their sides pass cyclically through a list of N points. We study this problem where the points are the vertices of a triangle and the circle is either the incircle or excircles.

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 th...

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 (...

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 cert...

We describe intriguing properties of a 1d family of triangles: two vertices are pinned to the boundary of an ellipse while a third one 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) over the family of said affine combinations, the centers of said loci swe...

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...

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 of properties of Steiner’s Curvature Centroid (Krümmungs-Schwerpunkt), proved in 1825. For case (ii) we prove area inv...

We describe a browser-based application built for the real-time visualization of the beauteous dynamic geometry of Poncelet 3-periodic families. The focus is on highly responsive, visually smooth, "live" experimentation with old and new phenomena involving loci of triangle centers and/or metric invariants. Another focus is on the production of beau...

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 confoc...

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...

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 (...

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 smo...

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...

We introduce 50+ new invariants manifested by the dynamic geometry of N-periodics in the Elliptic Billiard, detected with an experimental/interactive toolbox. These involve sums, products and ratios of distances, areas, angles, etc. Though curious in their manifestation, said invariants do all depend upon the two fundamental conserved quantities in...

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 the blased of a rotating propeller. Though each blade area is variable, their total area is invariant, even when the ellipse pair is not a...

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 Bro...

We study loci, and invariants of three Poncelet families associated with three distinct concentric Ellipse pairs: with-incircle, with-circumcircle, and homothetic. Most of their properties run parallel to those of 3 well-studied families: elliptic billiard (confocal pair), Chapple's poristic triangles, and the Brocard porism, allowing us to organiz...

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...

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-inter...

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 linear combination of barycenter and orthocenter, its locus is an ellipse; (ii) and that over the family of said linear combinations, the centers...

The Brocard porism is a 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 yi...

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 th...

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 fixed incircle and circumcircle, therefore non-concentric). In particular both systems conserved the ratio r/R and therefore the sum of cosines. Here we identify two new Poncele...

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) w...

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 Interestingly, the conic passing through the four arc endpoints and by P 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...

The Negative Pedal Curve (NPC) of the Ellipse with respect to a boundary point M is a 3-cusp closed-curve which is the affine image of the Steiner Deltoid. Over all M the family has invariant area and displays an array of interesting properties.

Discovered by William Chapple in 1746, the Poristic family is a set of variable-perimeter triangles with common Incircle and Circumcircle. By definition, the family has constant Inradius-to-Circumradius ratio. Interestingly, this invariance also holds for the family of 3-periodics in the Elliptic Billiard, though here Inradius and Circumradius are...

We introduce several-dozen experimentally-found invariants of Poncelet N-periodics in the elliptic billiard. Though these involve simple functions of angles, lengths, and areas of N-periodics and/or derived polygons, proofs already contributed (cited herein) have resorted to sophisticated algebraic and differential geometry. We do hope this article...

A Circumconic passes through a triangle's vertices. We define the Circumbilliard, a circumellipse to a generic triangle for which the latter is a 3-periodic. We study its properties and associated loci.

A Circumconic passes through a triangle's vertices; an Inconic is tangent to the sidelines. We study the variable geometry of certain conics derived from the 1d family of 3-periodics in the Elliptic Billiard. Some display intriguing invariances such as aspect ratio and pairwise ratio of focal lengths. We also define the Circumbilliard, a circumelli...

The dynamic geometry of the family of 3-periodics in the Elliptic Billiard is mystifying. Besides conserving perimeter and the ratio of inradius-to-circumradius, it has a stationary point. Furthermore, its triangle centers sweep out incredible loci including ellipses, quartics, circles, and a slew of other mesmerizing curves. Here we explore a bevy...

Can any secrets still be shed by that much studied, uniquely integrable, Elliptic Billiard? Starting by examining the family of 3-periodic trajectories and the loci of their Triangle Centers, one obtains a beautiful and variegated gallery of curves: ellipses, quartics, sextics, circles, and even a stationary point. Secondly, one notices this family...

A triangle center such as the incenter, barycenter, etc., is specified by a function thrice- and cyclically applied on sidelengths and/or angles. Consider the 1d family of 3-periodics in the elliptic billiard, and the loci of its triangle centers. Some will sweep ellipses, and others higher-degree algebraic curves. We propose two rigorous methods t...

In this paper we present invariants of the 1d family of 3-periodics (triangular orbits) in an Elliptic Billiard (EB), previously mentioned [14], including: (i) explicit Expressions classic invariants: perimeter and Joachimsthal's quantity; (ii) consevation of Inradius-to-Circumradius; (iii) stationarity of the Cosine Circle of the Excentral Triangl...

We analyze the family of 3-periodic trajectories in an El-liptic Billiard. Taken as a continuum of rotating triangles, its Triangle Centers (Incenter, Barycenter, etc.) sweep remarkable loci: ellipses, circles , quartics, sextics, and even a stationary point. Here we present a systematic method to prove 29 out of the first 100 Centers listed in Cla...

The Negative Pedal Curve (NPC) of the Ellipse with respect to a boundary point M is a 3-cusp closed-curve which is the affine image of the Steiner Deltoid. Over all M the family has invariant area and displays an array of interesting properties.