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Triangles and Ellipses: Pages and Media
Dan Reznik
Last update: 2021-02-20 12:30:53
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elliptic_billiards_v2.bib
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I’m going to assume that you love beautiful things and are curious to learn about them. The only things you will need on this journey are common sense and simple human curiosity. --Paul Lockhardt, "A mathematician’s lament" (2009)

Below we provide links to some of the artifacts (webpages, images, videos, applets, code) from ongoing experiments with triangles, billiards, ellipses, and other marvellous geometric objects. The fun and amazement with their beauty truly never ends.

Our Main Pages

  1. Invariants of 3- and N-Periodics in an Elliptic Billiard, with R. Garcia and J. Koiller.

  2. Loci of N-Periodics and their Derived and Internal Triangles

  3. Envelopes and Evolutes: beautiful curves enveloped by pairs $X_i,X_j$ of Triangle Centers.

  4. new Loci of Centers of Ellipse-Mounted Triangles: astounding loci generated by triangle centers when a vertex slides along and ellipse and the other two vertices are held fixed. With M. Helman.

  5. new Artsy Loci of Ellipse-Mounted Triangles: long, short

  6. new Extremal-Area Pedal and Antipedal Triangles: a discussion on maximal and minimal pedals and antipedals to a triangle. With M. Helman.

  7. new Loci of 3-periodics in the Elliptic Billiard: Why so many ellipses?: semi-axes and fit errors for 29 triangles centers (out of the first 100 in [@etc]) whose loci under 3-periodics in the elliptic billiard are ellipses. With R. Garcia and J. Koiller.

Videos N=3 {.tabset .tabset-fade .tabset-pills}

All videos are available as a single playlist.

Original 2011 {-}

Title sound Kimberling Centers Year urls
3-Periodic trajectories -- 2011 v1, v2, v3
Locus of incenter is elliptic for family of 3-periodics 1 2011 v1
Locus of the incircle touchpoints is a higher-order curve 1 2011 v1

Early Results {-}

Title sound Kimberling Centers Year urls
Elliptic Loci of X(1) to X(5) and Euler Line 1,2,3,4,5 2019 v1
Loci of Vertices of Medial, Intouch and Feuerbach Triangles is not elliptic T 1,5 2019 v1
Mittenpunkt is stationary at center of billiard 9 2019 v1, v2
Feuerbach Point Sweeps Billiard and its Anti-Complement and Extouch Points sweep caustic T 1,2,5,11,100 2019 v1
Conservation of Sum and Product of Cosines T -- 2019 v1

N=3 Loci {-}

Title sound Kimberling Centers Year urls
3-Periodics and Derived Triangles -- 2019 v1
Elliptic Loci of X(1) to X(5) and Euler Line 1,2,3,4,5 2019 v1
Locus of several triangular centers is elliptic 1,2,3,4,5 2019 v1
Locus of vertices of Feuerbach Triangle is non-elliptic -- 2019 v1
Non-Elliptic loci of vertices of Medial, Intouch and Feuerbach triangles T 1,5,11 2019 v1
Locus of Bevan Point X(40) is similar to billiard 40 2019 v1
Locus of Bevan Point X(40) identical to billiard when a/b=golden ratio 40 2020 v1
Locus of X(59) has 4 self-intersections 59 2020 v1
Anticomplementary triangle intouchpoints 1,2,7,9,100,144 2019 v1, v2
Anticomplementary, Medial Triangles and the Intouch Triangle 1,2,7,9,11,100,142,144 2019 v1, v2
The locus of X(140) is a circle over 3-periodics in the Elliptic Billiard T 3,5,140 2020 v1
Circular loci for X(140) and X(547) over 3-periodics in the Elliptic Billiard T 3,2,5,40,547 2020 v1

Feuerbach {-}

Title sound Kimberling Centers Year urls
Locus of Feuerbach point, its anticomplement and three extouchpoints T 1,2,5,11,100 2019 v1, v2, v3
Locus of Excentral and Anticomplementary Triangles and Objects 1,2,5,11,100 2019 v1, v2

Circumconics {-}

Title sound Kimberling Centers Year urls
The X(1)- and X(2)-centered circumellipses 1,2,9 2019 v1, v2
Locus of Intersection of X(1)- and X(2)-centered circumellipses 1,2,75,77,100,190 2019 v1
The X(100)-centered Excentral Jerabek Hyperbola 1,9,100 2019 v1
The Feuerbach and Excentral Hyperbolas 1,4,7,8,9,11,40, 100,144,1156 2019 v1, v2
The Jerabek Hyperbola and Circumbilliard of the Excentral Triangle 1,9,40,100, 168,1156 2019 v1
Peter Moses' Points on the X(9)-centered circumellipse 88, 100, 162, 190, 651, 653, 655, 658, 660, 662, 673, 771, 799, 823, 897, 1156, 1492, 1821, 2349, 2580, 2581, 3257, 4598, 4599, 4604, 4606, 4607, 8052, 20332, 23707, 24624, 27834, 32680 2019 v1, v2
Invariants of the X(1)-centered circumellipse 1,3,100 2019 v1
Invariants of the Steiner Circum and Inconics 2,190 2019 v1
The Yff Parabola, Contact Triangle & Loci of Vertex and Axis Foot 4,9,101,190,3234 2020 v1, v2
Every triangle has a unique Circumbilliard 9 2019 v1
Circumbilliard of anticomplementary triangle 1,2,5,7,11,100 2019 v1
Orthic Circumbilliard & Locus of Its Mittenpunkt 4,6,9 2020 v1
Circumbilliards of Triangles Derived from 3-Periodics 7,8,142,168 2020 v1
Feuerbach and Excentral Jerabek Circumhyperbolas: Invariant Focal Length Ratio T 1,4,9,11,40,100,1156 2020 v1, v2

Inconics {-}

Title sound Kimberling Centers Year urls
Excentral MacBeath Inconic: Invariant Aspect Ratio T 1,3,40,1742 2020 v1
X(3)-Centered Excentral Inconic: Invariant Aspect Ratio T 3,40,69,2951 2020 v1

Convex Combinations {-}

Title sound Kimberling Centers Year urls
Barycenter with Median, and Incenter with Intouchpoint 1,2 2019 v1
Orthocenter with one altitude foot, and Circumcenter with median 3,4 2019 v1
Excenter and its corresponding Extouch point -- 2019 v1

Orthic Phenomena {-}

Title sound Kimberling Centers Year urls
Locus orthic triangle's incenter is a 4-arc ellipse 4 2019 v1
Locus of orthocenter, orthic orthocenter, incenter, and orthic orthic's incenter 1,4 2019 v1
Excentral of Orthic for Acute and Obtuse Triangles 4 2019 v1

Stationary Circles {-}

Title sound Kimberling Centers Year urls
Cosine Circle of Excentral Triangle is Stationary 9,40 2019 v1
Locus of Intersection of Anti-Tangents is Stationary Circle 9 2019 v1
Intersections of Excentral Triangle and its Reflection is a Circle 9 2019 v1

Unrolled 3-Periodics {-}

Title sound Kimberling Centers Year urls
Fixed central billiard -- 2020 v1
Pin P1 and n1 -- 2020 v1
Pin P1 at origin -- 2020 v1
Pin P_1 at origin and P1'' vertically above it -- 2020 v1

Poncelet Family {-}

Title sound Kimberling Centers Year urls
Poncelet Triangle Inscribed in Ellipse and Circumscribed in Circle 1,2,3,4,5,11,100 2019 v1
Poncelet Family of Triangles over the Family of N=3 Caustics 1,2,3,4,5,11,9,100 2019 v1
Pencil of N=3 Poncelet Ellipse Pairs: Loci of Triangular Centers 1,2,3,4,5,6,7,8,9,11,100 2019 v1
Three Geometers Walk into a Bar: the 3-periodic Poncelet-Steiner family has invariant Brocard angle. T 2 2020 v1

Poristic {-}

Title sound Kimberling Centers Year urls
Chapple's Porism from (1746) and Weaver (1927) and Odehnal (2011) Invariants T 1,3,40,1155 2020 v1
Circumbilliard of the Poristic Triangle Family: Invariant Aspect Ratio T 1,3,9,40 2020 v1
Poristic Triangle Family and the Amazing Invariant Excentral X3-Centered Inconic T 1,3,9,40,100 2020 v1
Simson Lines from X100 and Excentral Medials are Parallel to L(X1,X3). T 1,3,40,100 2020 v1
X1-Centered Circumconic & X40-Centered (Excentral) Inconic: Identical Invariant Axes T 1,3,40,100 2020 v1, v2
Loci of center and foci of the Circumbilliard to the Poristic Family are circles. T 1,3,9,40,100 2020 v1
Aspect Ratios of X10- and Excentral X5-Centered Circumconics are Invariant & Equal T 1,3,10,40,100 2020 v1
Invariant aspect ratios for the Circumbilliard and Excentral X6-Ctr Circumconic T 1,3,9,40,100 2020 v1
Side-by-Side View of Poristic and 3-Periodic Families T 1,3,9,40,100 2020 v1
Feuerbach and Excentral Jerabek Hyperbolas to Poristic Family have invariant focal length ratio 1,3,9,11,100 2020 v1
Reference & Excentral Simson Lines have fixed points and are Orthogonal! T 1,3,40 2020 v1, v2

Misc {-}

Title sound Kimberling Centers Year urls
Loci of Outer Napoleon Equilateral Construction 13 2019 v1, v2
Conservation of Sum and Product of Cosines -- 2019 v1
The Miquel Point of the Extouch and Excentral Triangles 40 2019 v1, v2
An invariant in the parabolic pair associated with the N=3 family -- 2020 v1
Non-monotonic X(88) and the X(1)-X(100) envelope 1,88,100 2020 v1
The Thomson Cubic of 3-periodics 1,2,3,4,6,9 2020 v1
Locus and elliptic envelope of excircle tangents' hexagon (side touchpoints) T -- 2020 v1
Six-Point Conic passes through Sideline Tangents to Excircles T -- 2020 v1
Elliptic Billiard 3-Periodics: Invariants of the Focal Hyperbola T -- 2020 v1

Isogonal and Isotomic {-}

Title sound Kimberling Centers Year urls
Antiorthic Axis and 5 points on the Billiard 1,6,9,44,88,100 2019 v1
Isotomic and Isogonal Conjugates of Billiard with respect to the 3-periodic family 1,9,144 2019 v1

Envelopes {-}

Title sound Kimberling Centers Year urls
Envelope of Antiorthic and Gergonne Lines 9,44,1155,857,908 2020 v1
Evolute of Elliptic Billiard and Envelope of X(1)-X(5) 1,5,4,9, 2020 v1
Envelope of 3-Periodic Vertex with Triangle Center 1,2,3,4,5,6,7,8,10,11,12,20 2020 v1
Evolute Triangles of P1(t) with X(i) 1,3,5,20 2020 v1
Elliptic Envelope of P1(t) with P1(t+pi/2) -- 2020 v1
Envelope of 3-Periodic P1 and reflected P2 is Elliptic T -- 2020 v1
The Bat-Envelope of X(48) and X(37143) T 48,37143 2020 v1
Envelopes of Sides of Derived Triangles T -- 2020 v1
Envelope of Simson Lines from X100 and X99 to two N=3 Poncelet Families T 2,3,9,99,100 2020 v1

Swans {-}

Title sound Kimberling Centers Year urls
Motion of X(88) with respect to collinear X(100) and X(1) 1,88,100 2020 v1
Dance of the Swans: X(88) and X(162) T 9,88,162 2020 v1, v2

Pedal Invariants {-}

Title sound Kimberling Centers Year urls
Area Invariants of Pedal and Antipedal Polygons T -- 2020 v1

Ellipse-Mounted {-}

Title sound Kimberling Centers Year urls
Ellipse-Mounted Triangles: Elliptic locus of the Orthocenter X(4) and suprising area invariance! T 4 2020 v1
Circle-Mounted Triangles: Surprising Loci of the Brocard Points T -- 2020 v1

Poncelet {-}

Title sound Kimberling Centers Year urls
An N=3 Poncelet family (outer circle, inner ellipse) equivalent to Poristic Excentrals T 3,4,5 2020 v1
Between an Ellipse and a Concentric Circle: Poncelet 3-Periodics Identical to Poristic Triangles. T 1,3 2020 v1
Between a Circle and a Concentric Ellipse: Poncelet 3-Periodics Identical to Poristic Excentrals. T 3,4,5 2020 v1
3-Periodics in a Concentric Homothetic Poncelet Pair: Circular Loci of four Triangle Centers T 2,13,14,15,16 2020 v1
3-Periodics in a Homothetic-Rotated Poncelet Pair: stationary orthocenter and loci of X107 and X122 T 4,107,122 2020 v1
Poncelet 3-Periodic Invariants (Outer Circle, Inner Concentric Ellipse) of the Nine-Point Center II T 3,5,4 2020 v1
Isodynamic Pedals and Isogonic Antipedals: Equilaterals with Constant Area in the Homothetic Pair T 2,13,14,15,16,395,396,5463,5464 2020 v1

Brocard {-}

Title sound Kimberling Centers Year urls
Poncelet 3-Periodics of Homothetic Pair: Elliptic Loci of Brocard Pts + Vertices of 1st Brocard Tri T 2 2020 v1
It takes 2 to tango: Brocard-Poncelet Porism, stationary Brocard Points and invariant Brocard Angle T 3,6,39,182 2020 v1
Joined at the hip: Brocard Porism, Steiner Ellipses, and the Homothetic Poncelet Pair T 2,3,6,182 2020 v1
The Poncelet Homothetic Pair contains an Aspect-Ratio Invariant Brocard Inellipse T 2,39 2020 v1
Brocard Porism: Locus of 1st, 2nd, 5th, and 7th Brocard Triangles' Vertices are Circles T 3,6,39,182,9821 2020 v1
Russian-Doll nesting of Brocard porisms courtesy of the second Brocard triangle T 6,39,182 2020 v1
Rusian-doll nesting of Brocard porisms: concyclic sequence of Brocard points and the Beltrami points T 6,39,182 2020 v1
Brocard Porism: equilateral Isodynamic Pedals have invariant area ratio + circular centroidal locus T 3,6,15,16,39,182 2020 v1
Continuous Family of Brocard Porisms with Stationary Isodynamic Points X15 and X16 T 3,6,15,182 2020 v1
The Family of Second Brocard Triangles in the Brocard Porism T 3,6,15,39,182 2020 v1
Brocard Porism: Family of Second Brocard Triangles is a second Brocard Porism T 3,6,15,39,182,39498 2020 v1

Inversive Poncelet {-}

Title sound Kimberling Centers Year urls
Focus-Inversive N=3 Family in the Elliptic Billiard: Pascal Limaçon-Inscribed Billiard Triangles! T 9,7 2020 v1, v2
Focus-Inversive Polygons' Equi-Area Pedal Polygons (wrt foci) T -- 2020 v1

Ellipse-Inscribed Triangles {-}

Title sound Kimberling Centers Year urls
Loci of Ellipse-Inscribed Triangles I: Basic Phenomena T 2,3,4,5,381 2020 v1
Loci of Ellipse-Inscribed Triangles II: X() slides merrily along the Euler line T 2,4 2020 v1
Loci of Ellipse-Inscribed Triangles III: family of V1V2 parallels causes rigid locus translation T -- 2020 v1
Loci of Ellipse-Inscribed Triangles IV: Multiple Loci Over Parallel V1V2 T -- 2020 v1
Loci of Ellipse-Inscribed Triangles V: Circular Loci if V1V2 Horizontal or Vertical for Certain T -- 2020 v1
Ellipse-Inscribed Triangles VI: Envelope of X4 Loci is Area-Invariant and Cousin of Pascal's Limaçon T -- 2020 v1

Inversive N=3 {-}

Title sound Kimberling Centers Year urls
Circles Galore I: Loci of Focus-Inversive 3-Periodics in the Elliptic Billiard (11 notable centers) T 1, 2, 3, 4, 5, 9, 10, 11, 20, 40, 100 2020 v1
Circles Galore II: 29 Loci of Focus-Inversive 3-Periodics in the Elliptic Billiard T 1, 2, 3, 4, 5, 8, 9, 10, 11, 12, 20, 21, 35, 36, 40, 46, 55, 56, 57, 63, 65, 72, 78, 79, 80, 84, 90, 100 2020 v1
Circles Galore III: Loci of Focus-Inversive 3-Periodics in the Elliptic Billiard (9 notable centers) T 1,2,3,4,5,8,9,10,11 2020 v1

Locus App {-}

Title sound Kimberling Centers Year urls
Loci of Ellipse-Inscribed Triangles: Part 01 - Intro to the App T -- 2020 v1
Loci of Ellipse-Inscribed Triangles: Part 02 - The Homothetic Family T -- 2020 v1
Loci of Ellipse-Inscribed Triangles: Part 03 - Derived Triangles T -- 2020 v1
Loci of Ellipse-Inscribed Triangles: Part 04 - Locus Type T -- 2020 v1

Inverse Curvature {-}

Title sound Kimberling Centers Year urls
Algebraic Miracle: The Zero-Area N=3 Homothetic Poncelet Inverse-Curvature Polygon (ICP) T -- 2020 v1

Hyperbolic Billiard {-}

Title sound Kimberling Centers Year urls
3-Periodics in a Hyperbolic Billiard T -- 2020 v1

Videos N≥3 {.tabset .tabset-fade .tabset-pills}

Early Results {-}

Title sound N Year urls
Stationary Circle for N=5 T 5 2019 v1
Stationary Circles 3 to 8 2019 v1
Generalization of the Stationary Mittenpunkt and Caustic-Sweeping Extouchpoints 4,5 2019 v1

Inconics {-}

Title sound N Year urls
Excentral X(3)-Centered & MacBeath Inconics: Invariant Aspect Ratio T NULL 2020 v1

N>3 Periodics {-}

Title sound N Year urls
4-periodics and Monge's Orthoptic Circle 4 2019 v1, v2
4-periodics: Loci of Triangle Centers for Vertex Triad 4 2019 v1
5-periodics: locus of P1,P2,P3 triangle 5 2019 v1
5-periodics: locus of P1,P2,P4 triangle 5 2019 v1
5-periodics: Loci of Subtriangles (123 and 124) 5 2019 v1
Upright 5-periodic family 5 2019 v1
6-periodic family 6 2019 v1
Octagramma Mysticum 8 2019 v1, v2
Enagramma Mysticum: loci of side intersections 9 2019 v1
Mittenpunkt-like Construction of a Stationary Point 4 to 7 2019 v1
Generalized Mittenpunkt and On-Caustic Extouchpoints 4,5 2019 v1
Family of Orbits and Their Caustics 3 to 6 2019 v1
Ellipse-Inscribed Parallelogram: invariants of the Pedal Polygon w/ respect to boundary points T 4 2020 v1, v2
Elliptic Billiard with Perpendicular Reflection Rule T n/a 2020 v1
Invariant Area Ratios to Minimum-Area Steiner Pedal Polygons T 5 2020 v1
Circumcircles of Focus with Consecutive Vertices Homothetic to Focus Antipedal T 5,6 2020 v1
Incenters of Focus Triads: Invariant Area Ratio to N-Periodic and Elliptic Locus T 5 2020 v1
An Invariant Based on Inradii and Circumradii of Subtriangles in the Elliptic Billiard T 5 2020 v1

Self-Intersected {-}

Title sound N Year urls
Self-intersecting 4-periodics (bowtie and tangential polygon) 4 2020 v1, v2
Self-intersecting 5-periodics (pentagram) 5 2019 v1
Self-intersecting 5-periodics (pentagram): Locus of Internal Intersections 5 2019 v1
Elliptic Billiard: Self-Intersected 6-Periodics (type I) T 6 2020 v1
Elliptic Billiard: Self-Intersected 6-Periodics (type II) T 6 2020 v1
Self-Intersected 6-Periodics in the Elliptic Billiard: Invariant Perimeter Focus-Inversive Polygon T 6 2020 v1
Elliptic Billiard 8-Periodics: Null sum of double cosines of outer polygon 8 2020 v1
Family of Self-Intersecting 4-Periodics in the Elliptic Billiard: Inversive Polygon is a Segment T 4 2020 v1
Type II Self-Intersected 8-Periodics in the Elliptic Billiard + Outer & Inversive Polygons T 8 2020 v1
Type I Self-Intersected 8-Periodics in the Elliptic Billiard and the Inversive Polygon T 8 2020 v1
Elliptic Billiard Self-Intersected 7-Periodics, a/b=2: Invariant Perimeter Focus-Inversive Polygons T 7 2020 v1
Family of self-intersected N=8 w/ turning number 2 in the Elliptic Billiiard T 8 2020 v1
The two types of self-intersected 7-periodics in the Elliptic Billiard T 7 2020 v1
Elliptic Billiard: Vertices of Self-Intersected 4-Periodics & Outer Polygon are concyclic w/ foci T 4 2020 v1

Tangential Polygon {-}

Title sound N Year urls
Locus of Vertices of the Excentral Polygon 3 to 6 2019 v1
5-periodics and feet of excenters 5 2019 v1
Locus of meetpoints of Excentral-to-Orbit Perpendiculars 3,4,5 2019 v1, v2, v3

Pencil of Confocals {-}

Title sound N Year urls
Tangents from a point on boundary to caustics -- 2019 v1
Tangents to caustics from billiard's vertex lie on a single circle -- 2019 v1
Loci of tangents to confocals: point traverses entire elliptic boundary -- 2019 v1
Loci of tangents to confocals: point traverses neighborhood of right vertex -- 2019 v1
Locus of tangents from ellipse: -45,45 degrees starting points -- 2019 v1
Locus of tangents from ellipse: 5,95,-45,45 degrees starting points -- 2019 v1

Stationary Circles {-}

Title sound N Year urls
5-periodics and a stationary circle 5 2019 v1
Stationary circles for N=3 to 8 3 to 8 2019 v1

Misc {-}

Title sound N Year urls
Elliptic Billiards in Brazil T 4 2019 v1
Reuleaux Triangle: Properties of Negative Pedal Curve, and Exploring its Billiard Trajectories T n/a 2020 v1
Non-Concentric Circular Poncelet Pair: Invariant Sum of Japanese Theorem Inradii (A. Akopyan) T 5,6 2020 v1
Pascal's Limaçon as Envelope of Circles T -- 2020 v1

Pedal Invariants {-}

Title sound N Year urls
Concyclic Feet of Focal Pedals and Invariant Product of Sums of Lengths for odd N T 5,6 2020 v1
Invariant sum of squared altitudes from each focus to tangential polygon sides T 3,4,5,6,7,8 2020 v1
Altitude Invariants to N-Periodics and their Tangential Polygons (N=3,4) T 3,4 2020 v1
Altitude Invariants to N-Periodics and their Tangential Polygons (N=5,6) T 5,6 2020 v1
Sum of square altitudes from arbitrary point to N-periodic tangents is invariant T 5 2020 v1
Pedal polygons from each focus have invariant area product T 5 2020 v1
Pedal Polygons for the N-Periodic and its Tangent Polygon: Area Ratio Invariances T 5,6 2020 v1
Exploring Amazing Invariants of N-Periodics and their Pedal Polygons T 3--12 2020 v1
Centroid Stationarity (even N) T 4,6 2020 v1

Steiner's Hat {-}

Title sound N Year urls
Equal sum of distances from each focus to vertices of antipedal polygon T 3,4,5,6 2020 v1
Pedal Polygons to N-Periodics with respect to a Focus: Concyclic Vertices and Circular Caustic T 3,4,5,6 2020 v1
The Envelope of Ellipse Antipedals is a Constant-Area Deltoid T n/a 2020 v1, v2
A narrated tour of the Garcia Deltoid: Surprising Invariants and Properties T n/a 2020 v1
Properties of Osculating Circles to the Ellipse at the 3 Cusp Pre-Images T n/a 2020 v1
Locus of Cusps and Deltoid Center of Area T n/a 2020 v1
Concyclic pre-images, osculating circles, and 3 area-invariant triangles T n/a 2020 v1
Rotated Negative Pedal Curve of Ellipse is Area-Invariant T n/a 2020 v1

Area Invariants {-}

Title sound N Year urls
Amazing Ellipse Pedal and Contrapedal Curves: area invariance for all pedal points on a circle! T n/a 2020 v1
Regular Polygons: the Signed Area of the Antipedal Polygon Vanishes along a Circle? T 3,4,5,6,7,8 2020 v1
Steiner's Krümmungs-Schwerpunkt implies Area-Invariant Interpolated Pedal Curve over Circles T -- 2020 v1

Exotic Billiards {-}

Title sound N Year urls
Horizontal-Vertical Billiard in a Rhombus and Parallelogram: are there N-Periodics? T n/a 2020 v1

Poncelet {-}

Title sound N Year urls
Poncelet Family: Amazing Circular Locus of X3 and the Steiner's Curvature Centroid T 5 2020 v1
5-Periodic Poncelet Families and their Pedal Polygons with Respect to their Curvature Centroids T 5 2020 v1
Jean-Victor Poncelet & Jakob Steiner walk into a Bierhaus + discover many invariants! Prost! Santé! T 3,4,5,6 2020 v1
N-Periodics on a Homothetic-Rotated Poncelet Pair: All Altitudes Meet at the Center! T 3,4,5,6,7 2020 v1
5- and 7-Periodics on a Homothetic-Rotated Poncelet Pair: All Altitudes Meet at the Center T 5,7 2020 v1
Concentric Poncelet Pair w Incircle: Ratio of Sidelength Product to Perimeter is Invariant for odd N T 3,5 2020 v1
Concentric Poncelet Pair w Circumcircle: Locus of Pseudo-Orthocenter is Circle (odd N) + Invariants T 3,5 2020 v1
Poncelet Invariants: circular + point loci of the pseudo-circumcenter and pseudo-orthocenter, N=5,6 T 5,6 2020 v1
Family of 3-Periodics in Five Poncelet Pairs T 1,2,3,4,9 2020 v1
New Invariants of Poncelet N-Periodics in the Homothetic Pair T 5 2020 v1

Inversive Poncelet {-}

Title sound N Year urls
Elliptic Billiard N-Periodics: invariant sum of inverse focal distances & inversive Pascal Limaçon T 5 2020 v1
Inversive Elliptic Billiard N-Periodics are Circular Arcs Interscribed between two Pascal Limaçons T 5 2020 v1
Inversive Invariants of Elliptic Billiard N-Periodics Nestled within Pascal's Limaçon T 5 2020 v1
Invariants of Inversive, Polar, and Dual Polygons derived from N-Periodics in the Elliptic Billiard T 5 2020 v1, v2
Invariant Inversive Perimeter (all N) and Area Product (odd N) T 5 2020 v1
Invariant Area Ratio Between Focus-Inversive Polygons for all N T 5 2020 v1
Centers of Inversive Arcs area a Bicentric Poncelet Family w/ Invariants T 5 2020 v1
Invariant Inversive perimeter and N=6 a/b=2 Null Antipedal Area T 6 2020 v1
Loci of Invariant Inversive perimeter and N=6 a/b=2 Null Antipedal Area T 6 2020 v1
Elliptic Billiard Focus-Inversive N-periodics: Loci of Vertex, Perimeter, Area Centroids are Circles T 5 2020 v1
Self-Intersected 5-periodics in the Elliptic Billiard: Loci of Focus-Inversive Centroids are Circles T 5 2020 v1

Inverse Curvature {-}

Title sound N Year urls
Cremona-Inversive Polygon of Odd-N-Periodics in the Elliptic Billiard: Zero Signed Area T 5 2020 v1
Homothetic Poncelet Pair: Invariant-Area "Inverse Curvature" Polygons T 5 2020 v1
Homothetic Poncelet Pair: Zero-Area "Inverse Curvature" Polygons T 3,5,6,8 2020 v1
Homothetic Poncelet Pair: Invariant-Area N=5 Inverse Curvature Polygons and the Ellipse Evolute T 5 2020 v1

Interactive Applets {.tabset .tabset-fade .tabset-pills}

A p5.js interactive applet where you can easily inspect loci and other objects connected with the $N=3$ family.

The loci of X(i), i=1,5,10,11 and the many-cuspid envelope of X(88)~X(90)

(\#fig:unnamed-chunk-6)The loci of X(i), i=1,5,10,11 and the many-cuspid envelope of X(88)~X(90)

To simulate multiple ray bounces within an Elliptic Billiard, try our Mathematica CDF. It requires installation of the Wolfram Player.

Original 2011 {-}

Title N Year url
Dynamic Billiards in Ellipse -- 2011 applet

3-Periodics {-}

Title N Year url
3-Periodics in Elliptic Billiard: Experimental Playground 3 2019 applet
3-Periodics and Derived Triangles 3 2019 applet
Ellipse and notables 3 2019 applet

N-Periodics {-}

Title N Year url
Bouncing Rays in Elliptic Billiard -- 2019 applet
Visualizing 3- to 9-periodics, their excentral polygons and caustics 3 to 9 2019 applet
Self-Intersecting 4-Periodics (bowtie) 4 2019 applet
Self-Intersecting 5-Periodics (pentagram) 5 2019 applet
Relaxation Algorithm for Computing N-Periodics and Caustics -- 2019 applet
Octagramma Mysticum: Loci of Intersection of Sides for N Periodics 5 to 9 2019 applet
N-Periodics and their Envelopes 3 to 9 2019 applet

Loci of Subtriangles {-}

Title N Year url
3-Periodics & Loci. 3 2019 applet
4-Periodics & Loci. 4 2019 applet
5-Periodics & Loci. 5 2019 applet
6-Periodics & Loci. 6 2019 applet
7-Periodics & Loci. 7 2019 applet
8-Periodics & Loci. 8 2019 applet
9-Periodics & Loci. 9 2019 applet

Circumconics {-}

Title N Year url
Three points, their circumbilliard and the anticomplementary's circumbilliard 3 2019 applet
Peter Moses' Points on the X(9)-centered circumellipse 3 2019 applet
Isotomics of Peter Moses' Gergonne Line Points 3 2020 applet
Feuerbach and Excentral Jerabek Hyperbolas 3 2019 applet
Tangent at X(100) to X(1)-circumellipse intersects billiard at X(651) 3 2019 applet
Concentric Poncelet Circumellipses 3 2021 applet

Cosine Circle {-}

Title N Year url
Excentral Cosine Circle is stationary 3 2019 applet
Locus of Intersection of Symmetric Tangent with Excentral Triangle is Stationary Circle 3 2019 applet
Locus of Six Intersections of Excentral Triangle with its Reflection About the Symmedian is Stationary Cosine Circle 3 2019 applet

Isog. & Isot. Conjugs. {-}

Title N Year url
Inverting a point with respect to an ellipse 3 2019 applet
Isogonal Conjugate of a Point with Respect to a Triangle 3 2019 applet
Isotomic Conjugate of a Point with Respect to a Triangle 3 2019 applet
Isogonal (antiorthic) and Isotomic Axes 3 2019 applet

Tangents to Caustics {-}

Title N Year url
Locus of Tangents to Pencil of Confocals -- 2019 applet
Discrete Set of Tangents to Confocal Caustics -- 2019 applet

Misc {-}

Title N Year url
Elliptic Billiards in Brazil 4 2019 applet
Tangent to an Elliptic Billiard and Conservation of Momentum with Respect to Foci 3 2019 applet
Locus of Napoleon Equilateral Summits 3 2019 applet
Level Curves of Sum of Distances (or Squared Distances) from 3 points 3 2020 applet
Locus of free triangle vertex with two on the ellipse 3 2020 applet

Pedal Invariants {-}

Title N Year url
Demonstration of most N-Periodic Invariants including, pedal, antipedal, and centers of mass -- 2020 applet

Loci of Ellipse-Mounted Triangles {-}

Title N Year url
Single Mounting 3 2020 applet
Sixteen Mountings 3 2020 applet
p5.js Billiard and Mountings 3 2020 applet

N-gon anticevians {-}

Title N Year url
Anticevians of Regular 5-gon with forbideen zones 5 2021 applet

Image Galleries {.tabset .tabset-fade .tabset-pills}

Original 2011 {-}

Title N Year urls
N-Periodics in Elliptic Billiard -- 2011 img

3-Periodic Loci {-}

Title N Year urls
Loci of X(1)~X(100) for N-Periodics, Derived and Internal Tris 3 2019 img

Envelopes {-}

Title N Year urls
Envelopes of lines of center pairs [X(i),X(j)], 1=i=8, i 3 2020 img

Free Triangle Vertex on Ellipse {-}

Title N Year urls
Free Triangle Vertex on Ellipse 3 2020 img

Self-Intersected Polygons {-}

Title N Year urls
N=6 Permuted Polygons (60 in 7 inters. groups) 6 2020 img
N=8 Permuted Polygons (2520 in 18 inters. groups) 8 2020 img

Code and Data

  • R Simulation code can be found here
  • Wolfram Mathematica Notebooks for 3-Periodics and N-Periodics, and loci visualization.
  • Original (2011) interactive applet showing trajectories in ellipses can be found here
  • Excel spreadsheet for N-Periodic vertices vs starting angle for $N=3,4,\ldots,7, a/b=1.5$ available here

For comments, corrections, suggestions email me at: dreznik _theat_ gmail _thedot_ com.