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
Invariants of 3- and N-Periodics in an Elliptic Billiard, with R. Garcia and J. Koiller.
Loci of N-Periodics and their Derived and Internal Triangles
Envelopes and Evolutes: beautiful curves enveloped by pairs \(X_i,X_j\) of Triangle Centers.
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.
new Artsy Loci of Ellipse-Mounted Triangles: long, short
new Extremal-Area Pedal and Antipedal Triangles: a discussion on maximal and minimal pedals and antipedals to a triangle. With M. Helman.
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 [1]) whose loci under 3-periodics in the elliptic billiard are ellipses. With R. Garcia and J. Koiller.
Videos N=3
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
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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
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
A p5.js interactive applet where you can easily inspect loci and other objects connected with the \(N=3\) family.
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
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.
