1 Introduction

Consider an Elliptic Billiard (EB) with boundary given by, \(a>b>0\):

\[ f(x,y)=\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2-1=0 \]

Consider its 1d family of 3-periodics [1]. A pair of Kimberling Centers \([X_i,X_j],i{\neq}j\) defines a family of lines. Below we explore the caustic curves they envelop [2], i.e., are instantaneously tangent to. A few examples of loci and envelopes for basic Center pairs appear below (locis in red/green and envelope in purple):

Note: \(X_i,i=2,3,4,5\) are collinear and on the Euler Line [3]. Pairs thereof generate the (same) Euler Line envelope.

2 Envelope Carousels

We examine envelopes of lines generate by pairs \([X_i,X_j],i{\in}\{1,2,3,4,5,6,11\},j{\in}(i,100]\) for an EB with \(a/b{\simeq}1.618\), the Golden Ratio. Out of these we eliminate pairs collinear with \(X_9\), since the latter is stationary at the EB center [1].

Color coding:

  • Black: billiard
  • Red: locus of \(X_i\)
  • Green: locus of \(X_j\)
  • Blue (solid): envelope of \([X_i,X_j]\)
  • Blue (dashed): the evolute of the Billiard, i.e., the envelope of the boundary normals [3], a fixed astroid-like curve.

X(1): Incenter

X(2): Barycenter

X(3): Circumcenter

X(4): Orthocenter

X(5): 9-Point Center

X(6): Symmedian Point

X(11): Feuerbach Point

3 Toward a taxonomy

No tangency

Single tangency

Dual tangency

4 Hand-picked “Best-Of”

Heart-Shaped

Spikeys

X(92) Batman

Quasi-Diamond

Close-Tracking

Multiple Tangents

Misc

X(96) Brazil Flag

5 Swan Envelopes

Note: should a carousel below be empty, click on the right carousel arrow to bring in first image.

Let \(T\) be a triangle. The Isotomic Conjugate (I.C.) of a Circumellipse of \(T\) (an ellipse that passes thru its vertices) is a line which does not cross the circumcircle of \(T\) [3].

It turns out the I.C. of the EB (the \(X_9\)-centered circumellipse) is the Gergonne Line \(L_{55}\) which passes thru \(X_{514}\) and \(X_{661}\) [4]. Under \(X_9\) in [3] a list is provided of 57 Kimberling Centers discovered by Peter Moses which lie on \(L_{55}\), listed below.

Note: each appears either as (i) as a positive number followed by its isotomic conjugate in parenthesis, or (ii) as a negative number indicating its isotomic conjugate is not yet a listed Kimberling Center:

514 (190), 661 (799), 693 (100), -857, 908 (34234), -914, 1577 (662), 1959 (1821), -2084, 2582 (2580), 2583 (2581), 3239 (658), -3250, 3762 (3257), 3766 (660), 3835 (4598), 3904 (655), 3912 (673), 3936 (24624), -3948, -4129, 4358 (88), 4391 (651), 4462 (27834), -4468, -4486, 4728 (4607),-4766, -4776, -4789, 4791 (4604), 4801 (4604), -4823, -4978, -5074, -5179, 6332 (653), -6381, -6590, 8045 (653), 14206 (2349), -14207, 14208 (162), -14209, 14210 (897), -14281, -14349, -14350, -14963, -18669, -18715, 24018 (823), -30565, -30566, -30804, 30806 (36101), 32679 (32680).

Note: Moses has reported 30807 (36101) also lies on \(L_{55}\), to be confirmed.

We call the above swans since over the family of 3-periodics they glide over the EB’s boundary. Above certain thresholds, some can become non-monotonic with respect to the motion of 3-periodics’ vertices. The joint motion of swan pairs can become quite intricate, akin to a ballet [5].

Consider the line defined by \([X_i,X_k]\), where \(i=1{\ldots}100\) and \(k=1{\ldots}57\) indexes the isotomic cojugate of a Moses \(L_{55}\) point. Over the 3-periodic family, a choice of \(i,k\) defines a family of lines which envelopes a caustic.

Click on the blue tabs above for few nice-looking caustics swept by certain center-swan pairs, organized by appearance.

5.1 Greatest Hits: Center-Swan

note: If carousel appears empty click on left-right arrows to exhibit first picture.

Alien

Arrow Tips

Astroidal

Atomic Orbit

Batman

Beaded

Complex

Cushion

Devil Spikes

Dimpled Hearted

Lobed

Mini

Multi Tangent

Near Ellipse

Nippled

Non Compact

Serrated

Shalom Swan

Side Butts

Side Ears

Splitting Cells

Star

Straight Diamond

Touch Focus

Tracking

5.2 Greatest Hits: Swan-Swan

note: If carousel appears empty click on left-right arrows to exhibit first picture.

Astroidal

Atomic

Batman

Bejeweled

Complex

Cushion

Devil’s Vise

Ears

Inner Loop

Lobed

Mini

Multi Cell

Multi Tangent

Near Ellipse

Nippled

Non Compact

Pascal Lines

Rectilinear

Touch Focus

Tracking

6 Research Questions

Triple tangency

As shown below, for collinear Centers \(X_i,i=2,5,12\) the envelope is triply-tangent to their loci. However, it is not tangent to the locus of \(X_{11}\), also collinear.

Envelope of X(1),X(100),X(88)

These three points are known to be collinear. Shown elsewhere [6] was the fact that the locus of \(X_1\) is an ellipse and those of both \(X_{100}\) and \(X_{88}\) are on the Elliptic Billiard. Parametrize the 3-periodic family by its first vertex \(P_1(t)=[a\cos{t},b\sin{t}],0{\leq}2\pi\). As shown in this video, \(X_1\) will turn in the same direction as \(P_1\), whereas \(X_{100}\) will turn counter to \(P_1\). The nature of this direction is not understood. Furthermore, the motion of \(X_{88}\) is more complicated.

Let \(a_{88}=(\sqrt{2\sqrt{2}+6}\,)/2\simeq{1.485}\). The corresponding motion of \(X_{88}\) will be one of three types:

  • \(a/b<a_{88}\): monotonic and opposite to \(P_1(t)\), slowing down near EB vertices
  • \(a/b=a_{88}\): same, but with instantaneous stops at the EB vertices.
  • \(a/b>a_{88}\): same but with reverse velocity phases near the vertices.

Equivalently, \(X_{88}\) is stationary when the instantaneous envelope of line \(X_1X_{100}\) is \(X_{88}\) itself. Below we show that for \(a/b\) less than, equal, or greater than \(a_{88}\), the envelope will be entirely inside, touch the EB vertices, or pierce sideways through the Billiard, respectively, as shown on a video.

Envelope of Isogonal and Isotomic Axes

Given a triangle ABC, the isogonal and isotomic conjugate of any line on its plane is a circumellipse [3]. If the Elliptic Billiard is regarded as the stationary circumellipse to the 3-periodic family, the Antiorthic Axis and Gergonne Line [3] are its isogonal and isotomic conugates, respectively [4]. As shown on this video, these lines envelop the elliptic locus of \(X_{1155}\) and \(X_{908}\), respectively. This is remarkable since as seen above, most point pairs produce multi-cuspid envelopes (astroid-like).

6.1 Astroidal Ellipse Evolute

The evolute of an ellipse with axes \(a,b\) is the envelope of its normals. It is given by the following astroid-like curve [3]:

\[ \begin{align*} x(t)=&\frac{c^2}{a}\cos^3(t)\\ y(t)=&-\frac{c^2}{b}\sin^3(t)\\ c^2=&a^2-b^2 \end{align*} \]

We have found this evolute to be tangent to the locus of \(X_1\) for any Billiard aspect ratio. This could be related to the fact that the evolute is the envelope of lines \(P_1(t)X_1\), oriented as the normals.

We have found that for none of the envelopes \((X_1,X_j),1<j<100\) generated so far which are tangent to the locus of \(X_1\) coincide with the evolute. This could be caused by the fact that \(P_1(t)\) is not a Triangle Center.

Shown below is the envelope of \([X_1,X_5]\) (purple), doubly tangent to the two loci, and the EB’s evolute (dashed blue). Note they do not coincide.

7 List of Videos

Title sound Kimberling Centers Year urls
Envelope of Antiorthic and Gergonne Lines NA 9,44,1155,857,908 2020 v1
Evolute of Elliptic Billiard and Envelope of X(1)-X(5) NA 1,5,4,9, 2020 v1
Envelope of 3-Periodic Vertex with Triangle Center NA 1,2,3,4,5,6,7,8,10,11,12,20 2020 v1
Evolute Triangles of P1(t) with X(i) NA 1,3,5,20 2020 v1
Elliptic Envelope of P1(t) with P1(t+pi/2) NA – 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

References

[1] Reznik D, Garcia R, Koiller J. 2019.Math Intelligencer. Available from: https://arxiv.org/abs/1911.01515,.

[2] Weisstein E. 2019. Envelope of a curve. MathWorld–A Wolfram Web Resource. Available from: http://mathworld.wolfram.com/Envelope.html,.

[3] Weisstein E. 2019. Mathworld. MathWorld–A Wolfram Web Resource. Available from: http://mathworld.wolfram.com,.

[4] Kimberling C. 2019. Encyclopedia of triangle centers. Available from: https://faculty.evansville.edu/ck6/encyclopedia/ETC.html,.

[5] Reznik D, Garcia R, Koiller J. 2020. The ballet of triangle centers on the elliptic billiard..

[6] Garcia R, Reznik D, Koiller J. 2020. Loci of 3-periodics in an elliptic billiard: Why so many ellipses?.