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Figure 1. Three members (blue) of the 1d family of 3-periodic orbits. These are triangles inscribed in an ellipse, whose vertices are bisected by the local normals. Joachimsthal’s Integral [27] prescribes that all trajectory segments (closed or not) are tangent to a confocal Caustic (brown), i.e., this is a special case of Poncelet’s Porism [4]. Moreover, the entire family of 3-periodics is tangent to that same Caustic!

Another interesting observation was that a special Triangle Center, the 5 Mittenpunkt X9, is stationary at the EB center over the entire family [20], see [19, PL#02].

Results: Below we prove the following facts: (1) the 3-periodic family conserves the ratio of Inradius-to-Circumradius [19, PL#03] implying several corollaries; (2) the Cosine Circle of the Excentral Polygon is stationary, centered on the Mittenpunkt, and exterior to the EB [19, PL#04].

Some Related Work: The study of triangle families under certain con- straints is not new. In 1924, Weaver studied triangles with a ﬁxed Cir- cumcircle and 9-Point Circle [28], continued in [15, 29]. More recently, the triangle family with a common Incircle and Circumcircle [16] and that with a common Incircle and Centroid [17] have been studied, and the latter is shown to have vertices on a conic. Other examples include the family of triangles deﬁned by tangents to a circle [12], and that associated with two lines and a point not on them [26]. Also of interest is the family of rectangles inscribed in smooth curves [25].

Outline of the Paper: Section 2 provide explicit expressions for two classic billiard invariants. Section 3 proves the invariance of the ratio of Inradius- to-Circumradius and all its Corollaries. Section 4 describes the construction of a stationary circle as well as some its properties. A few generalizations for N > 3 as well as a list of the videos mentioned herein appear in Sec- tion 5. Appendix A reviews Elliptic Billiards, and Appendix B provides longer expressions used in the proofs.

5Point of concurrence of lines drawn from each Excenter through sides’ midpoints [30]. NEW PROPERTIES OF TRIANGULAR ORBITS IN ELLIPTIC BILLIARDS 3

2. Preliminaries: Classic Invariants Let the boundary of the EB satisfy:

x 2 y 2 (1) f(x,y)= + = 1, a > b. a b Joachimsthal’s Integral implies that every trajectory segment is tangent to the Caustic [27]. Equivalently, a positive quantity6 γ remains invariant7 at every bounce point Pi:

1 1 (2) γ = v.ˆ fi = fi cos α 2 ∇ 2|∇ | where vˆ is the unit incoming (or outgoing) velocity vector, and

xi yi fi = 2 , . ∇ a2 b2 Consider a starting point P1 = (x1,y1) on the boundary of the EB. The the exit angle α (measured with respect to the normal at P1) required for the trajectory to close after 3 bounces is given by [6, 21]:

2 2 2 a b √2δ a b 2 2 2 (3) cos α = − − , c = a b , δ = a4 a2b2 + b4 . 2 4 2 2 p c a c x1 − − p − An explicit expression for γ can be derived based8 on Equations (2),(3):

√2δ a2 b2 (4) γ = − − c2 When a = b, γ = √3/2 and when a/b , γ 0. Using explicit expressions for the orbit vertices [6] (reproduced in→∞ Appendix→ B), we derive the perimeter L = s1 + s2 + s3:

2 2 (5) L = 2(δ + a + b )γ

3. Proof of Invariance of r/R

Referring to Figure 2, let r, R, and r9 denote the radii of Incircle, Circum- circle, and 9-Point Circle [30] of a 3-periodic orbit, respectively. These are centered on X1, X3, and X5, respectively. The Mittenpunk X9, is stationary at the EB center over the 3-periodic family [20].

6This is called J in [1, 2]. 7The trajectory need not be periodic. 8 Use P1 =(a, 0) as a convenient location. 4 RONALDO GARCIA, DAN REZNIK, AND JAIR KOILLER

Figure 2. A 3-periodic (blue), a starting vertex P1, the Incircle (green), Cir- cumcircle (purple), and 9-Point Circle (pink), whose centers are X1, X3, and X5, and radii are the inradius r, circumradius R, and 9-Point Circle radius r9. The Mittenpunkt X9 is stationary at the Billiard center [20].

3.1. Experimental Exploration. Our process of discovery of the constancy of r/R started with picking a particular a/b and plotting R, r9 and r vs. the t parameter in P1(t), Figure 3 (top). This conﬁrmed the well-known rela- tion R/r9 = 2 valid for any triangle [30]. It also suggested R might be proportional to r, not true for any triangle family. To investigate this potential relationship, we produced a scatter plot of orbit triangles for a discrete set of a/b in r vs R space. There one notices triangles corresponding to individual a/b fall on straight-line segments, and that all segments pass through the origin, suggesting r/R is a constant, Fig- ure 3 (bottom). One also notices each segment has endpoints (Rmin, rmin) and (Rmax, rmax). These are produced at isosceles orbit conﬁgurations, Fig- ure 4 and are of the form:

b2(δ b2) a2 + δ rmin = − ,Rmin = c2a 2a a2(a2 δ) b2 + δ (6) rmax = − ,Rmax = c2b 2b NEW PROPERTIES OF TRIANGULAR ORBITS IN ELLIPTIC BILLIARDS 5

Figure 3. Top: the three radii (Inradius r, Circumradius R and radius of the 9-Point Circle r9 plotted vs t of P1(t) = (a cos t, sin t). R/r9 =2 holds for any triangle, though R/r is not constant in general. Bottom: Scatter plot in r vs. R space: each dot is a 3-periodic in a discrete set of a/b families. Each family of triangles organizes along a straight line, suggesting the r/R ratio is constant. The minimum and maximum (r, R) that can occur in a family are bound by two continuous curves (dotted black), see (6). 6 RONALDO GARCIA, DAN REZNIK, AND JAIR KOILLER

Figure 4. Left: When the 3-periodic (blue) is a sideways isosceles (one of its vertices is at either horizontal EB vertex), r and R are minimal. The Incircle and Circumcircle are shown (green and purple, respectively), as are the loci of the Incenter and Circumcenter (dashed green and dashed purple, respectively). Right: When the orbit is an upright isosceles (one vertex on the top or bottom EB vertex), r and R are maximal.

We now prove a result announced in [20]: Theorem 1. r/R is invariant over the 3-periodic family and given by r 2(δ b2)(a2 δ) (7) = − − . R (a2 b2)2 − Proof. Let r and R be the radius of the incircle and circumcirle respectively. For any triangle [30]:

s1s2s3 rR = 2L · Where L = s1 +s2 +s3 is the perimeter, constant for 3-periodics, (5). There- fore:

r 1 s1s2s3 (8) = R 2L R2 Obtain a candidate expression for r/R for an orbit with P1 = (a, 0). This yields (7) exactly. Using explicit expressions for orbit vertices (Appendix B, derive an expression for the square9 of the right-hand-side of (8) as a function of x1 and subtract from it the square of (7). With a Computer Algebra 10 System (CAS) we showed this is magically zero for any value of x1 ( a, a). ∈ − For illustration Figure 5 shows the monotonically-decreasing dependence of r/R vs a/b.

9 2 2 The quantity s1s2s3/R is rational, see [21]. 10 We used R =(s1s2s3)/(4A), A = triangle area, for simpliﬁcation. NEW PROPERTIES OF TRIANGULAR ORBITS IN ELLIPTIC BILLIARDS 7

Figure 5. Dependence of r/R on EB aspect ratio a/b. Maximum is achieved when EB is a circle. Curve contains an inﬂection point still lacking a geometric interpretation.

3.2. Corollaries. The relations below are valid for any triangle [30, 8]:

3 r (9) cos θ = 1+ X i R i=1 3 ′ r (10) cos θ = Y | i| 4R i=1 A r (11) = A′ 2R

′ Where θi are the angles internal to the orbit, θi are those of the Excentral ′ Triangle (opposite to orbit θi), and A (resp. A ) is the area of the orbit (resp. Excentral Triangle), Figure 6.

Corollary 1. The sum of orbit cosines, the product of excentral cosines11, and the ratio of excentral-to-orbit areas are all constant.

In [20] we reported experiments that showed that the left hand sides of (9), (10) were constant for all N-periodics whereas (11) was constant for odd N only. These generalizations were subsequently proven [1, 2].

11The absolute value in (10) can be dropped as the Excentral family is acute [20]. 8 RONALDO GARCIA, DAN REZNIK, AND JAIR KOILLER

Figure 6. An a/b = 1.25 EB is shown (black) as well as a 3-periodic orbit ′ ′ ′ ′ T = P1P2P3 (blue). The orbit’s Excentral Triangle T = P1P2P3 (green) has vertices at the intersections of exterior bisectors (therefore it is tangent to the ′′ ′′ ′′ ′′ EB at the orbit vertices). The Extouch Triangle T = P1 P2 P3 (red) has vertices at where Excircles (dashed green) touch each side. There are known to lie on the orbit’s Caustic (brown) as it is its Mandart Inellipse [30]. Let A, A′, A′′ be the areas of T,T ′,T ′′, respectively Video: [19, PL#06]

3.3. Excentral-to-Extouch Area Ratio. Let A′′ denote the area of the Extouch Triangle, whose vertices are where each Excircle touches a side [30], Figure 6. Observation 1. The vertices of the orbit’s Extouch Triangle lie on the Caus- tic. A well-known result is that those lie on the Mandart Inellipse, whose center is the Mittenpunkt X9 [30], therefore it must be the Caustic, see [19, PL#06]. Theorem 2. A′/A′′ is invariant and equal to (2R/r)2. Proof. For any triangle, the ratios A′/A and A/A′′ are equal12. Since A′/A = r/(2R), and A′/A′′ = (A′/A)(A/A′′) the result follows. 3.4. Billiard-to-Caustic Area Ratio. The N = 3 Caustic semiaxes are given by13 [6, 21]:

a δ b2 b a2 δ (12) ac = − , bc = − c2 c2 · 12 ′ ′′ 2 Actually, A /A = A/A =(s1s2s3)/(r L), where si are the sides and L the perimeter [30, Excentral,Extouch]. 13Two concentric, axis-aligned ellipses can generate a 3-periodic Poncelet family if and only if a/ac + b/bc = 1 [7], which holds above. NEW PROPERTIES OF TRIANGULAR ORBITS IN ELLIPTIC BILLIARDS 9

Theorem 3. Let Ab and Ac be the areas of Billiard and Caustic ellipses respectively. Then:

A r c = Ab 2R·

Proof. As Ac = πacbc and Ab = πab the result follows from Equation 7.

Notice the Caustic-to-Billiard area ratio is equal to the Orbit-to-Excentral area ratio, Equation 11.

4. Cosine Circle is External to Billiard The Cosine Circle14 [30] of a Triangle passes through 6 points: the 3 pairs of intersections of sides with lines drawn through the Symmedian X6 15 parallel to sides of the Orthic Triangle . Its center is X6 [30]. If one takes the Excentral Triangle of an orbit as the reference triangle, it is easy to see its Orthic is the orbit itself, Figure 4:

Theorem 4. The Cosine Circle of the Excentral Triangle of a 3-periodic orbit is invariant over the family. Its radius r∗ is constant and it is concentric and external to the EB.

∗ Proof. Once again, we set P1 = (a, 0), derive a candidate expression for r , and with a CAS, check if it holds for all x1 ( a, a). This yields: ∈ −

2 2 ∗ a b (13) r = − √2δ a2 b2 − − More generally a simple expression for r∗ valid for all N was kindly contributed by S. Tabachnikov [20]:

∗ (14) r = 1/γ

2 Let a>b> 0 and δ = √a4 a2b2 + b4. As 0 < (a2 b2) < δ2 it follows that − − 2 2 ∗ 2 a + b + 2δ (r ) = 3 a2 + b2 + 2(a2 b2) > − 3 2 > a .

As the Excentral Cosine Circle and the EB are concentric, the proof is complete.

14Also known as the Second Lemoine Circle. 15The Orthic Triangle has vertices at altitude feet. 10 RONALDO GARCIA, DAN REZNIK, AND JAIR KOILLER

Figure 7. Given a 3-periodic orbit (blue), the Cosine Circle (red) of the Ex- central Triangle (green) passes through the 3 pairs of intersections of the dashed lines with the Excentral Triangle. These are lines parallel to the orbit’s sides drawn through the Excentral’s Symmedian Point X6, congruent with the orbit’s Mittenpunkt X9. This circle is stationary across the 3-periodic family and always exterior to the Billiard. Video: [19, PL#04].

4.1. YouTube Mathematics. In response to a posting of one of our videos on YouTube [19, PL#00], D. Laurain contributed an alternate expression for r∗ [13]:

∗ 2L (15) r = r/R + 4 Combining this with r∗ = 1/γ (14), we obtain: Theorem 5. r/R = γL 4, where γ and L are Joachimsthal’s constant and perimeter L of the orbit.− This motivated our conjecture that for all N:

N r cos θi =1+ = γL N X R − i=1 And this was recently proven [1, 2].

5. Conclusion One key result ﬁrst observed experimentally and then proven is that r/R is invariant over the 3-periodic family. In Section 3 we demonstrate that this ﬁrst fact implies that 3-periodics conserve the sum of cosines of orbit angles. This motivated a conjecture that the sum is conserved for all N, a fact recently proven [1, 2]. NEW PROPERTIES OF TRIANGULAR ORBITS IN ELLIPTIC BILLIARDS 11

Also observed was that A′/A and A/A′′ are invariant for odd N and that A′/A′′ is invariant for all N, quite a surprise! Indeed, several other new invariants with readily inspectable manifesta- tions have been observed, and will be reported in [22]. This stream of results deriving from the humble triangle family has surprised us. This also reinforced for us the idea that experimentation is a very valuable tool for Mathematical discovery. 5.1. Video List. All videos mentioned above have been placed on a playlist [19]. Table 1 contains quick-reference links, with column “PL#” providing video number within the playlist.

PL# Title Section 01 Locus of Incenter and Intouchpoint 1 02 Mittenpunkt stationary at EB center 1 03 Constant cosine sum and product 3 04 Excentral Cosine Circle is 4 stationary and exterior to EB 05 Alternate Cosine Circle video 4 with D. Laurain’s r∗ formula (15) 06 Extouchpoints lie on N = 3 Caustic 3 Table 1. Videos mentioned in the paper. Column “PL#” indicates the entry within the playlist [19].

Acknowledgments

We would like to thank Sergei Tabachnikov, Richard Schwartz, Arseniy Akopyan, Olga Romaskevich, Corentin Fierobe, Ethan Cotterill, Dominique Laurain, and Mark Helman for generously helping us during this research. The second author is fellow of CNPq and coordinator of Project PRONEX, CNPq, FAPEG 2017 10 26 7000 508. The third author thanks the Federal University of Juiz de Fora for a 2018- 2019 fellowship. References

[1] Akopyan, A., Schwartz, R., Tabachnikov, S.: Billiards in ellipses revisited (2020). URL https://arxiv.org/abs/2001.02934. ArXiv [2] Bialy, M., Tabachnikov, S.: Dan Reznik’s identities and more (2020). URL https://arxiv.org/abs/2001.08469. ArXiv [3] Dragović, V., Radnović, M.: Poncelet Porisms and Beyond: Integrable Billiards, Hyperelliptic Jacobians and Pencils of Quadrics. Frontiers in Mathematics. Springer, Basel (2011). URL https://books.google.com.br/books?id=QcOmDAEACAAJ [4] Dragović, V., Radnović, M.: Caustics of Poncelet polygons and classical ex- tremal polynomials. Regul. Chaotic Dyn. 24(1), 1–35 (2019). DOI 10.1134/ S1560354719010015. URL https://doi.org/10.1134/S1560354719010015 [5] Fierobe, C.: On the circumcenters of triangular orbits in elliptic billiard (2018). URL https://arxiv.org/pdf/1807.11903.pdf. Submitted [6] Garcia, R.: Elliptic billiards and ellipses associated to the 3-periodic orbits. American Mathematical Monthly 126(06), 491–504 (2019). URL https://doi.org/10.1080/00029890.2019.1593087 12 RONALDO GARCIA, DAN REZNIK, AND JAIR KOILLER

[7] Griffiths, P., Harris, J.: On Cayley’s explicit solution to Poncelet’s porism. Enseign. Math. (2) 24(1-2), 31–40 (1978) [8] Johnson, R.A.: Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Houghton Mifflin, Boston, MA (1929) [9] Kaloshin, V., Sorrentino, A.: On the integrability of Birkhoff billiards. Phil. Trans. R. Soc. A(376) (2018). DOI https://doi.org/10.1098/rsta.2017.0419 [10] Kimberling, C.: Major centers of triangles. The American Mathematical Monthly 104(5), 431–438 (1997) [11] Kimberling, C.: Encyclopedia of triangle centers (2019). URL https://faculty.evansville.edu/ck6/encyclopedia/ETC.html [12] Kovačević, N., Sliepčević, A.: On the certain families of triangles. KoG-Zagreb 16, 21–27 (2012) [13] Laurain, D.: Formula for the radius of the orbits’ excentral cosine circle. Private Communication (August, 2019) [14] Levi, M., Tabachnikov, S.: The Poncelet grid and billiards in ellipses. The Amer- ican Mathematical Monthly 114(10), 895–908 (2007). DOI 10.1080/00029890.2007. 11920482. URL https://doi.org/10.1080/00029890.2007.11920482 [15] Murnaghan, F.D.: Discussions: Note on mr. weaver’s paper "a system of triangles related to a poristic system" (1924, 337-340). The American Mathematical Monthly 32(1), 37–41 (1925). DOI 10.2307/2300090. URL www.jstor.org/stable/2300090 [16] Odehnal, B.: Poristic loci of triangle centers. Journal for Geometry and Graphics 15(1), 45–67 (2011) [17] Pamfilos, P.: Triangles with given incircle and centroid. Forum Geometricorum 11, 27–51 (2011) [18] Reznik, D.: Triangular orbits in elliptic bil- lards: Loci of points X(1) X(100) (2019). URL https://dan-reznik.github.io/Elliptical-Billiards-Triangular-Orbits/loci_6tri.html [19] Reznik, D.: Playlist for “New Properties of Triangular Orbits in an Elliptic Billiard” (2020). URL https://bit.ly/379mk1I [20] Reznik, D., Garcia, R., Koiller, J.: Can the elliptic billiard still surprise us? Mathematical Intelligencer (2019). DOI 10.1007/s00283-019-09951-2. URL https://arxiv.org/pdf/1911.01515.pdf [21] Reznik, D., Garcia, R., Koiller, J.: Loci of triangular orbits in elliptic billiards (2020). DOI 10.1007/s00283-019-09951-2. To appear [22] Reznik, D., Garcia, R., Koiller, J.: New invariants of n-periodics in elliptic billiards (2020). In preparation [23] Romaskevich, O.: On the incenters of triangular orbits on elliptic billiards. Enseign. Math. 60(3-4), 247–255 (2014). DOI 10.4171/LEM/60-3/4-2. URL https://arxiv.org/pdf/1304.7588.pdf [24] Rozikov, U.A.: An Introduction To Mathematical Billiards. World Scientific Publish- ing Company (2018) [25] Schwartz, R.: Rectangle coincidences and sweepouts (2019). URL https://arxiv.org/abs/1809.03070. ArXiv [26] Sliepčević, A., Halas, H.: Family of triangles and related curves. Rad HAZU 515, 203–2010 (2013) [27] Tabachnikov, S.: Geometry and Billiards, Student Mathematical Library, vol. 30. American Mathematical Society, Providence, RI (2005). DOI 10.1090/stml/030. URL http://www.personal.psu.edu/sot2/books/billiardsgeometry.pdf. Mathematics Advanced Study Semesters, University Park, PA [28] Weaver, J.H.: A system of triangles related to a poristic system. The Amer- ican Mathematical Monthly 31(7), 337–340 (1924). DOI 10.2307/2299387. URL www.jstor.org/stable/2299387 [29] Weaver, J.H.: Curves determined by a one-parameter family of triangles. The American Mathematical Monthly 40(2), 85–91 (1933). DOI 10.2307/2300940. URL www.jstor.org/stable/2300940 [30] Weisstein, E.: Mathworld (2019). URL http://mathworld.wolfram.com NEW PROPERTIES OF TRIANGULAR ORBITS IN ELLIPTIC BILLIARDS 13

Appendix A. Review: Elliptic Billiards An Elliptic Billiard (EB) is a particle moving with constant velocity in the interior of an ellipse, undergoing elastic collisions against its boundary [27, 24], Figure 8. For any boundary location, a given exit angle (e.g., measured from the normal) may give rise to either a aperiodic or N-periodic trajectory [27], where N is the number of bounces before the particle returns to its starting location. The EB is the only known integrable Billiard in the plane [9]. It satisﬁes two important integrals of motion: (i) Energy, since particle velocity has constant modulus and collisions are elastic, and (ii) Joachimsthal’s, implying that all trajectory segments are tangent to a confocal Caustic [27]. The EB is a special case of Poncelet’s Porism [3]: if one N-periodic trajectory can be found departing from some boundary point, any other such point will initiate an N-periodic, i.e., a 1d family of such orbits will exist. A striking consequence of Integrability16 is that for a given N, all N-periodics have the same perimeter and the same confocal Caustic [27], Figure 1.

Figure 8. Trajectory regimes in an Elliptic Billiard. Top left: first four segments of a trajectory departing at P1 and toward P2, bouncing at Pi,i = 2, 3, 4. At each bounce the normal nî bisects incoming and outgoing segments. Joachimsthal’s integral [27] means all segments are tangent to a confocal Caus- tic (brown). Top right: a3-periodic trajectory. All 3-periodics in this Billiard will be tangent to a special confocal Caustic (brown). Bottom: first 50 segments of a non-periodic trajectory starting at P1 and directed toward P2. Seg- ments are tangent to a confocal ellipse (left) or hyperbola (right). The former (resp. latter) occurs if P1P2 passes outside (resp. between) the EB’s foci (black dots).

16Integrability implies that closed orbits have two rationally-related frequencies, shar- ing therefore a common period. Since the velocity is a ﬁxed constant, the total length of N-periodic orbits with the same winding number is also constant [27]. 14 RONALDO GARCIA, DAN REZNIK, AND JAIR KOILLER

Appendix B. Orbit Vertices Let the boundary of the Billiard satisfy Equation (1). Assume, without loss of generality, that a b. ≥ Given a starting vertex P1 and an exit angle α (as above), the orbit P1P2P3 will be such that [6]: vertex P2 will be given by (p2x,p2y)/q2, with

4 2 2 2 2 3 4 2 2 p2x = b a + b cos α a x1 2 a b cos α sin αx1y1 − 4 2 2 2 − 2 − 2 6 3 + a (a 3 b ) cos α + b x1 y1 2a cos α sin αy1, 6 − 3 4 2 2 −2 2 2 p2y =2b cos α sin αx1 + b (b 3 a ) cos α + a x1y1 2 4 2 −4 2 2 2 2 3 + 2 a b cos α sin αx1y1 a a + b cos α b y1 4 2 2 2 2 4− 2 2 2 2 − q2 =b a c cos α x1 + a b + c cos α y1 2−2 2 2 a b c cos α sin αx1 y1. − and vertex P3 will be given by (p3x,p3y)/q3, with:

4 2 2 2 2 3 4 2 2 p3x =b a b + a cos αx1 + 2 a b cos α sin αx1y1 4 − 2 2 2 2 2 6 3 + a cos α a 3 b + b x1 y1 + 2 a cos α sin αy1 6 −3 4 2 2 2 2 2 p3y = 2 b cos α sin αx1 + b a + b 3 a cos α x1 y1 − 2 4 2 4 2 − 2 2 2 3 2 a b cos α sin αx1y1 + a b b + a cos α y1, −4 2 2 2 2 4 2 2− 2 2 q3 =b a c cos α x1 + a b + c cos α y1 2−2 2 + 2 a b c cos α sin αx1 y1.

The sides of the triangular orbit have lengths s1 = P3 P2 , s2 = P3 P1 | − | | − | and s3 = P2 P1 . | − | Ronaldo Garcia, Inst. de MatemÃątica e EstatÃŋstica,, Univ. Federal de GoiÃąs,, GoiÃćnia, GO, Brazil E-mail address: [email protected]

Dan Reznik, Data Science Consulting,, Rio de Janeiro, RJ, Brazil E-mail address: [email protected]

Jair Koiller,, Dept. de MatemÃątica,, Univ. Federal de Juiz de Fora,, Juiz de Fora, MG, Brazil E-mail address: [email protected]