Family Ties: Relating Poncelet 3-Periodics by their Properties
aa r X i v : . [ m a t h . M G ] D ec FAMILY TIES: RELATING PONCELET3-PERIODICS BY THEIR PROPERTIES
RONALDO GARCIA AND DAN REZNIK
Abstract.
We compare loci types and invariants across Poncelet familiesinterscribed in three distinct concentric Ellipse pairs: (i) ellipse-incircle, (ii)circumcircle-inellipse, and (iii) homothetic. Their metric properties are mostlyidentical to those of 3 well-studied families: elliptic billiard (confocal pair),Chapple’s poristic triangles, and the Brocard porism. We therefore organizedthem in three related groups.
Keywords invariant, elliptic, billiard, locus.
MSC Introduction
We have been studying loci and invariants of Poncelet 3-periodics in the confocalellipse pair (elliptic billiard). Classic invariants include Joachmisthal’s constant J (all trajectory segments are tangent to a confocal caustic) and perimeter L [25].A few properties detected experimentally [20] and later proved can be dividedinto two groups: (i) loci of triangle centers (we use the X k notation in [16]), and(ii) invariants.In terms of loci, the following results have been proved: (i) the locus of theincenter [22, 9], barycenter [24], circumcenter [9, 7], orthocenter [11] and manyothers are ellipses; (ii) a special triangle center known as the Mittenpunkt X isstationary [23].For invariants we chiefly have (i) the sum of cosines [1, 2], (ii) the product ofouter polygon cosines, and (iii) outer-to-3-periodic area ratio [4].We continue our inquiry into loci and invariants by now considering 3-periodicfamilies three other non-confocal though concentric ellipse pairs. Referring to Fig-ure 1: • Family I: outer ellipse and incircle, incenter X is stationary. • Family II: outer circumcircle and inellipse, circumcenter X is stationary • Family III: an axis aligned pair of homothetic ellipses, the barycenter X is stationary.One goal is to identify properties of the above common with previously-studied3-periodic families, namely, (i) the confocal pair (elliptic billiard), (ii) Chapple’sporism [8] and (iii) the so-called Brocard Porism [3, 14]. A quick review of theirgeometry appears in Section 2. Date : December, 2020.
Figure 1.
Poncelet 3-periodic families in the various concentric ellipse pairs studied in the article.Properties and loci of the confocal pair (elliptic billiard) were studied in [20, 12, 11]. For eachfamily the particular triangle center which is stationary is indicated.
Main Results.
Here are our main results: • Family I – It conserves the circumradius, the sum of cosines, and the sum ofsidelengths divided by their product. – Its sum of cosines is identical to the confocal pair which is its affineimage. – The family is the image of the Chapple’s poristic family [18] under avariable rigid rotation. – The poristic family is the image of the confocal family under a variablesimilarity transform [10]. So family I retains several all scale-free in-variants identified for the elliptic billiard, including the sum of cosines. • Family II – It conserves the cosine product and the sum of squared sidelengths. – Its product of cosines is identical to the excentrals’ in the confocal pairwhich is its affine image. – In the elliptic billiard the locus of the incenter (resp. symmedian point)is an ellipse (resp. quartic) [11]. Here the roles swap: the incenterdescribes a quartic, and the symmedian is an ellipse.
AMILY TIES: RELATING PONCELET 3-PERIODICS BY THEIR PROPERTIES 3 – The orthic triangles to the family are the image of the poristic family’sunder a variable rigid rotation. • Family III – It conserves area, sum of sidelengths squared, sum of cotangents (thelatter implies the Brocard angle is invariant). – Again in contradistinction with the elliptic billiard, the locus of theincenter X is non-elliptic while that of X is an ellipse. – The locus of irrational triangle centers X k , k = – As shown in [19], this family is the image of Brocard porism triangles[3] under a variable similarity transform.Thus the following group Poncelet families is proposed with mostly identicalproperties: (i) family I: confocal, poristics; (ii) family II: confocal excentrals, poris-tic excentrals; (iii) family III: Brocard porism. Table 1 shows how loci types areshared and/or differ across families, and Figure 10 gives a bird eye’s view of thekinship across these families via various transformations.
Related Work.
Romaskevich proved the locus of the incenter X over the confocalfamily is an ellipse [22]. Schwartz and Tabachnikov showed the locus of barycenterand area centers of Poncelet trajectories describe ellipses though the perimetercentroid in general does not [24]. For N = 3 the former correspond to X andthe latter to the Spieker center X . Garcia [9] and Fierobe [7] the locus of thecircumcenter of 3-periodics in the elliptic billiard are ellipses. Indeed, 29 out of thefirst 100 triangle centers listed in [16] are ellipses [11]. Tabachnikov and Tsukerman[26] and Chavez-Caliz studied [4] properties andloci of the circumcenters of massof Poncelet N-periodics.The following invariants for N-periodics in the elliptic billiard have been proved:(i) sum of cosines [1, 2], (ii) product of cosines of the outer polygons [1, 2], and (iii)area ratios and products of N-periodics and their polar polygons (excentral trianglefor N=3); interestingly, these depend on the parity of N [4, 2]. Result (i) also holdsfor the Poncelet family interscribed between an ellipse and a concentric circle [1,Corollary 6.4]. Article structure.
We start by reviewing the confocal, Chapple’s and Brocardporisms in Section 2. We then describe properties, invariants and transformations offamilies I, II, and III in Sections 3, 4, and 5, respectively. We summarize all resultsin Section 6. Highlights include (i) a graph representing affine and/or similarityrelationships between the various families (Figure 10), (ii) a table of conservedquantities which we have found to continue to hold for
N > (proof pending), and(iii) a table with links to videos illustrating some phenomena herein.2. Review of Classic Porisms and Proof Method
Grave’s Theorem affirms that given a confocal pair ( E , E ′′ ) , the two tangents to E ′′ from a point P on E will be bisected by the normal of E at P [17]. A consequenceis that any closed Poncelet polygon interscribed in such a pair, if regarded as thepath of a moving particle bouncing elastically against the boundary, will be N-periodic . For this reason, this pair is termed the elliptic billiard ; [25] is the seminalwork. It is conjectured as the only integrable planar billiard [15]. One consequence,
RONALDO GARCIA AND DAN REZNIK
Figure 2.
The poristic triangle family (blue) [8] has a fixed incircle (green) and circumcircle(purple). Let r, R denote their radii. Its excentral triangles (green) are inscribed in a circleof radius R centered on the Bevan point X and circumscribe the MacBeath inconic (dashedorange) [27], centered on X with foci on X and X . A second configuration is also shown(dashed blue and dashed green). Video mentioned above, is that it conserves perimeter L . An explicit parametrization for3-periodic vertices appears in Appendix A.1.Referring to Figure 2, poristic triangles are a 1d Poncelet family with fixed incir-cle and circumcircle. Discovered in 1746 by William Chapple. Recently, Odehnal[18] has studied loci of its triangle centers. showing many of are either station-ary, ellipses, or circles. Surprisingly, the poristic family is the image of billiard3-periodics under a variable similarity transform [10], and these two families sharemany properties and invariants.Referring to Figure 3, the Brocard porism [3] is a family of triangles inscribed ina circle and circumscribed about a special inellipse known as the “Brocard inellipse”[27, Brocard Inellipse]. Notably the family’s Brocard points are stationary andcoincide with the foci of the inellipse. Also remarkable is the fact that the Brocardangle ω is invariant [14]. In [19] we show this family is the image of family IIItriangles under a variable similarity transform. A word about our proof method.
We omit some proofs below as they areobtained from a consistent method used previously in [11]: (i) depart from symbolicexpressions for the vertices of an isosceles 3-periodic (see Appendix A); (ii) obtain asymbolic expression for the invariant of interest; (iii) simplify it assisted by a CAS,arriving at a “candidate” symbolic expression for the invariant; (iv) verify the latterholds for any (non-isosceles) N-periodic and/or Poncelet pair aspect ratios and if itdoes, declare it as provably invariant.3.
Family I: Outer Ellipse, Inner Circle
Here we study a Poncelet family inscribed in an ellipse centered at O with semi-axes ( a, b ) and circumscribes an concentric circle of radius r , Figure 4 (left). Anexplicit parametrization is provided in Appendix A.2. AMILY TIES: RELATING PONCELET 3-PERIODICS BY THEIR PROPERTIES 5
Figure 3.
The Brocard porism [3] is a 1d Poncelet family of triangles (blue) inscribed in a circle(black, upper half shown) and circumscribed about the Brocard inellipse [27] centered on X andwith foci on the stationary Brocard points Ω and Ω of the family. The Brocard angle is invariant[14]. Video Cayley’s closure condition [6] assumes a simple form for 3-periodics in a concen-tric, axis-aligned pair of ellipses [13]:
Proposition 1.
For 3-periodics in an axis-aligned, concentric ellipse pair: (1) a ′ a + b ′ b = 1 Corollary 1.
For Family I 3-periodics, the radius r of the fixed incircle is givenby: r = aba + b Proposition 2.
In the Family I 3-periodics the locus of the barycenter X is anellipse of axes a = a ( a − b ) / (3 a + 3 b ) and b = b ( a − b ) / (3 a + 3 b ) centered about O = X . Theorem 1.
Family I 3-periodics have invariant circumradius R = ( a + b ) / .Furthermore, the locus of the circumcenter X is a circle of radius d = R − b = a − R about O = X .Proof. Consider the explicit expressions derived for 3-periodic vertices in Appen-dix A.2. Let a first vertex P = ( x , y ) . From this obtain the center X of theorbit’s circumcircle: X = − x ( a − b ) (cid:16) − x ( a + b ) + a b (2 a + b ) (cid:17) a (( a − b ) x + a b ) , ( a − b ) (cid:16) x ( a + b ) − a b (cid:17) y b ( a x + b ( a − x )) and radius ( a + b ) / . Also obtain that the locus of X is a circle with center (0 , and radius ( a − b ) / . (cid:3) RONALDO GARCIA AND DAN REZNIK
Proposition 3.
Over Family I 3-periodics the locus of the orthocenter X is anellipse of axes a = ( a − b ) b/ ( a + b ) and b = ( a − b ) a/ ( a + b ) centered about O = X . Proposition 4.
Over family I 3-periodics the locus of the X triangle center is acircle of radius d = ( a − b ) a + b ) about O = X . Proposition 5.
The power of O with respect to the circumcircle is invariant andequal to − ab .Proof. Direct, analogous to [12, Thm 3]. (cid:3)
Proposition 6.
Over family I 3-periodics, the locus of X is a quartic given by: X : (cid:0) b ( b + 2 a ) (cid:0) a + 2 ab + 3 b (cid:1) x + a ( a + 2 b ) (cid:0) a + 2 ab + b (cid:1) y (cid:1) − a b ( a − b ) (cid:16) b ( b + 2 a ) x + a ( a + 2 b ) y (cid:17) = 0 Connection with the Poristic family.
Below we show that Family I 3-periodics is the image of the Poristic family [18] under a variable rigid rotationabout X .Recall the Poristic family of triangles with fixed, non-concentric incircle andcircumcircle with centers separated by d = p R ( R − r ) [8, 18]. Let I be a (moving)reference frame centered on X with one axis oriented toward X . Referring toFigure 4 (right): Theorem 2.
With respect to I , Family I 3-periodics are the Poristic triangle family(modulo a rigid rotation about X ).Proof. This stems from the fact that R , r , and d are constant. (cid:3) As proved in [10, Thm 3]:
Observation 1.
The X -centered circumconic to the Poristic family is a rigidly-rotating ellipse with axes R + d and R − d . Since this circumellipse is identical (up to rotation) to the outer ellipse of FamilyI, then R + d = a which is coherent with Proposition 1.Furthermore, because poristic triangles are the image of Billiard 3-periodics un-der a (varying) affine transform [10, Thm 4], it displays the same scale-free invari-ants. Corollary 2.
Family I 3-periodics conserve the sum of cosines, product of half-sines, and all scale-free invariants. X i =1 cos θ i = a + 4 ab + b ( a + b ) (2) Y i =1 sin θ i ab a + b ) Note that invariant sum of cosines for family I N-periodics was proved for all N in [1, Corollary 6.4]. In fact: AMILY TIES: RELATING PONCELET 3-PERIODICS BY THEIR PROPERTIES 7
Figure 4.
Family I 3-periodics (left) are identical (up to rotation) to the family of Poristic triangles(right) [8], if the former is observed with respect to a reference system where X and X are fixed.The fixed incircle (resp. circumcircle) are shown purple (resp. blue). The original outer ellipse(black on both drawings) becomes the X -centered circumellipse in the Poristic case. Over thefamily, this ellipse is known to rigidly rotate about X with axes R + d, R − d , where d = | X − X | [10]. Video Theorem 3.
Let ( E I , E ′′ I ) be a confocal pair of ellipses which is an affine image ofa family I pair. Both families have invariant and identical sums of cosines.Proof. Let α, β and α ′′ , β ′′ denote the semi-axes of E I and E ′′ I , respectively. For thepair to admit a 3-periodic family, the latter are given by [9]: α ′′ = α ( δ − β ) α − β , β ′′ = β ( α − δ ) α − β Consider the following affine transformation: T ( x, y ) = ( β ′′ α ′′ x, y ) . This takes E I to an ellipse with semi-axes ( a, b ) , a = α β ′′ α ′′ and b = β and thecaustic E ′′ I to a concentric circle of radius β ′′ .In [12, Thm. 1] the following expression was given for invariant r/R in theconfocal pair:(3) rR = 2( δ − β )( α − δ )( α − β ) , δ = p α − α β + β Recall that for any triangle, P i =1 cos θ i = 1 + r/R [27, Circumradius, Eqn. 4].Plugging a = α β ′′ α ′′ and b = β into to (2) yields (3) plus one. (cid:3) It turns out the proof of [1, Corollary 6.4] implies that for all N, the cosinesum for family I N-periodics is invariant and identical to the one obtained with itsconfocal affine image [1].A known relation for triangles is that
R r = ( s s s ) / (4 s ) , where s , s , s aresidelengths and s = ( s + s + s ) / is the semiperimeter. Since both R and r areconserved: RONALDO GARCIA AND DAN REZNIK
Figure 5.
Family II, the N = 3 case: the loci of both orthocenter X (pink) and nine-point center X (olive green) are concentric with the external circle (black), with radii d ′ and d ′ , respectively.I.e., | X − X | = d ′ . In contradistinction to the elliptic billiard, the locus of the incenter X (dashed brown) is non-elliptic while that of the symmedian point X (dashed blue) is an ellipse.Video. Corollary 3.
The quantity ( s s s ) / (4 s ) is conserved and is equal to ab/ . Family II: Outer Circle, Inner Ellipse
This family is inscribed in a circle of radius R centered on O and circumscribesa concentric ellipse with semi-axes a, b ; see Figure 5. An explicit parametrizationappears in Appendix A.3.For the N = 3 case, (1) implies R = a + b . By definition X is stationary at O and R is the (invariant) circumradius. As shown in Figure 5: Proposition 7.
Over family II 3-periodics, the loci of the orthocenter X and Eulercenter X are concentric circles about X = O , with radii d ′ and d ′ respectively,where d ′ = ( a − b ) / .Proof. CAS-assisted algebraic simplification. (cid:3)
Recall that in the confocal pair the loci of X (resp. X ) is an ellipse (resp. aquartic) [11]; see Appendix C. Interestingly: Proposition 8.
Over family II 3-periodics, the locus of the symmedian point X (resp. the incenter X ) is an ellipse (resp. the convex component of a quartic).These are given by: AMILY TIES: RELATING PONCELET 3-PERIODICS BY THEIR PROPERTIES 9 X : x a + y b = 1 , a = a − b a + 2 b , b = a − b a + bX : (cid:0) x + y (cid:1) − a + 3 b ) ( a + b ) x − a + b ) (3 a + b ) y + (cid:0) a − b (cid:1) = 0 Proof.
CAS-assisted simplification. (cid:3)
Let s i denote the sidelengths of an N -periodic. Theorem 4.
Family II 3-periodics conserve L = P i =1 s i = 4( a + 2 b )(2 a + b ) .Proof. Direct, using the parametrization for vertices in Appendix A.3. (cid:3)
Note: the above is true for all N [1, Thm. 8, corollary].4.1. Family II and the Poristic family.
Below we show that the orthic trianglesof Family II 3-periodics are the image of the Poristic family [18] under a variablerigid rotation about X . Lemma 1.
Family II 3-periodics conserve the product of cosines, given by: Y i =1 cos θ i = ab a + b ) Proof.
CAS-assisted simplification. (cid:3)
The orthic triangle has vertices at the feet a triangle’s altitudes [27]. Let R h denote its circumradius. A known property is that R h = R/ [27, Orthic Triangle,Eqn. 7], therefore it is invariant over family II 3-periodics. Referring to Figure 6(left): Proposition 9.
The inradius r h of family II orthics is invariant and given by r h = ab/ ( a + b ) .Proof. r h = 2 R Q i =1 cos θ i [27, Orthic Triangle, Eqn. 5]. Referring to Lemma 1completes the proof. (cid:3) Let ( E II , E ′′ II ) denote the confocal pair which is an affine image of a circle-inellipseconcentric pair. Let α, β and α ′′ , β ′′ denote the semi-axes of E II , and E ′′ II , respec-tively. Theorem 5.
The invariant product of cosines for family II triangles is identicalto the one obtained from excentral triangles of 3-periodics in ( E II , E ′′ II ) .Proof. Excentrals in the confocal pair conserve the product of cosines [12, Corollary2]. Recall that for any triangle: Y i =1 | cos θ ′ i | = r R where θ ′ i are the angles of the excentral triangle. Plugging a = α ′′ and b = αβ β ′′ into(1) yields four times the above identity when r/R is computed as in (3), completingthe proof. (cid:3) Lemma 2.
Family II 3-periodics are always acute.
Figure 6.
Left:
Family II 3-periodics (blue), and their orthic triangle (red). The latter’s inra-dius and circumradius are invariant. The orthic’s incircle and circumcircle (both dashed red) arecentered on the 3-periodic’s orthocenter X and Euler center X , respectively. Also shown is therigidly-rotating MacBeath inellipse (dashed green), centered on X with foci at X and X . Right:
Family II orthics are identical (up to a variable rotation), to the Poristic triangles (red) [18]. Equiv-alently, the original family is the of poristic excentral triangles (blue), for which both incircle andcircumcircle (solid red) are stationary. Also stationary is the excentral MacBeath inellipse (green),i.e., it is the caustic [10], with center X and foci X , and X , respectively. The original outercircle (black on both images) is also stationary on the poristic case, however the inner ellipse in thePoncelet pair (purple) becomes a rigidly-rotating X -centered excentral inellipse (dashed purple),whose axes are R + d ′ and R − d ′ . Video 1, Video 2 Proof.
Since X is the common center and is internal to the inner circle, it is alwaysinterior to Family II 3-periodics, i.e., the latter are acute. (cid:3) Let I ′ be a (moving) reference frame centered on X with one axis orientedtoward X (or X as these 3 are collinear). Referring to Figure 4 (right): Theorem 6.
With respect to I ′ , family II 3-periodics are the excentral trianglesto the Poristic family (modulo a rigid rotation about X ). Equivalently, family IIorthics are identical (up to said variable rotation) to the poristic triangles.Proof. X is the orthic’s X [16]. Since the family is always acute (Lemma 2), X is the orthic’s X [5]. By Proposition 7, d ′ = | X − X | is invariant, i.e., thedistance between X and X of the orthic is invariant. The claim follows fromnoting X , X , X are collinear [27] and that the orthic inradius and circumradiusare invariant, Proposition 9. (cid:3) Recall from [10, Thm 2]:
Observation 2.
The X -centered inconic to the Poristic excentral triangles is arigidly-rotating ellipse with axes R + d ′ and R − d ′ . Which makes sense when one considers the rotating reference frame. Also recallfrom [10, Thm 1] that:
Observation 3.
The MacBeath Inconic to the excentrals is stationary with axes R and √ R − d ′ . AMILY TIES: RELATING PONCELET 3-PERIODICS BY THEIR PROPERTIES 11
Figure 7.
Family III (Homothetic pair) 3-periodics (blue). Video
Therefore its focal length is simply d ′ = | X − X | . Furthermore, because poris-tic triangles are the image of billiard 3-periodics under a (varying) affine transform[10, Thm 4], Family II 3-periodics will share all scale-free invariants with billiardexcentrals, such as product of cosines, ratio of area to its orthic, etc., see [21].5. Family III: Homothetic
This family is inscribed in an ellipse centered on O with semi-axes ( a, b ) andcircumscribes an homothetic, axis-aligned, concentric ellipse with semi-axes ( a ′′ , b ′′ ) ;see Figure 7. An explicit parametrization is provided in Appendix A.4. Proposition 10.
For Family III 3-periodics, a ′′ = a/ and b ′′ = b/ , the Barycen-ter X is stationary at O and the area A is invariant and given by: A = 3 √ ab Proof.
Family III is the affine image of a family of equilaterals interscribed withintwo concentric circles. The inradius of such a family is half its circumradius.Amongst triangle centers, the barycenter X is uniquely invariant under affine trans-formations; it lies at the origin for an equilateral. Affine transformations preservearea ratios. A is the area of an equilateral triangle inscribed in a unit circle scaledby the Jacobian ab . This completes the proof. (cid:3) A known result is that the cotangent of the Brocard angle cot( ω ) of a triangle isequal to the sum of the cotangents of its three internal angles [27, Brocard Angle,Eqn. 1]. Surprisingly: Proposition 11.
Family III 3-periodics have invariant ω given by: cot ω = X i =1 cot θ i = √ a + b ab Proof.
Direct calculations using the explicit parametrization of vertices in AppendixA.4. (cid:3)
A known relation is cot ω = ( P i =1 s i ) / (4 A ) [27, Brocard Angle, Eqn. 2]. Therefore: Corollary 4.
The sum of squared sidelengths s i is invariant and given by: X i =1 s i = 92 (cid:0) a + b (cid:1) As mentioned above, in the confocal pair the loci of X (resp. X ) is an ellipse(resp. a quartic) [11]; see Appendix C. Interestingly: Proposition 12.
For family III, the locus of the incenter X (resp. symmedianpoint X ) is a quartic (resp. an ellipse). These are given by: X : 16 (cid:0) a y + b x (cid:1) (cid:0) a x + b y (cid:1) − b (cid:0) a + 5 a b + 2 b (cid:1) x − a (cid:0) a + 5 a b + b (cid:1) y + a b (cid:0) a − b (cid:1) = 0 X : x a + y b = 1 , a = a ( a − b )2( a + b ) , b = b ( a − b )2( a + b ) Proof.
CAS-assisted simplification. (cid:3)
Surprising Circular Loci.
The two isodynamic points X and X as wellas the two isogonic points X and X have trilinear coordinates which are irra-tional on a triangle’s sidelengths [16]. In the elliptic billiard their loci are non-elliptic. Indeed, in the elliptic billiard we haven’t yet found any triangle centerswith a conic locus whose trilinears are irrational. Referring to Figure 8, for familyIII this is a surprising fact: Proposition 13.
The loci of of X k , k = ( a − b ) / , ( a + b ) / , ( a − b ) /z , and ( a + b ) /z , respectively, where z = 2( a + b ) . Observation 4.
Over all a/b , the radius of X is minimum when a/b = 3 . Family III and the Brocard Porism.
The Brocard porism [3] is a family oftriangles inscribed in a circle and circumscribed about a special inellipse known asthe “Brocard inellipse” [27, Brocard Inellipse]. Its foci coincide with the stationaryBrocard points of the family. Furthermore, this family conserves the Brocard angle ω . Referring to Figure 7, we showed that over the homothetic family, the aspectratio of the Brocard inellipse is invariant. [19]. This leads to the following result,reproduced from [19, Theorem 3]: Theorem 7.
The 3-periodic family in a homothetic pair and that of the Brocardporisms are images of one another under a variable similarity transform.
As shown in Appendix B.1, the locus of the center X of the Brocard inellipseis an ellipse (it is stationary on the Brocard porism). AMILY TIES: RELATING PONCELET 3-PERIODICS BY THEIR PROPERTIES 13
Figure 8.
Circular loci of the first and second Fermat points X and X (red and green) as wellas the first and second isodynamic points X and X (purple and orange) for two aspect ratiosof the homothetic pair: a/b = 3 (left) and a/b = 5 (right). The radius of the X locus is minimalat the first case. Video Figure 9.
Family III triangles (blue) are the image of Brocard porism triangles under a variablesimilarity transform [19]. This stems fromthe fact that the family’s Brocard inellipse (purple),centered on X and with foci on the Brocard points Ω , Ω , has a fixed aspect ratio. Also shownis the elliptic locus of X . Video Summary
Table 1 summarizes the types of loci (point, circle, ellipse, etc.) for severaltriangle centers for all families mentioned above. These are organized within threegroups A,B,C with closely related loci types. Exceptions are also indicated thoughwe still lack a theory for it.
Group A Group B Group CConf. F.I Por. Conf.Exc F.II Por.Exc. F.III Broc. X E P P X P X 4 X X E E C E C P P C X E C P E P P E P X E E C E C P E C X E C C E C P E C X E P E C E P X E E C X X X X X X E E C X X X X X X P E C X X X X X X E E C X X X X X X E ′′ C ′′ C ′′ X X C X X X E C C X X X X X X X X X X X X C C X X X X X X X C C X X X X X X X C P X X X X X X X C P X X X C ′ X C ′ C ′ E ′ C ′ X E ′ E ′ C ′ X C ′ C ′ X C ′ X X X C ′ E ′ C ′ C ′ X C ′ Table 1.
Types of loci for several triangle centers over several Poncelet triangle families, divided in3 groups A,B,C with closely-related metric phenomena: (i) confocal, fam. I, poristics; (ii) confocalexcentral, fam. II, poristic excentrals; (iii) fam. III and Brocard porism. Symbols P, C, E, and Xindicate point, circle, ellipse, and non-elliptic (degree not yet derived) loci, respectively. A numberrefers to the degree of the non-elliptic implicit, e.g., ’4’ for quartic. A singly (resp. doubly) primedletter indicates a perfect match with the outer (resp. inner) conic in the pair. The symbol C refers to the Euler circle. The boldface entries indicate a discrepancy in the group (see text). Note: X n for the confocal excentrals and poristic excentrals refer to triangle centers of the family itself(not of their reference triangles). The plethora of circles in the Poristic family had already been shown in [18]. Anabove-than-expected frequency of ellipses for the confocal pair was signalled in [11].As mentioned above, irrational centers X k , k ∈ [13 , sweep out circles for thehomothetic pair. X and X are known to be stationary over the Brocard family[3], however X and X are circles! Also noticeable is the fact that (i) though inthe confocal pair X and X is an ellipse and 4, respectively, in both family II andfamily III said locus types are swapped. The reasons remain mysterious.It is well-known that there is a projective transformation that takes any Ponceletfamily to the confocal pair, [6]. In this case only projective properties are preserved.If one restricts the set of possible transformations to either affine ones or similarities(which include rigid transformations), one can construct the two-clique graph ofinterrelations shown in Figure 10.As mentioned above, the confocal family is the affine image of either family Ior family II. In the first (resp. second) case the caustic (resp. outer ellipse) issent to a circle. Though affine group is not conformal, we showed above that bothfamilies conserve their sum of cosines (Theorem 3). One way to see this is that AMILY TIES: RELATING PONCELET 3-PERIODICS BY THEIR PROPERTIES 15
Fam I (Incircle) (X1) (cid:1)(cid:2)(cid:3)
Poris (cid:1) c (X1,X3) (cid:1)(cid:2)(cid:3) Confocal (Billiard) (X9) (cid:4) (cid:1)(cid:2)(cid:3)
Confocal Excentrals(X6) (cid:1)(cid:2)(cid:3)
Fam II (Circumc.) (X3) (cid:1)(cid:2)(cid:3) (cid:5) (cid:6)(cid:7)
Poris (cid:0) c Excentrals(X5,X4,X1) (cid:1)(cid:2)(cid:3) (cid:5) (cid:6)(cid:7)
Rigid Rot SimilarityRigid Rot SimilarityA (cid:2) ne IA (cid:3) ne II Orthic/ExcentralFam II Orthics (X40) (cid:1)(cid:2)(cid:3)
Orthic/Excentral Rigid RotFam III (Homot.) (X2) A, (cid:5) (cid:6)(cid:7) (cid:1)(cid:2)(cid:8) (cid:9) Brocard Porism (X39, X3, Ω , Ω ) (cid:1)(cid:2)(cid:8) , (cid:9) Similarity
Figure 10.
Diagram of transformations that take one 3-periodic family into another. The familiesare specified in each box while the transformations label the arrows. The second (resp. third) linein each box lists the stationary points(s) (resp. main invariants) in the family. there is an alternate, conformal path which takes family I triangles to the confocalones, namely a rigid rotation (yielding poristic triangles), followed by a variablesimilarity (yielding the confocals).A similar argument is valid for family II triangles: there is an affine path (non-conformal) to the confocal family though both conserve the product of cosines(Theorem 5). Notice an alternate conformal composition of rotation (yielding poris-tic excentrals) and a variable similarity (yielding confocal excentrals). All in thispath conserve the product of cosines.Finally, family III and Brocard porism triangles form an isolated clique. Asmentioned in [19] these are variable similarity images of one another but cannot bemappable to the other families via similarities nor affinely.Table 2 summarizes some properties of 3-periodics mentioned herein. The lastcolumn reveals that many of invariants continue to hold for N>3. Animationsillustrating some focus-inversive phenomena are listed on Table 3. fam. pair N=3outer conic N=3inner conic N=3invariants N>3billiard ellipse ( a, b ) confocal caustic L, J, r/R, P cos L, J, P cos I inner circle ellipse ( a, b ) circle r = aba + b R, r/R, P cos P cos II outer circle R = ( a + b ) ellipse ( a, b ) P s i , Q cos P s i , Q cos III homothetic ellipse ( a, b ) ellipse ( a/ , b/ A, P s i , ω, P cot A, P s i , P cot Table 2.
Summary of properties across different concentric Poncelet families. The last columnshows some invariants which continue to hold for N>3. id family N Title youtu.be/...
01 all 3 Concentric Poncelet families
02 por. 3 Chapple’s Poristic family & excentrals
DS4ryndDK6Qo
03 por. 3 Poristics are image of billiard 3-periodics
NvjrX6XKSFw
04 I 3 Side-by-side w/ the poristic family
ML_AZoX736w
05 I 3,5 Circular loci of X3 & Steiner’s curvature centroid
06 I 3,5 Invariant ratio of sidelength product to sum
07 II 3 Family is image of poristic excentrals wUu2iMesv3U
08 II 3 Side-by-side w/the poristic family xM1SAZO9bDc
09 II 3,5 Circular locus of generalized orthocenter
10 III 3 Stationary X and invariant Brocard angle
11 III 3 Loci of X k , k = ZwTfwaJJitE
12 III 3 Family is image of Brocard porism h3GZz7pcJp0
13 I,II 5,6 Locus of generalized circum- and orthocenter
ZfQEDujbirQ
14 I,II 5 Locus of generalized circumcenter
RP18B827l5I
15 I,II 5 Generalized circumcenter (Steiner’s curv. centroid)
RP18B827l5I
16 dual 3 The dual pair: stationary orthocenter fpd_Zot5cKk
17 dual 3–8 Generalized stationary orthocenter ttKjzWeG5B8
18 dual 5,7 Generalized stationary orthocenter gNHiZvBhKF8
Table 3.
Videos illustrating some phenomena mentioned herein. The last column is clickable andprovides the YouTube code.
Appendix A. Explicit 3-Periodic Vertices
A.1.
Pair 0: Confocal.
Let ( a, b ) be the semi-axes of the external ellipse. Let P i = ( x i , y i ) /q i , i = 1 , , , denote the 3-periodic vertices, given by [9]: AMILY TIES: RELATING PONCELET 3-PERIODICS BY THEIR PROPERTIES 17 q = 1 x = − b (cid:0)(cid:0) a + b (cid:1) k − a (cid:1) x − a b k x y + a (cid:0) ( a − b ) k + b (cid:1) x y − a k y y = 2 b k x + b (cid:0) ( b − a ) k + a (cid:1) x y + 2 a b k x y − a (cid:0)(cid:0) a + b (cid:1) k − b (cid:1) y q = b (cid:0) a − c k (cid:1) x + a (cid:0) b + c k (cid:1) y − a b c k x y x = b (cid:0) a − (cid:0) b + a (cid:1)(cid:1) k x + 2 a b k x y + a (cid:0) k (cid:0) a − b (cid:1) + b (cid:1) x y + 2 a k y y = − b k x + b (cid:0) a + (cid:0) b − a (cid:1) k (cid:1) x y − a b k x y + a (cid:0) b − (cid:0) b + a (cid:1) k (cid:1) y ,q = b (cid:0) a − c k (cid:1) x + a (cid:0) b + c k (cid:1) y + 2 a b c k x y . where: k = d δ d = cos α,k = δ d d q d − d δ = sin α cos αc = a − b , d = ( a b/c ) , d = b x + a y δ = p a + b − a b , δ = p δ − a − b where α , though not used here, is the angle of segment P P (and P P ) withrespect to the normal at P .A.2. Pair I: Incircle. P ( t ) = ( x , y ) = ( a cos t, b sin t ) .Then P i = ( x i , y i ) , i = 2 , are: x =2 a b (cid:0) − a bx + k y (cid:1) /q y = − ab (cid:0) a by + k x (cid:1) /q x = − a b (cid:0) a bx + k y (cid:1) /q x =2 b a (cid:0) − a by + k x (cid:1) /q k = q a ( a + 2 b ) x + a b (2 a + b ) y q =2 b ( a + b )(( a − b ) x + a b ) q = (cid:0) b a − y a + 2 a b + a b x − x b (cid:1) ( a + b ) A.3.
Pair II: Inellipse. P ( t ) = ( x , y ) = R (cos t, sin t ) with R = a + b . Then the P i = ( x i , y i ) , i = 2 , are given by: x = (cid:0) − b x + y s x (cid:1) k x y = − (cid:0) y a + x s y (cid:1) k y x = − (cid:0) b x + y s x (cid:1) k x y = (cid:0) − y a + x s y (cid:1) k y s x = q a ( a + 2 b ) − ( a − b ) x s y = q ( a − b ) y + b (2 a + b ) k x = a ( − a + b ) x + a ( a + b ) k y = b ( a − b ) y + b ( a + b ) A.4.
Pair III: Homothetic. P ( t ) = ( x , y ) = ( a cos t, b sin t ) .Then P i = ( x i , y i ) , i = 2 , are: ( x , y ) = √ ay − bx b , −√ bx − ay a ! ( x , y ) = −√ ay − bx b , √ bx − ay a ! Appendix B. Elliptic Loci
Below we list triangle centers amongst X k , k = 1 , . . . , for each of the Ponceletpairs mentioned in this article, whose loci are either ellipses or circles. •
0. Confocal pair (stationary X ) – Ellipses: 1, 2, 3, 4, 5, 7, 8, 10, 11, 12, 20, 21, 35, 36, 40, 46, 55, 56, 57,63, 65, 72, 78, 79, 80, 84, 88, 90, 100, 104, 119, 140, 142, 144, 145, 149,153, 162, 165, 190, 191, 200. Note: the first 29 in the list were provedin [11]. – Circles: n/a • I. Incircle: (stationary X ) – Ellipses: 2, 3, 4, 7, 8, 9, 10, 20, 21, 30, 63, 72, 78, 79, 84, 90, 100, 104,140, 142, 144, 145, 149, 153, 170, 176, 191, 200. – Circles: 3, 5, 11, 12, 35, 36, 40, 46, 55, 56, 57, 65, 80, 119, 165. • II. Inellipse: (stationary X ) – Ellipses: 6, 15, 21, 27, 28, 39, 49, 51, 52, 54, 58, 61, 64, 66, 67, 68, 69,70, 113, 125, 141, 143, 146, 154, 155, 159, 161, 182, 184, 185, 193, 195,199. – Circles: 2, 4, 5, 20, 22, 23, 24, 25, 26, 74, 98, 99, 100, 101, 102, 103,104, 105, 106, 107, 108, 109, 110, 111, 112, 140, 156, 186, 201. • III. Homothetic: (stationary X ) – Ellipses: 3, 4, 5, 6, 17, 18, 20, 32, 39, 61, 62, 69, 76, 83, 98, 99, 114,115, 140, 141, 147, 148, 182, 187, 190, 193, 194. – Circles: 13, 14, 15, 16.
AMILY TIES: RELATING PONCELET 3-PERIODICS BY THEIR PROPERTIES 19
B.1.
Homothetic pair: semi-axes of elliptic loci.
Below are the semi-axeslengths of triangle centers for 3-periodics in System III (homothetic pair) whoseloci are ellipses. a = a − b a , b = a − b ba = a − b a , b = a − b ba = a − b a , b = a − b ba = a ( a − b )2( a + b ) , b = b ( a − b )2( a + b ) a = a − b , b = a − b a = a + b , b = a + b a = ( a − b ) a + b ) , b = ( a − b ) a + b ) a = ( a + b ) a − b ) , b = ( a + b ) a − b ) a = a − b a + 6 b ) , b = a − b a + 2 b ) a = a − b a − b ) , b = a − b a − b ) a = a − b a , b = a − b ba = a (cid:0) a − b (cid:1) (cid:0) a + 5 b (cid:1) a + 2 a b + 3 b ) b = b (cid:0) a − b (cid:1) (cid:0) a + 5 b (cid:1) a + 2 a b + 3 b ) a = (cid:0) a − b (cid:1) a a + 3 b ) , b = (cid:0) a − b (cid:1) b a + b ) a = (cid:0) a − b (cid:1) (3 a − b )2(3 a + 2 ab + 3 b ) b = (cid:0) a − b (cid:1) ( a − b )2(3 a + 2 ab + 3 b ) a = (cid:0) a − b (cid:1) (3 a + b )2(3 a − ab + 3 b ) b = (cid:0) a − b (cid:1) ( a + 3 b )2(3 a − ab + 3 b ) a = (cid:0) a − b (cid:1) aa + b , b = (cid:0) a − b (cid:1) ba + b a = (cid:0) a − b (cid:1) aa + 3 b , b = (cid:0) a − b (cid:1) b a + b AMILY TIES: RELATING PONCELET 3-PERIODICS BY THEIR PROPERTIES 21 a = (cid:0) a − b (cid:1) a a + 3 b , b = (cid:0) a − b (cid:1) b a + 5 b a = a + b a , b = a + b ba = a, b = ba = a + b a , b = a + b ba = a , b = b a = a − b a , b = a − b ba = ( a − b ) a a + b ) , b = ( a − b ) b a + b ) a = a + b a , b = a + b ba =2 a, b = 2 ba = ( a − b ) a ( a + b ) , b = ( a − b ) b ( a + b ) a = a ( a + 3 b )2( a − b ) , b = b (3 a + b )2( a − b ) a = a, b = ba = 2( a − b ) aa + b , b = 2( a − b ) ba + b a = 2( a − b ) aa + 3 b , b = 2( a − b ) b a + b Appendix C. Loci of Incenter and Symmedian in the Elliptic Billiard
Over 3-periodics in the elliptic billiard ,the locus of the incenter X is an origincentered ellipse with axes a , b given by [9]: a = δ − b a , b = a − δb Over the same family, the locus of X is a convex quartic given by [11, Theorem2]: X ( x, y ) = c x + c y + c x y + c x + c y = 0 where: c = b (5 δ − a − b ) δ − a b ) c = a (5 δ + 4( a − b ) δ − a b ) c = 2 a b ( a b + 3 δ ) c = a b (3 b + 2(2 a − b ) δ − δ ) c = a b (3 a + 2(2 b − a ) δ − δ ) δ = √ a − a b + b Appendix D. Table of Symbols symbol meaning note O center of concentric pair a, b ellipse semi-axes s i , s sidelength and semiperimeter i = 1 , . . . Nθ i internal angle L perimeter P i s i L sum of squared sidelengths P i s i K Steiner’s Curvature Centroid P i w i P i / P i w i w i = sin(2 θ i ) r, R inradius, circumradius d ′ | X − X | r h , R h inradius, circumradius of ortic ω Brocard angle tan( ω ) = 4 A/L X incenter X barycenter X circumcenter X orthocenter X center of 9-point circle Table 4.
Symbols used.
References [1] Akopyan, A., Schwartz, R., Tabachnikov, S. (2020). Billiards in ellipses revisited.
Eur. J.Math. doi.org/10.1007/s40879-020-00426-9 . 1, 3, 6, 7, 9[2] Bialy, M., Tabachnikov, S. (2020). Dan Reznik’s identities and more.
Eur. J. Math. doi.org/10.1007/s40879-020-00428-7 . 1, 3[3] Bradley, C., Smith, G. (2007). On a construction of Hagge.
Forum Geometricorum , 7: 231––247. forumgeom.fau.edu/FG2007volume7/FG200730.pdf . 1, 3, 4, 5, 12, 14[4] Chavez-Caliz, A. (2020). More about areas and centers of Poncelet polygons.
Arnold Math J. doi.org/10.1007/s40598-020-00154-8 . 1, 3[5] Coxeter, H. S. M., Greitzer, S. L. (1967).
Geometry Revisited , vol. 19 of
New MathematicalLibrary . New York: Random House, Inc. 10[6] Dragovi´c, V., Radnovi´c, M. (2011).
Poncelet Porisms and Beyond: Integrable Billiards, Hy-perelliptic Jacobians and Pencils of Quadrics . Frontiers in Mathematics. Basel: Springer. books.google.com.br/books?id=QcOmDAEACAAJ . 5, 14[7] Fierobe, C. (2018). On the circumcenters of triangular orbits in elliptic billards.arXiv:1807.11903. 1, 3[8] Gallatly, W. (1914).
The modern geometry of the triangle . Francis Hodgson. 1, 4, 6, 7[9] Garcia, R. (2019). Elliptic billiards and ellipses associated to the 3-periodic orbits.
AmericanMathematical Monthly , 126(06): 491–504. 1, 3, 7, 16, 21[10] Garcia, R., Reznik, D. (2020). Related by similarity I: Poristic triangles and 3-periodics inthe elliptic billiard. Submitted. 2, 4, 6, 7, 10, 11[11] Garcia, R., Reznik, D., Koiller, J. (2020). Loci of 3-periodics in an elliptic billiard: why somany ellipses? Submitted. 1, 2, 3, 4, 8, 12, 14, 18, 21[12] Garcia, R., Reznik, D., Koiller, J. (2020). New properties of triangular orbits in ellipticbilliards.
Amer. Math. Monthly , to appear. 2, 6, 7, 9[13] Georgiev, V., Nedyalkova, V. (2012). Poncelet’s porism and periodic triangles in ellipse.
Dy-namat . . 5 AMILY TIES: RELATING PONCELET 3-PERIODICS BY THEIR PROPERTIES 23 [14] Johnson, R. A. (1929).
Modern Geometry: An Elementary Treatise on the Geometry of theTriangle and the Circle . Boston, MA: Houghton Mifflin. 1, 4, 5[15] Kaloshin, V., Sorrentino, A. (2018). On the integrability of Birkhoff billiards.
Phil. Trans. R.Soc. , A(376). 3[16] Kimberling, C. (2020). Encyclopedia of triangle centers.
ETC . faculty.evansville.edu/ck6/encyclopedia/ETC.html . 1, 3, 10, 12[17] Miller, N. (1925). A Generalization of a Property of Confocal Conics. Amer. Math. Monthly ,32(4): 178–180. https://doi.org/10.2307/2300245 . 3[18] Odehnal, B. (2011). Poristic loci of triangle centers.
J. Geom. Graph. , 15(1): 45–67. 2, 4, 6,9, 10, 14[19] Reznik, D., Garcia, R. (2020). Related by similarity II: Poncelet 3-periodics in the homotheticpair and the brocard porism. Submitted. 3, 4, 12, 13, 15[20] Reznik, D., Garcia, R., Koiller, J. (2020). Can the elliptic billiard still surprise us?
MathIntelligencer , 42: 6–17. rdcu.be/b2cg1 . 1, 2[21] Reznik, D., Garcia, R., Koiller, J. (2021). Eighty new invariants of n-periodics in the ellipticbilliard.
Arnold Math. J.
ArXiv:2004.12497. 11[22] Romaskevich, O. (2014). On the incenters of triangular orbits on elliptic billiards.
Enseign.Math. , 60(3-4): 247–255. arxiv.org/pdf/1304.7588.pdf . 1, 3[23] Romaskevich, O. (2019). Proof the mittenpunkt is stationary. Private Communication. 1[24] Schwartz, R., Tabachnikov, S. (2016). Centers of mass of Poncelet polygons, 200 years after.
Math. Intelligencer , 38(2): 29–34. . 1, 3[25] Tabachnikov, S. (2005).
Geometry and Billiards , vol. 30 of
StudentMathematical Library . Providence, RI: American Mathematical Society. . Mathematics AdvancedStudy Semesters, University Park, PA. 1, 3[26] Tabachnikov, S., Tsukerman, E. (2015). Remarks on the circumcenter of mass.
Arnold Math-ematical Journal , 1: 101–112. 3[27] Weisstein, E. (2019). Mathworld.
MathWorld–A Wolfram Web Resource . mathworld.wolfram.commathworld.wolfram.com