Poncelet Propellers: Invariant Total Blade Area
aa r X i v : . [ m a t h . M G ] J a n PONCELET PROPELLERS:INVARIANT TOTAL BLADE AREA
DOMINIQUE LAURAIN, DANIEL JAUD, AND DAN REZNIK
Abstract.
Given a triangle, a trio of circumellipses can be defined, eachcentered on an excenter. Over the family of Poncelet 3-periodics (triangles)in a concentric ellipse pair (axis-aligned or not), the trio resembles a rotatingpropeller, where each “blade” has variable area. Amazingly, their total areais invariant, even when the ellipse pair is not axis-aligned. We also prove aclosely-related invariant involving the sum of blade-to-excircle area ratios. Introduction
Invariants of Poncelet N-periodics in various ellipse pairs have been a recent focusof research [9, 6, 5]. For the confocal pair alone (elliptic billiard), 80+ invariantshave been catalogued [10], and several proofs have ensued [1, 3, 4].We start by describing an invariant manifested by Poncelet 3-periodics (triangles)inscribed in an ellipse and circumscribed about a concentric circle; see Figure 1. Forsuch a pair to admit a 3-periodic family, its axes are constrained in a simple way,explained below.
Date : January, 2021.
Figure 1.
A Poncelet 3-periodic (blue) inscribed in an outer ellipse (black) and circumscribedabout an inner concentric circle (brown). Over the family, the circumrcircle (dashed blue) hasconstant circumradius R = ( a + b ) / [5, Thm 1]. Video, app1 DOMINIQUE LAURAIN, DANIEL JAUD, AND DAN REZNIK Figure 2.
A Poncelet 3-periodic (blue) is shown interscribed between an outer ellipse (black) andan inner concentric circle (brown). By definition, the incenter X is stationary at the commoncenter. The excentral triangle (green) is the anti-cevian with respect to X . A circumellipse(orange) centered on an excenter P ′ is shown. Video By definition, the incenter X of this family is stationary at the common centerand the inradius r is fixed. Remarkably, the circumradius R is also invariant, andthis implies the sum of cosines of its internal angles is as well [5].Referring to Figure 2, consider a circumellipse [11, Circumellipse] of a 3-periodicin the incircle pair centered on one of the excenters, i.e., a vertex of the excentraltriangle [11, Excentral Triangle]. Main Result.
Referring to Figure 3, over 3-periodics in the concentric ellipse pairwith an incircle, the total area of the three excenter-centered circumellipses (bladesof a “Poncelet propeller”) is invariant.We then extend this property to a 3-periodic family interscribed in any pair ofconcentric ellipses, including non-axis-aligned; see Figure 4. The only proviso isthat the circumellipses be centered on the vertices of the anticevian triangle withrespect to the common center [11, Anticevian Triangle].
Structure of Article.
In Section 2 we derive the areas for the 3 circumellipsescentered on the vertices of the anticevian triangle with respect to some point X . InSection 3 we apply those formulas for 3-periodics in a pair with incircle. In Section 4we generalize the result to an unaligned concentric ellipse pair. In Section 5 weprove a related invariant contributed by L. Gheorghe [8] involving area ratios ofcircumellipses and excircles. We encourage the reader to watch some of the videosmentioned herein listed in Section 6. ONCELET PROPELLERS: INVARIANT TOTAL BLADE AREA 3
Figure 3.
The three excenter-centered circumellipses (propeller blades) are shown (orange, pink,light blue) centered on the 3 excenters P ′ i , i = 1 , , . Our main result is that over the 3-periodicPoncelet “incircle” family, the sum of their areas is invariant. Video Areas of Anticevian Circumellipses
Let T = P i , i = 1 , , be a reference triangle. Let C X denote the circumellipse[11, Circumellipse] centered on point X interior to T and let ∆ X denote its area.Let the trilinear coordinates of X be [ u, v, w ] [11, Trilinear Coordinates]. Let T X denote the anticevian triangle of T wrt X , and P ′ i its vertices [11, AnticevianTriangle]. Let C ′ i denote the three circumellipses centered on P ′ i . Let ∆ i denotetheir areas, respectively. DOMINIQUE LAURAIN, DANIEL JAUD, AND DAN REZNIK
Lemma 1.
The areas of the aforementioned circumellipses are given by: ∆ X = zs s s uvws u + s v + s w (1) ∆ = zs s s uvw − s u + s v + s w ∆ = zs s s uvws u − s v + s w ∆ = zs s s uvws u + s v − s w z = π s ( s + s + s )( − s + s + s )( s − s + s )( s + s − s )( s u + s v + s w )( − s u + s v + s w )( s u − s v + s w )( s u + s v − s w ) Proof.
Let the trilinears of X be [ u : v : w ] . The trilinears of P ′ i are known tobe [ − u : v : w ] , [ u : − v : w ] and [ u : v : − w ] , see [11, Anticevian Triangle]. Theexpressions are obtained from the formula of the area of a circumellipse centeredon X = [ u : v : w ] [11, Circumellipse]. ∆ X = zs s s uvws u + s v + s w (cid:3) The above also implies:
Corollary 1.
For any triangle the following relation holds: o = 1∆ + 1∆ + 1∆ Note: this is reminiscent of the well-known property /r = P /r i where r is theinradius and r i are the exradii of a triangle [11, Excircles].From direct calculations: Lemma 2.
Let Σ X denote the sum ∆ + ∆ + ∆ . This is given by: (2) Σ X = ( s u + s v + s w − s s uv − s s uw − s s vw ) s s s uvwz ( s u + s v − s w )( s u − s v + s w )( s u − s v − s w ) Poncelet Family with Incircle
Let F denote the 1d Poncelet family of 3-periodics inscribed in an outer ellipse E with semi-axes a, b and circumscribed about a concentric circle E ′ with radius r .Assume a > b > r > . Let O denote the common center.Recall Cayley’s condition for the existence of a 3-periodic family interscribedbetween two concentric, axis-aligned ellipses [7]: a c a + b c b = 1 where a c and b c are the major and minor semi-axes of the inscribed ellipse corre-sponding to the caustic of the Poncelet family. For F , a c = b c = r , then ar + br = 1 ,i.e.: r = aba + b . ONCELET PROPELLERS: INVARIANT TOTAL BLADE AREA 5
By definition, the incenter X of F triangles is stationary at O . Also by definition,the inradius r is fixed.Also known is the fact that the circumradius R of F triangles is invariant andgiven by [5]: R = ab r = a + b . Let ρ = rR denote the ratio r/R . From the above it is invariant and given by: ρ = rR = 2 r ab . Taking X = O , note that the area ∆ o = πab of E is by definition, invariant. Alsonote the anticevian T o in this case is the excentral triangle [11, Excentral Triangle]. Lemma 3.
Over F , Σ o is invariant and given by: Σ o = (cid:18) ρ (cid:19) ∆ o Proof.
By using well-known formulas for r and R [11, Inradius,Circumradius], onecan express ρ in terms of the sidelengths: ρ = ( s + s − s )( s − s + s )( − s + s + s )2 s s s Using the trilinears for the incenter O = X = [ u, v, w ] = [1 : 1 : 1] in (2), directcalculations yield the claim. (cid:3) Generalizing the Result
It turns out Lemma 3 can be generalized to a larger class of Poncelet 3-periodicfamilies.Let ( E , E ′ ) be a generic pair of concentric ellipses (axis-aligned or not), admittinga Poncelet 3-periodic family. Let O be their common center. Let C ′ i be the circum-ellipses centered on the vertices of the anticevian with respect to O . Referring toFigure 4: Theorem 1.
Over the 3-periodic family interscribed in ( E , E ′ ) , Σ o is invariant.Proof. ( E , E ′ ) can be regarded as an affine image of the original family F (withincircle). Let matrix A represent the required affine transform. Since conics areequivariant under affine transformations of the 5 constraints that define them (inour case, passing through the vertices of the 3-periodic and being centered on ananticevian vertex) [2], the area of each circumellipse will scale by det( A ) , so byLemma 3 the result follows. (cid:3) In Figure 5, a few other concentric, axis-aligned pairs are shown which illustratethe above corollary.
Observation 1. If ( E , E ′ ) are homothetic and concentric, then each of the ∆ i areconstant. In fact, they are equivariant under projective transformations [2].
DOMINIQUE LAURAIN, DANIEL JAUD, AND DAN REZNIK
Figure 4.
Consider Poncelet 3-periodics (blue) interscribed in a concentric, pair of ellipses whichin general is not axis-aligned. The total area of the three circumellipses (orange, light blue, pink)centered on the vertices of the O -anticevian triangle (green) is invariant. Video This arises from the fact that this pair is affinely-related to a pair of concentriccircles. The associated Poncelet 3-periodic is then given by an equilateral trianglewhere ∆ = ∆ = ∆ .5. Circumellipses Meet Excircles
Referring to Figure 6, L. Gheorghe detexted experimentally that the sum of arearatios of excircles to excentral circumellipses is invariant for two Poncelet families(see below) [8]. Below we prove this and derive explicit values for the invariants.As before, let ∆ i denote the area of a circumellipse centered on the ith excenter.Let Ω i denote the area of the ith excircle. Theorem 2.
Over the 3-periodic family F (with incircle): ∆ Ω + ∆ Ω + ∆ Ω = 2 ρ Proof.
The exradii are given by [11, Excircles]: r = Ss − s , r = Ss − s , r = Ss − s where s = s + s + s and S are semi-perimeter and area of triangle P P P , respec-tively. From them obtain the excircle areas Ω i : ONCELET PROPELLERS: INVARIANT TOTAL BLADE AREA 7
Figure 5.
A picture of four 3-periodic families. The circumellipses are centered on vertices of theanticevian with respect to the center. Since each case is affinely related to F , the total circumellipsearea is invariant. For the homothetic pair (bottom left), the area of each circumellipse is invariant.Video Figure 6.
Left : The pair with incircle centered on X . Shown also are a 3-periodic (blue), theexcentral triangle (solid green), the excircles (dashed green), and the three excentral circumellipses(magenta, light blue, and orange). Right : the same arrangement for the confocal pair, centeredon X . DOMINIQUE LAURAIN, DANIEL JAUD, AND DAN REZNIK Ω = π s ( s − s )( s − s ) s − s Ω = π s ( s − s )( s − s ) s − s Ω = π s ( s − s )( s − s ) s − s For i = 1 , , , the following can be derived: ∆ i Ω i = µ ( s − s i ) with µ = s s s s ( s − s )( s − s )( s − s ) . Therefore: X i =1 ∆ i Ω i = µs = 2 ρ Recall ρ is constant for family F so the result for the pair incircle follows. (cid:3) Using an analogous proof method:
Theorem 3.
Over the 3-periodic family interscribed in the confocal the followingquantity is also invariant: ∆ Ω + ∆ Ω + ∆ Ω = 2 ρ Observation 2.
Although the pair with incircle and the confocal pair are affinely-related, neither Theorem 2 nor 3 for any other ellipse pair in the affine continuum. List of Videos
Animations illustrating some of the above phenomena are listed on Table 1. id Title youtu.be/<.>
01 3-Periodic incircle family has invariant R eIxb1so6ORo
02 Family of incircle 3-periodics and one excentral circumellipse
JUCmAMsfdkI
03 Three excentral circumellipses with invariant total area ub4wAv8Hgb0
04 3-periodics in 5 concentric, axis-aligned ellipse pairs
05 Incircle family tHUDfx9o0Wg
06 Four Poncelet families crXxPJ93ZDk
07 Non-concentric ellipse pair
FJXMpUcslaA
Table 1.
Videos of some focus-inversive phenomena. The last column is clickable and providesthe YouTube code.
We would like to thank Liliana Gheorghe, Ronaldo Garcia, and Arseniy Akopyanfor valuable discussions and contributions.
ONCELET PROPELLERS: INVARIANT TOTAL BLADE AREA 9
Appendix A. Table of Symbols symbol meaning E , E c outer and inner ellipses a, b outer ellipse semi-axes’ lengths a c , b c inner ellipse semi-axes’ lengths O common center of ellipses P , P , P s , s , s P ′ , P ′ , P ′ vertices of the anticevian wrt O C X , ∆ X X -centered circumellipse and its area C ′ i , ∆ i P ′ i -centered circumellipse and its area T X anticevian wrt X Σ X area sum of 3 circumellipses centeredon the vertices of T X r, R ρ ratio r/R [ u : v : w ] trilinear coordinates Ω i area of the excircle corresponding to P i Table 2.
Symbols of euclidean geometry used
References [1] Akopyan, A., Schwartz, R., Tabachnikov, S. (2020). Billiards in ellipses revisited.
Eur. J.Math. doi:10.1007/s40879-020-00426-9 . 1[2] Akopyan, A. V., Zaslavsky, A. A. (2007).
Geometry of Conics . Providence, RI: Amer. Math.Soc. 5[3] Bialy, M., Tabachnikov, S. (2020). Dan Reznik’s identities and more.
Eur. J. Math. doi:10.1007/s40879-020-00428-7 . 1[4] Chavez-Caliz, A. (2020). More about areas and centers of Poncelet polygons.
Arnold Math J. doi:10.1007/s40598-020-00154-8 . 1[5] Garcia, R., Reznik, D. (2020). Family ties: Relating poncelet 3-periodics by their properties.arXiv:2012.11270. 1, 2, 5[6] Garcia, R., Reznik, D., Koiller, J. (2020). New properties of triangular orbits in ellipticbilliards. arxiv.org/abs/2001.08054 . 1[7] Georgiev, V., Nedyalkova, V. (2012). Poncelet’s porism and periodic triangles in ellipse.
Dy-namat . . 4[8] Gheorghe, L. (2020). Personal communication. 2, 6[9] Reznik, D., Garcia, R., Koiller, J. (2019). Can the elliptic billiard still surprise us? MathIntelligencer , 42. rdcu.be/b2cg1 . 1[10] Reznik, D., Garcia, R., Koiller, J. (2021). Eighty new invariants of n-periodics in the ellipticbilliard.
Arnold Math. J.
ArXiv:2004.12497. 1[11] Weisstein, E. (2019). Mathworld.
MathWorld–A Wolfram Web Resource . mathworld.wolfram.commathworld.wolfram.com