Bar Billiards and Poncelet's Porism

Bar Billiards and Poncelet's Porism

WITOLDMOZGAWA(*)

ABSTRACT- Weprovethat for a given ovalCthere exist ovalsCinandCout, insideand outsideof C, such that thepairs (C, Cin) and (Cout,C) satisfy the Poncelet's porism for almost any number of reflections in their bar billiards.

1. Introduction.

Let us consider two ovalsC1 andC2, i.e., two smooth strictly convex closed plane curves. Let the ovalC₂lieinsidethesecond ovalC₁. From any point onC₁, draw a tangent to C₂ and extend it toC₁ in theoppositedi- rection. From this point we draw another tangent, etc. For all tangents, the resulting Poncelet's transverse will be called a bar billiard since it is similar to a traditional game played years ago. In general in this game players scored points by knocking balls into the holes while avoiding toppling a skittlein themiddleof thetable. Heretheroleof theskittleis played by theovalC₂ and it must be ``toppled'' by the tangent line. The general be- havior of such billiards is worth investigating but in this paper we will concentrate on the Poncelet's porism for this specific setting. Namely, we say that a bar billiard has a Poncelet's porism property if the following is true: if, on the oval C₁, there is one point of origin for which a Poncelet transverse is closed, then the transverse will also close for any other point of origin on the oval. One can find an extensive bibliography on Poncelet's porism in [4]; a nice introduction to theory of billiards is provided in [3].

In this paper we prove that for a given ovalC, in a suitably smallC²- neighborhood there exist ovalsC_inandC_outinsideand outsideofC, such that for a suitable large number of segments the pairs (C;C_in) and (C_out;C)

(*)Indirizzo dell'A.:InstytutMatematyki, UniwersytetMarii Curie-Sklodowskiej,⁶ pl. Marii Curie-Sklodowskiej 1, 20-031 Lublin, Poland⁶

E-mail: [email protected]

1991 Mathematics Subject Classification: 53A04, 51M15, 37E99.

satisfy Poncelet's porism in the above sense. In the course of the proof we bring into play the isoptics introduced and developed in [1] and [2].

2. Outer oval.

Let us begin with a simple algebraic lemma

LEMMA2.1. Let v;w2R²andhv;wiand[v;w]denote the dot product and the determinant of these vectors. Then

(1) vhv;wi whw;vi ˆ[v;w]J(w) (2) hv;wi ˆ[J(w);v],

where J is the positive rotation of anglep=2.

Next we consider an operator D

Daof theform Dp(t)

Da ˆp(t‡a) p(t) cosa sina

acting on real functions defined on allR. From the elementary calculus it is clear that, for anyC¹-functionp,

lima!0

Dp(t) Da ˆp⁰(t) for anyt2R.

In what follows we give a preview of certain facts concerning isoptics as thefundamental tool in our constructions.

DEFINITION2.1. A plane, closed, simple, positively oriented, curve of positive curvature is called an oval.

Wetakea coordinatesystem with originOin theinterior ofC. Le tp(t), t2[0;2p], bethedistancefromOto thesupport linel(t) ofCperpendicular to thevectore^itˆ cost‡isint. It is well-known thatp(t) is of classC¹and that theparametrization ofCin terms ofp(t) is given by the formula

z(t)ˆp(t)e^it‡p⁰(t)ie^it; (2:1)

where ie^itˆ sint‡icost. Notethat thesupport functionp can beex- tended to a periodic function onRwith theperiod 2p.

DEFINITION 2.2. Let Ca be a locus of vertices of a fixed angle p a formed by two support lines of the oval C. The curve Cawill be called ana- isoptic of C.

It is convenient to parametrize thea-isopticCa by thesameangletso that theequation ofCatakes the form

za(t)ˆp(t)e^it‡Dp(t) Da ie^it: (2:2)

Since the functions involved are at least of classC¹ weobservethat the given parametrization of an isoptic is of classC¹. It can beshown that it is a regular curve. Note that

za(t)ˆz(t)‡l(t;a)ie^itˆz(t‡a)‡m(t;a)ie^i(t‡a) (2:3)

z⁰_a(t)ˆ l(t;a)e^it‡r(t;a)ie^it (2:4)

for suitablefunctionsl,mandr. Moreover, we have l(t;a)ˆDp(t)

Da p⁰(t);

(2:5)

r(t;a)ˆp(t)‡Dp⁰(t) Da : (2:6)

COROLLARY2.1. With the notations above we have 1) lim

a!0z_a(t)ˆz(t);

2) lim

a!0l(t;a)ˆ0;

3) lim

a!0r(t;a)ˆR(t), where R(t)ˆp(t)‡p⁰⁰(t) is the curvature ra- dius of the oval C at t.

LEMMA2.2. The function

k_a(t)ˆ ka(t); 0<a<p k(t); aˆ0

(2:7)

is continuous from the right ona, wherek_a(t)is the curvature of the isoptic za(t)at t.

PROOF. It follows immediately from the equation of an isoptic that z⁰_a(t)ˆ l(t;a)e^it‡r(t;a)ie^it

(2:8)

and

jz⁰_a(t)j²ˆl²(t;a)‡r²(t;a):

(2:9)

The latter formula ensures that the isoptics of ovals are always regular curves. By the above considerations we get

a!0limz⁰_a(t)ˆlim

a!0 p⁰(t) Dp(t) Da

e^it‡ p(t)‡Dp⁰ Da

ie^it

(2:10)

ˆ…p(t)‡p⁰⁰(t)†ie^itˆz⁰(t) and

lima!0jz⁰_a(t)j² ˆlim

a!0 p⁰(t) Dp(t) Da

₂

‡ p(t)‡Dp⁰ Da

₂

" #

(2:11)

ˆ…p(t)‡p⁰⁰(t)†²ˆ jz⁰(t)j²: Theformula for curvaturek_a(t) ([2]) of an isoptic curvecan betransformed to thefollowing

k_a(t)ˆ 1 l²‡r²

³₂ 2l²‡2r² R(t)r‡lR(t) cota lR(t‡a) sina

(2:12)

ˆ 1

l²‡r²

³₂ 2l²‡2r² R(t)r lDR(t) Da

:

Hence,

lima!0k_a(t)ˆ 1

R(t)ˆk(t):

(2:13)

p COROLLARY 2.2. There exists e>0 such that if 0<a<e then ka(t)>0for any t2R.

THEOREM2.1. For any oval C there exists an oval O containing oval C inside such that the corresponding bar billiard has the Poncelet's porism property.

PROOF. Le taˆ2p

n bean anglewith 0<a<efor theabovee. Then we can consider a bar billiard such that the starting oval isC₂and itsa-isoptic isC₁. In this case, for anytPoncelet's transverses close afternreflections and form ann-gon with or without self intersections. p

3. Inner oval.

In this section we consider the reverse problem: for a given ovalC, does there exist an oval Oinside Csuch that for a corresponding bar billiard Poncelet's porism holds?

First, weassumethat theovalCis parametrized as in formula (2.1) and that the origin of coordinates is placed at the Steiner point of C. Fur- thermore, the coordinate system is chosen in such a way that the tangent lineat z(0) is perpendicular to the x-axis. Then, in particular, we have z(0)ˆ(a;0) for some a>0 and p⁰(0)ˆ0. If weintroducethefollowing notation

q(t)ˆ(q1(t);q2(t))ˆz(t) z(t‡a) (3:1)

then the linel(t) through thepointsz(t) andz(t‡a) is given by l(t): yq₁(t)‡xq₂(t) [z(t);q(t)]ˆ0:

(3:2)

Hence, we easily calculateh(t), its distancefrom theoriginOˆ(0;0) h(t)ˆ [z(t);q(t)]

jq(t)j : (3:3)

LetW(t) be the angle contained between the positive direction of the axisOx and thevectorq(t), andx(t) be the angle between the positive direction ofOx andOO⁰, the shortest segment joiningOwith theline(3.2). Evidently, we havetherelation

x(t)ˆp 2‡W(t):

(3:4)

Notethat thefollowing formulas aretrueforW(t) cosW(t)ˆhz(0);q(t)i

jq(t)kz(0)j (3:5)

sinW(t)ˆ[z(0);q(t)]

jq(t)kz(0)j: (3:6)

Differentiating both sides of formula (3.5) we obtain W⁰(t)ˆ[q(t);q⁰(t)]

jq(t)j² (3:7)

and, after some further manipulations we get W⁰(t)ˆl(t;a)R(t‡a) m(t;a)R(t)

sina(l²(t;a)‡r²(t;a)) >0;

(3:8)

wherel(t;a) andm(t;a) are the same as in formulas (2.3) and (2.4). Moreover from the geometric interpretation of these functions, we see that they are nonzero and thatm<0. Thus, it follows from elementary calculus that there exists an inverse functionc(x)ˆW ¹ x p

2

ˆt. This function allows us to define a new support function

P(x)ˆh(c(x)):

(3:9)

NotethatjOO⁰j ˆP(x) and recall thatxis the angle between the positive direction of axisOxandOOƒƒ!⁰

. This construction will be used to find an en- velopez_izoof thefamily of lines (3.2) in theform

z_izo(x)ˆP(x)e^ix‡dP(x) dx ie^ix: (3:10)

After some computations we have dP(x)

dx ˆdh

dt(c(x)) dc(x)

dx ˆ [z⁰;q]hq;qi ‡ hq;zi[q;q⁰] jqj[q;q⁰] (c(x)):

(3:11)

In this frameworktis a generic parameter and we may consider an equation of the envelope in the form

z_izo(t)

ˆP p 2‡W(t)

e^i…p2‡W(t)† ‡dP dx

p 2‡W(t)

ie^i…p2‡W(t)† (3:12)

ˆP(t)E^it‡P⁰_x(t)iE^it; where

E^itˆ [z(0);q(t)]

jq(t)kz(0)j;hz(0);q(t)i jq(t)kz(0)j

; (3:13)

iE^itˆ hz(0);q(t)i

jq(t)kz(0)j;[z(0);q(t)]

jq(t)kz(0)j

: (3:14)

Ultimately, the above considerations lead to zizo(t)ˆ [z(t);q(t)]

jq(t)j E^it (3:15)

[z⁰(t);q(t)]hq(t);q(t)i ‡ hq(t);z(t)i[q(t);q⁰(t)]

jq(t)j[q(t);q⁰(t)] iE^it; wherez⁰(t)ˆdz

dt(t) andq⁰(t)ˆdq

dt(t). Let us make the convention that in the reminder of this paper prime will denote differentiation with respect tot.

By taking the derivative, we obtain

LEMMA3.1.

E^it0

ˆiE^it[q(t);q⁰(t)]

jq(t)j² ; (3:16)

iE^it₀

ˆ E^it[q(t);q⁰(t)]

jq(t)j² : (3:17)

THEOREM3.1. The curve zizo(t)is the closed envelope of the family of linesfl(t): t2Rg.

PROOF. By lemma (3.1) it follows that z⁰_izo(t)ˆ P(t)[q(t);q⁰(t)]

jq(t)j² ‡ dP dx(t) ₀!

iE^it (3:18)

and from (3.14) weobtain that

iE^itˆa q(t) jq(t)j: (3:19)

This means that the tangent vector toz⁰_izo(t) is linearly dependent onq(t).

Note, that we don't claim that it is a nonzero vector. p We do not yet know whetherz_izo(t)!z(t) whena!0. To this end we prove

LEMMA3.2.

a!0limz_izo(t)ˆz(t):

(3:20)

PROOF. Write [z(0);q(t)]

jq(t)kz(0)jˆ(rsina lcosa) sint(lsina‡rcosa) cost l²‡r²

q ;

(3:21)

hz(0);q(t)i

jq(t)kz(0)j ˆ(rsina lcosa) cost‡(lsina‡rcosa) cost l²‡r²

q ;

(3:22)

wherelˆl(t;a) andrˆr(t;a). By passing to thelimit, weobtain lima!0E^itˆe^it;

(3:23)

lima!0iE^itˆie^it: (3:24)

Similarly, since q(t)

jq(t)jˆ(rsina lcosa)e^it(lsina‡rcosa)ie^it l²‡r²

q ;

(3:25)

weget

lima!0P(t)ˆ lim

a!0

[z(t);q(t)]

jq(t)j ˆp(t):

(3:26)

When we analyze formula (3.25) we easily see that

a!0lim q(t)

sinaˆ R(t)ie^it; (3:27)

a!0lim q⁰(t)

sinaˆR(t)e^it R⁰(t)ie^it (3:28)

Theaboveformulas and theformula (3.15) give

a!0lim dP

dx(t)ˆp⁰(t);

(3:29)

which implies that

lima!0z_izo(t)ˆz(t):

(3:30)

p Our next task is to prove

a!0limz⁰_izo(t)ˆz⁰(t):

(3:31) Recall that

dP

dx(t)ˆh⁰(t) jq(t)j² [q(t);q⁰(t)]; (3:32)

so that

z⁰_izo(t)ˆ h(t)[q(t);q⁰(t)]

jq(t)j² ‡ h⁰(t) jq(t)j² [q(t);q⁰(t)]

!₀! iE^it: (3:33)

In order to determine lim

a!0z⁰_izo(t) we need only two facts:

lima!0

[q(t);q⁰(t)]

jq(t)j² ˆ1;

(3:34)

lima!0

q⁰(t)

sinaˆ2R⁰(t)e^it‡(R(t) R⁰⁰(t))ie^it: (3:35)

Hence, after some tedious computations we get

a!0lim h⁰(t) jq(t)j² [q(t);q⁰(t)]

!₀

ˆp⁰⁰(t);

(3:36) that implies

LEMMA3.3. We have

lima!0z⁰_izo(t)ˆz⁰(t):

(3:37)

In particular, the envelope which can be called an inner isoptic is a regular curve ifais sufficiently small.

Notethat for somevalues ofathe inner isoptic can develop cusps.

Our last problem before formulating the final theorem is to find lima!0z⁰⁰_izo(t).

LEMMA3.4. Under the above notations

lima!0z⁰⁰_izo(t)ˆR⁰(t)ie^it R(t)e^itˆz⁰⁰(t):

(3:38)

In particular, the function

k_izo(t)ˆ k_izo(t); 0<a<p;

k(t); aˆ0;

(3:39)

wherek_izo(t)denotes the curvature of the inner isoptic z_izo(t), is continuous from the right.

PROOF. Write

z⁰⁰_izo(t)ˆ h⁰(t)[q(t);q⁰(t)]

jq(t)j² ‡h(t) [q(t);q⁰(t)]

jq(t)j²

!₀ (

(3:40)

‡ h⁰(t) jq(t)j² [q(t);q⁰(t)]

!₀₀) iE^it

h(t)[q(t);q⁰(t)]

jq(t)j² ‡ h⁰(t) jq(t)j² [q(t);q⁰(t)]

!₀

( )

[q(t);q⁰(t)]

jq(t)j² E^it: Note that in the above representation the second term tends to R(t)e^itand h⁰(t)[q(t);q⁰(t)]

jq(t)j² !p⁰(t). It remains to find two other limits in the first term.

This is a very long and cumbersome job, and after computations similar to those we have done earlier, we obtain

lima!0

[q(t);q⁰(t)]

jq(t)j²

!₀

ˆ0;

(3:41)

lima!0 h⁰(t) jq(t)j² [q(t);q⁰(t)]

!₀₀

ˆp⁰⁰⁰(t):

(3:42)

Hence, the lemma holds. p

COROLLARY 3.1. There exists e>0 such that if 0<a<e then k_izo(t)>0for anyt2R.

THEOREM3.2. For any oval C there exists an oval O, contained in C such that corresponding bar billiard has Poncelet's porism property.

PROOF. Le t aˆ2p

n bean anglesuch that 0<a<e for theabovee.

Then we can consider a bar billiard with the starting ovalC₁and its inner isoptic z_izo(t) take n forC₂. In this case, for any t, the Poncelet's trans- verses close after n reflections and form an n-gon with or without self

intersections. p

REFERENCES

[1] K. BENKO - W. CIESÂLAK - S. GOÂZÂDZÂ - W. MOZGAWA, On isoptic curves, An.

Stiint. Univ. Al. I. Cuza Iasi, Ser. Noua, Mat.,36, No. 1 (1990), pp. 47±54.

[2] W. CIESÂLAK - A. MIERNOWSKI - W. MOZGAWA, Isoptics of a closed strictly convexcurve, Global differential geometry and global analysis, Proc. Conf., Berlin/Ger. 1990, Lect. Notes Math.,1481(1991), pp. 28±35.

[3] S. TABACHNIKOV,Geometry and billiards, Student Mathematical Library 30.

Providence, RI: American Mathematical Society, (2005).

[4] H. BOS- C. KERS- F. OORT- D. RAVEN,Poncelet's closure theorem, Expos.

Math.,5(1987), pp. 289±364

[5] E. W. WEISSTEIN, Poncelet's Porism, From MathWorld±A Wolfram Web Resource. http://mathworld.wolfram.com/PonceletsPorism.html

Manoscritto pervenuto in redazione il 12 maggio 2007.