Probablty of an Acute Trangle n the Two-dmensonal Spaces of Constant Curvature Afflated Hgh School of SCNU, Canton, Chna Student: Lv Zyuan, Guo Yuyu Advsor: We Jzhu - 54 -
Probablty of an Acute Trangle n the Two-dmensonal Spaces of Constant Curvature Abstract The nterest n the statstcal theory of shape has arsen snce Kendall found that the metrc geometry of spaces s precsely the requred tool for the systematc comparson and classfcaton of varous shapes n 980s. The statstcal theory of shape s wdely used n the felds such as quantum physcs, bology, and medcal scence. Ths paper concerns the probablty of acute trangles on the spaces of constant curvature. We prove the followng results:. On the unt sphere S, the probablty for a trangle formed by choosng three ponts at random to be an acute one s. 6. On Poncaré dsc D, the probablty for a trangle formed by choosng three 5 ponts at random to be an acute one s. 8 The paper nvolves many dscplnes, ncludng probablty, geometry. The hghlght s the dea of reducng the gven probablty problem to a queston of sold geometry. Key words: Random trangle, Geometrc probablty, Non-Eucldean geometry, Spaces of constant curvature, Rgdty theorem. - 55 -
Probablty of an Acute Trangle n the Two-dmensonal Spaces of Constant Curvature Introducton The nterest n the statstcal theory of shape has arsen snce Kendall found that the metrc geometry of spaces s precsely the requred tool for the systematc comparson and classfcaton of varous shapes n 980s. The statstcal theory of shape s wdely used n the felds such as quantum physcs, bology and medcal scence. ( [],[],[],[4],[6]). The statstcal theory of shape can be traced back from 89 when Charles Dodgson (Lews Carroll) proposed the followng queston. Queston: Fnd the probablty that a trangle formed by choosng three ponts at random on an nfnte plane would have an obtuse trangle. In [5], by ntroducng the Cartesan coordnates of the three ponts, S. Portnoy argued that the set of trangles can be dentfed wth the sx-dmensonal Eucldean space R 6. And the set T O of obtuse trangles s a double cone. Also, he clamed that the requrement of takng three ponts at random n the plane can be understood as the nduced probablty dstrbuton n R 6 beng sphercally symmetrc. Hence, the condtonal dstrbuton gven the dstance from the orgn s unform on the approprate sphere. S. Portnoy proved the probablty of formng obtuse trangle s 4. Ths paper concerns the probablty of an acute trangles n the two dmensonal spaces of constant curvature. By choosng the values of nteror angles as coordnates, the set of trangles can be dentfed wth a regon S n the three-dmensonal Eucldean space R. Also, the requrement of takng three ponts at random n the two-dmensonal spaces of constant curvature s understood as the pont n the set S unformly dstrbuted. In ths way, we can compute the probablty of an acute trangle n the two- dmensonal spaces of constant curvature. In partcular, n the Eucldean case, we obtan the same result as S. Portnoy. - 56 -
Prelmnary and Man results It s well known that Eucldean geometry s based on fve postulates. The Eucld s ffth postulate, called the parallel postulate, can be expressed as follow: The parallel postulate: There s at least one lne L and at least one pont P not on L, such that one lne can be drawn through P coplanar wth but not meetng L. Durng a long perod of tme, people attempted to prove that the parallel postulate could be deduced from the other four postulates and found that the parallel postulate s equvalent to the fact that the sum of nteror angles of a trangle equals π (represented by radan). In nneteenth century, Gauss, Bolya, Lobachevsky found that the parallel postulate was ndependent of the other four postulates. By replacng the ffth postulate wth one of the followng two postulates whle keepng the other four postulates unchanged, the sphercal geometry and hyperbolc geometry may be establshed respectvely. The parallel postulate n sphercal geometry: Gven a lne L and a pont P not on t, there s no lne can be drawn through the pont P whch s parallel to the gven lne L (that s, all lnes through the pont P ntersect wth the gven lne L). The parallel postulate n hyperbolc geometry: There s at least one lne L and at least one pont P, not on L, such that two lnes can be drawn through P coplanar wth but not meetng L. The unt sphere n three-dmensonal Eucldean space S {( x, y, z) R x y z } can be regarded as a model of sphercal geometry, where lnes are defned as great crcles (As the shortest dstance between two ponts n a sphere s the nferor great crcular arc whch s analogue to the fact n Eucldean geometry that the shortest dstance between two ponts s a lne segment.) Defnton : Let A, B, C be three ponts n the sphere whch are not on the same great crcle. The sde AB of the sphercal trangle ABC s defned to be the nferor great crcular arc jonng A and B. The angle A wth vertce A s defned to be the angle formed by the tangent vectors AX and AY of the sdes AB and AC respectvely. See - 57 -
Fgure. Fgure Defnton : The sphercal trangle ABC s called an acute trangle f A B C are all acute angles. From the defnton of dhedral angle, t s easy to see that XAY s the plane angle <B-OA-C> of the dhedral angle B-OA-C,.e. A=<B-OA-C>. Smlarly, B=<C-OB-A>, C=<A-OC-B>. Let ABC be a sphercal trangle n S²and let A, B, C (0,, ). Set ', ', '..e. α,β,γ are the exteror angle of ABC. Then by the specal case of Gauss-Bonnet Theorem (Theorem.0 n [7],) we have ' ' ' S ' ' ' S ABC ABC () where S ABC s the area of the trangle ABC. Gauss-Bonnet theorem suggests that the sphercal trangle s, up to an sometry of the sphere, unquely determned by ts angles. Actually we have the followng rgdty theorem: Rgdty theorem: f α,β,γ satsfy the nequaltes ( ' ' ' ) () ( ' ' ') () ( ' ' ') (4) ( ' ' ') (5) 4-58 -
Then up to an sometry of the sphere, there exsts a unque trangle ABC wth α,β, γ as ts nteror angles. The above Theorem s stated on page 6, [7], and a proof s ndcated on page 66. For hyperbolc geometry, the unt dsc n complex-plane C D z C z x y R x y { } {(, ) } can be regarded as ts model, where lnes are the dameters of D and arcs of crcles n D that are orthogonal to the unt crcle D { z D z }. The model s called Poncaré dsc. Defnton : Gven three ponts A,B,C n the Poncarédsc whch are not on the same lne, the sde AB of the hyperbolc trangle ABC s defned to be the arc of crcle n D jonng A and B and orthogonal to the unt crcle D { z D z }. The angle A wth vertce A s defned to be the angle formed by the tangent vector AX and AY of the sdes AB and AC respectvely. See Fgure. Fgure Defnton 4: The hyperbolc trangle ABC s called an acute trangle f A B C are all acute angles. From the specal case of Gauss-Bonnet theorem (Theorem.0, [7]), we know that, Hence ( ) ( ) ( ) S ABC S ABC (6) Gauss-Bonnet theorem suggests that the hyperbolc trangle s, up to an sometry of the Poncarédsc, unquely determned by ts angles. Actually we have the followng 5-59 -
rgdty theorem: Rgdty theorem: If α,β,γ satsfy the nequaltes, 0,, Then up to an sometry of the Poncare dsc, there exsts a unque trangle (7) ABC wth α, β,γ as ts nteror angles. Fgure Note: The above Theorem s a specal case of Theorem.8 n [7], where the exstence s proved and the proof of unqueness s ndcated on page 66. For reader s convenence, we gve a detaled proof here. Proof : For exstence, t s suffcent to look for the desred trangle n the class of trangles admttng an nscrbed crcle. For postve number r 0, 0,,,, consder the quadrlateral Q,,, as n fgure : We need only to prove that there exsts r 0 such that (8) In fact, f such an r>0 can be found, then the problem s solved: one can smply lay the quadrlaterals Q, Q, Qone besde the other, successvely jonng them along the sdes equal to r, then the resultng trangle s the desred one. Note that when r s small enough, Q, =,, can be approxmately regarded as the fgure n the Eucldean plane. Therefore as r 0+, ( ) 0,whch mples that ( ) ( ). 6-60 -
On the other hand, as r, 0. Snce s a contnuous functon of r, accordng to the ntermedate value theorem of contnuous functon, there exsts r satsfyng. We have Ths fnshes the proof of exstence. The unqueness n rgdty theorem can be proved as follow: By the dual cosne theorem of hyperbolc geometry, Here a s the length of sde BC n Thus cosha cos cos cos sn sn cosh a, cos cos cos cosh a. sn sn ABC s determned by α,β,γ. a a e e, also cosh a. x x x x e e e e Now consder the functon f( x), we have f '( x). When x 0, f '( x) 0,hence f(x) s monotonc ncreasng n (0, ). As a consequence, a s unquely determned bycosha. Smlarly, b,c s unquely determned. Therefore sdes a,b,c of hyperbolc trangle can be unquely determned by α,β,γ. Next we prove the trangle s unquely determned up to sometry. By Cosne Law n hyperbolc geometry: We have, for fxed b, c>0, cosh a cosh b cosh c snh b snh c cos (9) cosha s monotoncally ncreasng. Hence for fxed a, b, c>0, there exsts a unque angle α satsfyng (9). As a consequence, determned up to sometry. The unqueness s proved. 7 ABC s unquely - 6 -
From the vewpont of dfferental geometry, Eucldean plane E, unt sphere S n three-dmensonal Eucldean space and the Poncarédsc (endowed wth sutable metrc) have Gaussan curvature 0, +, - respectvely. So they are generally called two-dmensonal spaces of constant curvature. The followng specal case of Toponogov s trangle comparson theorem n Remannan geometry s ntutvely clear. Toponogov s trangle comparson theorem (Specal Case) Gven a,b,c>0, Let T, T, T be the trangles wth sde lengths a,b,c n S, E and D respectvely, and let A, B, C, A, B, C, A, B, C. be the correspondng angles, then we have A >A >A B >B >B C >C >C Fgure 4 By the above theorem, t s reasonable to clam the followng. The probablty of acute trangles n the Eucldean plane E s greater than that of acute trangles n unt Sphere S, whle less than that of acute trangles n Poncarédsc. We verfy the above clam and calculate the correspondng probablty of an acute trangle. More precsely, we have the followng: Man Results. On the unt sphere S, the probablty for a trangle formed by choosng three ponts at random to be an acute one s. 6. On Poncaré dsc D, the probablty for a trangle formed by choosng three 8-6 -
5 ponts at random to be an acute one s. 8 Proof of the Man Results. Consder frst the Eucldean plane E², n ths case the Gaussan curvature K=0. Assumng <),then ABC s a random trangle n E², and A=, B=, C= (0<,, 0,, (0) Fgure 5 Takng α,β,γ as Cartesan coordnates, as shown n fgure 5. Snce ABC s randomly chosen on E², we may assume the ponts wth coordnates (α,β,γ) s unformly dstrbuted n the regon determned by (0), whch corresponds to the set of Eucldean trangles. The necessary and suffcent condton for 0,, ABC to be an acute trangle s: () corresponds to G H I n fgure 5, whch s obtaned by cuttng the cube OD G E -F H J I by the trangle A B C. Snce α,β,γ obeys unform dstrbuton n () A B C, therefore, SG P( ABC s an acute trangle)= H I S A B C. 4 9-6 -
. Next, we consder the unt Sphere S², n ths case the Gaussan curvature K=. As n Secton, let ', ', ' be the exteror angles of the trangle ABC. Snce 0<,, <,we have 0<,, <. Takng,, as Cartesan coordnates, as n fgure 6: Fgure 6 where D,E,F are mdponts of G H,G I,H I respectvely. Snce ABC s randomly chosen on S², we may assume the ponts wth coordnates (α,β,γ) s unformly dstrbuted n the regon determned by ()-(5), whch corresponds to the set of sphercal trangles. The regon determned by ()~(5) s the nteror of the tetrahedron O-I G H, whch can be obtaned from the cube OA G B -C H J I by elmnatng the tetrahedra C -OI H,A -OG H,B -OG I,J -I G H, as shown n Fgure 6. Hence the volume of the tetrahedron O-I G H can be computed as V V ( V V V V ) OI G H OA G B C H J I A OG H B OG I C OI H J I G H 4 Note that the trangle ABC s an acute one f and only f ', ', '. The regon determned by ()-(5) and the condton ', ', ' s the ntersecton of 0-64 -
the cube N D K E -F L J M and the tetrahedron O-I G H, whch s also a tetrahedron N -D E F. The volume of the tetrahedron N -D E F s ( ) V N D E F 48 and VN P( ABC s an acute trangle)= D E F. VO I 6 GH. Fnally, we consder the Poncarédsc D, n ths case the Gaussan curvature K= - Let ABC be a random trangle on П², and let A=, B=, C= (0<,, <). Takng,, as Cartesan coordnates, as shown n fgure 7: Fgure 7 In fgure 7, the regon determned by (7) s the nteror of tetrahedron O-A B C Note that the trangle ABC s an acute one f and only f 0,,. The regon determned by (7) and the condton T of the cube OG D H -I E J F and the tetrahedron O-A B C. Note that T can be obtaned from the cube OG D H -I E J F by elmnatng the tetrahedron J -E D F. The volumes of the tetrahedron O-A B C and T can be computed as: 0,, s the ntersecton V O - ABC π π π 6, - 65 -
V T π π - π 5π 48 and VT 5 P( ABC s an acute trangle)=. V 8 O A B C Thus the theorem s proved. 4 Research Prospectve In ths paper, we only nvestgate the probablty of an acute trangle n the two-dmensonal spaces of constant curvature. It s natural to contnue our research on surfaces of varable curvature. For nstance, we may study the probablty of an acute trangle on the parabolod of revoluton z=x +y. Reference [] Aste T, BooséD and Rver N, From one cell to the whole froth:a dynamcal map, Phys. Rev. E 5 68-9, 996 [] Atyah M and Sutclffe P, The geometry of pont partcles Proc. R. Soc. London A 458 089-5, 00 [] Battye R A,Gbbon G W and Sutclffe P M, Central confguraton n three Dmensons,Proc. R. Soc London A 459 9-4, 00 [4] Brody D C,Shapes of Quantum States,J. Phys. A:Math, Gen. 7 5-57, 004 [5] Portnoy S,A Lews Carroll Pllow Problem: Probablty of an Obtuse Trangle, Statstcal Scence vol.9, no. 79-84, 994 [6] Small C G,The Statstcal Theory of Shape, Sprnger-Verlag New York, Inc., 996 [7] Vnberg E.B. (ed.), Geometry II Spaces of Constant Curvature, Sprnger-Verlag Berln Hedelberg, 99-66 -