The paper makes an attempt to visualize one of the
homogeneous geometries, the it Sol geometry, by illustrating first
the geodesic curves and spheres then the so-called translation
curves and spheres. We\u27ve collected their basic properties, too.Ovaj članak je pokušaj vizualizacije jedne od homogenih
geometrija, Sol geometrije. Prvo se ilustriraju geodetske krivulje
i sfere, a zatim i tzv. translatirajuće krivulje i sfere.
Također su navedena njihova osnovna svojstva

Attila Bölcskei

Brigitta Szilágyi

English

HRČAK - Portal of Croatian Scientific and Professional Journals

KoG•10–2006 A. Bölcskei, B. Szilágyi: Visualization of Curves and Spheres inSolGeometryOriginal scientific paperAccepted 21. 12. 2006.ATTILA BÖLCSKEI∗BRIGITTA SZILÁGYIVisualization of Curves and Spheres inSol GeometryVisualization of Curves and Spheres inSol GeometryABSTRACTThe paper makes an attempt to visualize one of the ho-mogeneous geometries, the Sol geometry, by illustratingfirst the geodesic curves and spheres then the so-calledtranslation curves and spheres. We’ve collected their ba-sic properties, too.Key words: visualization of Thurston’s geometriesMSC 2000: 53B20, 53C30Vizualizacije krivulja i ploha u Sol geometrijiSAŽETAKOvaj članak je pokušaj vizualizacije jedne od homogenihgeometrija, Sol geometrije. Prvo se ilustriraju geodetskekrivulje i sfere, a zatim i tzv. translatirajuće krivulje i sfere.Takod-er su navedena njihova osnovna svojstva.Ključne riječi: vizualizacija Thurstonovih geometrija“This (the Sol geometry)is the real weird. Unlike theprevious geometries, solve geometry isn’t even rotationallysymmetric. I don’t know any good intrinsic way to under-stand it.” (J. R. WEEKS) [7]1 IntroductionIn [5] W. P. Thurston formulated a geometrization conjec-ture for three-manifolds which states that every compactorientable three-manifold has a canonical decompositioninto pieces, each of which admits a canonical geometricstructure from among the 8 maximal simply connected ho-mogeneous Riemannian 3-geometriesE3, H3, S3, S2×R,H2 ×R, S̃L(2,R), Nil and Sol. Obviously, the Poincaréconjecture (a compact three-manifold with trivial funda-mental group is necessarily homeomorphic to the 3-sphere)is a special case of the Thurston conjecture. In the pastthirty years, many mathematicians have contributed to theunderstanding of this problem, maybe the most importantattempts are due to R. Hamilton. In 2006 a scoop wentround the world claiming that a Russian mathematician,G. I. Perelman could give a complete proof of the Thurstonconjecture and so the Poincare conjecture, too. Followedby the complex and knotty proof (using modern differentialgeometry of Ricci flows) the interest has turned to homo-geneous spaces. This paper tries to help in understandingone of the above geometries, theSol.Let (M,g) be a Riemannian manifold. If for anyx,y∈ Mthere does exist an isometryΦ : M →M such thaty= Φ(x),then the Riemannian manifold is calledhomogeneous.The visualization of the three possible two-dimensionalhomogeneous Riemann geometriesE2, H2, S2 is famil-iar to anyone, but in higher dimensions we face a lot ofopen questions. Even in three dimensions, where first timeanisotropic cases also appear we have difficulties in theimagination. No doubt, the standard models work forE3,H3, S3, moreover real-time interactive graphics algorithmshave been developed by J. R. WEEKS that can be extendedeven more for the product spacesS2×R, H2×R [8]. Theremaining three Thurston‘s homogeneous 3-dimensionalgeometriesS̃L(2,R), Nil andSol, however are difficult tohandle. From these the twisted spaces̃SL(2,R) and Nilneed multiple imaging and there are just a few results aboutthem, whilest theSol (mentioned also asolv in the litera-ture) is the most unusual as our motto above indicates, aswell (for more information consult [5], [6], [4]). We notethat in the paper [2] Emil MOLNÁR elaborated the projec-tive interpretations of all the eight geometries, we only citehis model forSol.Sol geometry can be obtained by giving a group structureT to be a semi-direct productR⋉ R2 as follows:(1 a b c)1 x y z0 e−z 0 00 0 ez 00 0 0 1=(1 x+ae−z y+bez z+c)∗ Supported by the Öveges József Programme OMFB-01525/200627KoG•10–2006 A. Bölcskei, B. Szilágyi: Visualization of Curves and Spheres inSolGeometryis the right action by a translation(x,y,z) on an affine point(a,b,c) yielding also a point ofSol expressed in homoge-neous (projective) coordinates after choosing a fixed originO(1,0,0,0).Then an invariant metric onSol(O,T) is given by(ds)2 = e2z(dx)2 +e−2z(dy)2 +(dz)2,as infinitesimal arc length square, now in any point(1,x,y,z) [5], [2].2 Geodesics and their representationIn the following we briefly recall from [1] the standard pro-cedure yielding the geodesics ofSol.Consider first the fundamental (metric) tensor from theabove mentioned equation(gi j ) =e2z 0 00 e−2z 00 0 1 .The well-known equation of geodesicsd2ukdt2+ Γki jduidtdujdt= 0containing the Christoffel symbols of second kind:Γki j =12(∂g jl∂ui +∂gli∂u j −∂gi j∂ul)glk turns [1] toẍ+2ẋż= 0ÿ−2ẏż= 0z̈−e2z(ẋ)2 +e−2z(ẏ)2 = 0.Solving this differential equation system as a Cauchy prob-lemx(0) = 0 ẋ(0) = uy(0) = 0 and ẏ(0) = vz(0) = 0 ż(0) = wu2 +v2+w2 = 1we could arrange the following table that contains our re-sults:(1) x(t) = u∫ t0 e−2z(τ)dτu 6= 0 y(t) = v∫ t0 e2z(τ)dτv 6= 0 z(t) comes from the separable differential equation0 < |w| =√1−u2−v2 < 1 dz±√1−u2e−2z−v2e2z= dt, for w ≷ 0whose solution is non-elementary function.(2) u 6= 0 x(t) = utv 6= 0 y(t) = vtw = 0 z(t) = 0(3) v = 0 x(t) = usinhtcosht +wsinhty(t) = 00 < |w| =√1−u2 < 1 z(t) = ln(cosht +wsinht)(4) u = 0 x(t) = 0y(t) = vsinhtcosht −wsinht0 < |w| =√1−v2 < 1 z(t) = − ln(cosht −wsinht)(5) u = 0 x(t) = 0v = 0 y(t) = 0|w| = 1 z(t) = ±t, for w = ±1Table 1: Table of geodesics in Sol geometry, depending on the initialvelocityparameters(u,v,w), u2 +v2 +w2 = 1.28KoG•10–2006 A. Bölcskei, B. Szilágyi: Visualization of Curves and Spheres inSolGeometryThe forthcoming pictures try to visualize the most generalcases (1) and (3). As we easily see the change(u,v,w)↔ (v,u,−w) leads to the isometry of the correspondinggeodesic curves.0,60,50,42,50,320,21,510,10,500000,100,20,30,40,50,50,60,410,81,51,221,62,5Fig. 1:The approximate view of the most general geodesiccurve with initial velocity parameters u= 0.9 andv = 0.25 in the parameter interval t∈ [0,2]. Thefirst picture shows the curve in a general view, theother from the direction of z-axis.-0,4-0,20000,20,10,20,40,30,40,50,60,40,81,21,6Fig. 2:Geodesic curve starting in the [x,z] coordinateplane with u= 0.9 in a general view in the param-eter interval t∈ [0,2].Clearly, the geodesic sphere unfortunatelly can not be ex-pressed in a closed explicite form. We give approximationsby plotting the endpoints of many geodesic curves (of thefirst type) with different initial unit velocities, according togeographic parametersu = cosϑcosϕ −π ≤ ϕ ≤ πv = cosϑsinϕ − π2≤ ϑ ≤ π2w = sinϑ .That means, ifϑ is fixed andϕ varies, then the endpoints ofgeodesics describe an altitude circle. Similarly we get lon-gitude half-circle for fixedϕ. The following figures showour results.-0,1-0,050-0,1 0,050,10,050-0,05-0,1-0,05 0,10,051-0,1-0,05-0,10,10,050,050-0,050-0,050,1-0,1 0,051-0,1-0,1 -0,1-0,05-0,05-0,0500 00,050,050,050,10,10,1-0,1-0,05-0,050,1 0,05 00,10,050-0,05-0,1-0,10,051Fig. 3: Geodesic sphere of radius 0.1. The first picture shows the sphre in a general view, then from the direction of axesz, y and x, respectively.29KoG•10–2006 A. Bölcskei, B. Szilágyi: Visualization of Curves and Spheres inSolGeometry-1-1 -1-0,5-0,5-0,500 00,50,50,5111-1-0,50-0,5-1-0,5-1110,50,500,51 -1-0,50 -0,5-0,5-1010,5-11 0,50,51-1-0,5-0,5-1-0,50,50-1110,500,51Fig. 4: The same arrangement as in Fig. 4 but now the radius is 1.-3-3-2-2-2-1-1-10 001 112 23 32-210-132 3121-1-2-30-30-212 -3210-1-23 2 1 0 -1 -2 -3021123-3210-1-23210-1-2 0-321123Fig. 5: Geodesic sphere of radius 2.3 Translation curves and spheresA Riemannian manifold with a transitive group of isome-tries is called homogeneous. In a homogeneous space thereare postulated isometries, mapping each point to any point.Translations can be introduced in a natural way. Considera unit vector at the origin. Translations, postulated at thebeginning carry this vector to any point by its tangent map-ping. If a curvet 7→ (x(t),y(t),z(t)) has just the translatedvector as tangent vector in each point, then the curve iscalled atranslation curve. This assumption leads to a sys-tem of first order differential equations, thus translationcurves are simpler than geodesics and differ from them inmost cases (except in spaces of constant curvature).From [3] we have already known the solution of the abovedefined systemẋ(t) = ue−z(t) ,ẏ(t) = vez(t) ,ż(t) = w ,of differential equation which holds for a curve starting atthe origin in direction(u,v,w):x(t) = − uw(e−wt −1).y(t) =vw(ewt −1),z(t) = wt ,.In the following -as illustration- we show how a translationcurve looks like.00 0 0,10,20,10,20,30,20,40,40,30,50,60,40,60,70,50,80,610,71,20,70,60,50,40,30,21,20,1010,80,60,40,2000,1234567Fig. 6:Translation curve with the same initial velocity pa-rameters as for geodesics above, (u= 0.9 and v=0.25) in the parameter interval t∈ [0,2]. The firstpicture shows the curve in a general view, the otherfrom the direction of z-axis.30KoG•10–2006 A. Bölcskei, B. Szilágyi: Visualization of Curves and Spheres inSolGeometryWith unit velocity translation curves we can define thetranslation sphereof radiusr with centre in the origin ofusual longitude and altidude parametersϕ andϑ, respec-tively ( [3]):u = cosϑcosϕ −π ≤ ϕ ≤ πv = cosϑsinϕ − π2≤ ϑ ≤ π2w = sinϑ ;x(ϑ,ϕ) = −cotϑcosϕ(e−r sinϑ −1)y(ϑ,ϕ) = cotϑsinϕ(er sinϑ −1)z(ϑ,ϕ) = r sinϑ .As illustrations we give the following nice pictures.-0,1-0,1-0,1-0,05-0,05-0,050000,050,050,050,10,10,1-0,1-0,050,10,10,050,050-0,050-0,1-0,05-0,10,051-0,1-0,050,1 0,05 -0,1-0,05-0,10,10,050-0,050,050,1-0,1-0,050,10,050-0,050,10,05-0,05-0,1-0,10,050,1Fig. 7: Translation sphere of radius 0.1. The first picture shows thesphere in a general view, then from the direction ofaxes z, y and x, respectively.-1-1-1-0,5-0,5-0,50000,50,50,5111-1-0,5110,50,50-0,50-1-0,5-10,51-1-0,51 00,5 0 -0,510-0,5-10,5-10,51-1-0,510,500-0,5-10,51-0,5-10,51Fig. 8: The same arrangement as above but now the radius is 1.-3-2-2-3-2-1-1-100011213223-2103322110-1-20-3-1-2-312-321210-1-2-23 2 01 0 -1 -3123-321210-11-232-1-2-3 0123Fig. 9: Translation sphere of radius 2.31KoG•10–2006 A. Bölcskei, B. Szilágyi: Visualization of Curves and Spheres inSolGeometryReferences[1] BÖLCSKEI A., SZILÁGYI B., Frenet Formulas andGeodesics in Sol geometry, (to appear in Beiträge zurAlgebra und Geometrie)[2] M OLNÁR, E., The projective interpretation ofthe eight 3-dimensional homogeneous geometries,Beiträge zur Algebra und Geometrie 38 (1997), 261-288. p[3] M OLNÁR, E., SZILÁGYI , B., Translation curvesand their spheres in homogeneous geometries,(manuscript)[4] SCOTT, P., The geometries of 3-manifolds, Bull. ofthe Lon. Math. Soc. 15 (1983), 401-487.[5] THURSTON, W.P., Three dimensional manifolds,Kleinian groups and hyperbolic geometry, Bull.Amer. Math. Soc. 6 (1982), 357-381.[6] THURSTON, W.P., Three-Dimensional Geometryand Topology, Vol. 1 (edited by S. Levy) PrincetonUniversity Press, Princeton, New Yersey (1997)[7] WEEKS, J.F., The Shape of Space, Marcel DekkerInc. (2001)[8] WEEKS, J.F., Real-Time Animation in Hyperbolic,Spherical and Product Geometries, Non-EuclideanGeometries (edited by A. Prékopa and E. Molnár)Mathematics and Its Applications 581, Springer,(2006), 287-306.Attila BölcskeiSzent István UniversityYbl Miklós CollegeDept. of Representation and InformaticsH 1146 Budapest XIV. Thököly I. u. 74.e-mail: bolcskei.attila@ymmfk.szie.huBrigitta SzilágyiBudapest University of Technology and EconomicsDept. of GeometryH 1521 Budapest XI. Egry J. u. 1, H.II.22.e-mail: szilagyi@math.bme.hu32

Vizualizacije krivulja i ploha u Sol geometriji

Abstract

Similar works

Full text

Available Versions

HRČAK - Portal of Croatian Scientific and Professional Journals

Hrčak - Portal of scientific journals of Croatia