Pseudo Spheres. A Sample of Electronic Lecture Notes in Mathematics. Eberhard Malkowsky.

Transcription

1 Pseudo Spheres A Sample of Eletroni Leture Notes in Mathematis Eberhard Malkowsky Mathematishes Institut Justus Liebig Universität Gießen Arndtstraße D-3539 Gießen Germany /o Shool of Informatis Computing German Jordanian University P.0. Box 3547 Amman 80 Jordan eberhard.malkowsky@math.uni-giessen.de ema@bankerinter.net

2 LIST OF FIGURES LIST OF FIGURES Contents 0 The Desription Use of the Leture Notes 3 A Study of Pseudo Spheres 4. Intodution. 4. The Tratrix Pseudo Spheres. 5.3 Aymptoti Lines on a Pseudo Sphere. 9.4 The Vetors of the Trihedra the Curvature along an Asymptoti Line..5 The Osulating Cirles Spheres along an Asymptoti Line on a Pseudo Sphere. 4.6 Geodesi Lines on the Pseudo-Sphere. 8 List of Figures Figure. The onstrution of a tratrix 5 Figure. A family of tratries 5 Figure 3. A pseudo sphere. 6 Figure 4. Spherial surfaes 7 Figure 5. Pseudo-spherial surfaes.8 Figure 6. Hyperboli spherial pseudo spherial. 9 surfaes with minimal maximal radii of u lines Figure 7. Families of asymptoti lines on a pseudo sphere. 0 Figure 8. The vetors of a trihedron Figure 9. Osulating plane irle. 5 Figure 0. Osulating plane sphere. 6 Figure. Osulating sphere 6 Figure. Surfae generated by the osulating planes. 7 Figure 3. Envelope of the osulating spheres. 7 Figure 4. A geodesi line 8 Figure 5. Families of geodesi lines on a pseudo sphere 8

3 0 THE DESCRIPTION AND USE OF THE LECTURE NOTES List of Animations Constrution of a tratrix Spherial surfaes Paraboli pseudo spherial surfaes Hyperboli pseudo spherial surfaes Ellipti pseudo spherial surfaes Vetors of the trihedra along an asymptoti line of a pseudo-sphere Osulating plane osulating irle at a point of an asymptoti line on a pseudo-sphere Osulating plane osulating sphere at a point of an asymptoti line on a pseudo-sphere 3 0 The Desription Use of the Leture Notes The eletroni leture notes are typeset by A M S L A TEX; it seems that L A TEX yields the best quality in display of mathematial formulae. We reated the PDF file diretly from the L A TEX soure file by \pdflatex filename. The graphis are inluded in PNG format in the L A TEX soure ode; the animations are in HTM format an be linked to. All our graphis animations were reated exlusively by the use of our own software pakage ([3,,, 4]) in Borl PASCAL, also in DELPHI, whih provides a user interfae for interative elements. Our graphis reated in PASCAL an be exported to various formats, inluding BMP, PS, PLT JVX. These formats an then be onverted by any graphis onverter software to an EPS file in TEX or L A TEX a PDF or PNG file a GIF file in HTML or WORD douments We use the software pakage Animagi GIF 3 to reate an animation in animated GIF format from a sequene of GIF files of our graphis, inlude the animation as an animated GIF image in an HTML file. We deided to use small sized PNG graphis in the leture notes to have better ontrol of the their plaement in the text, to redue the time to open the leture notes. The full sized PDF graphis an be linked to by liking on the PNG piture. To return to the leture notes, we reommend to lik on Close in the navigation bar at the top of the full sized piture.

4 A STUDY OF PSEUDO SPHERES A Study of Pseudo Spheres Here we study the most important geometrial properties of pseudo spheres. Our animations are linked to by liking on the following ion The interative elements an be opened by liking on Finally we mention that the best quality of viewing is obtained in Full Sreen mode.. Intodution The study of pseudo spheres is part of every ourse on Differential Geometry. We introdue pseudo spheres as surfaes of rotation generated by tratries (Definition.). A parametri representation a tratrix is derived in Proposition. from its geometri definition in Definition.. In Proposition.3, we establish a parametri representation for pseudo spheres. We also give a haraterisation of pseudo spheres as surfaes of rotation with onstant negative Gaussian urvature (Remark.4), of the related hyperboli, ellipti spherial, paraboli, hyperboli ellipti pseudo spherial surfaes. Furthermore, we solve the differential equation for asymptoti lines on pseudo spheres (Proposition.5), give a parametri representation for the asymptoti lines with respet to their ar lenghts (Proposition.6). This parametri representation is used to ompute the vetors of the trihedra, the urvature torsion along the asymptoti line (Proposition.8), to determine the irles (Proposition.0 (a)) spheres (Proposition.0 (b)). Finally, we solve the differential equations for the geodesi lines on pseudo spheres (Proposition.). 4

5 . The Tratrix Pseudo Spheres A STUDY OF PSEUDO SPHERES. The Tratrix Pseudo Spheres Pseudo spheres are generated by the rotation of so-alled tratries in the x x 3 plane about the x 3 axis. Definition. The tratrix Let γ γ be a urve a straight line in a plane, respetively, that have no points of intersetion. Given a point P on γ, letp denote the point of intersetion of γ with the tangent to γ at P. If the distanes between the points P P have a onstant value d, the urve γ is alled a tratrix (Figure ). Proof. Let x(t) {t, y(t)} be a parametri representation of γ. Then the tangent to γ at t is given by x(t) + λ{, y (t)} (λ IR), so we obtain for P OP {0, p } {t, y(t)} + λ{, y (t)}, that is λ t. We observe that γ γ implies d > 0, hene we have d d(p, P ) x(t) ( x(t) t{, y (t))} t + (y (t)) so, sine t > 0, Figure The onstrution of a tratrix Figure A familiy of tratries y (t) ± d t t for t < d. (.) We hoose the upper sign in (.) t (0, d), obtain d t y(t) dt. (.3) t Sine t (0, d), we may put t d sin φ for φ (0, π/) this yields We derive a parametri representation for the tratrix. Proposition. We introdue a Cartesian oordinatesystem in the plane suh that the straight line γ is the y axis. Then γ has a parametri representation x(φ) { ( ( ( )) )} φ d sin φ, d log tan + os φ (φ (0, π/)). (.) where I d t d d dt sin φ d os φ dφ t d sin φ os φ d sin φ dφ d (I + I ), (.4) sin φ dφ os φ I dφ sin φ sin φ d t + (.5) d 5

6 . The Tratrix Pseudo Spheres A STUDY OF PSEUDO SPHERES where is a onstant of integration. To evaluate the integral I, we make the substitution z tan (φ/). Then z (0, ) we obtain dφ dz d ( artan z) dz + z, sin φ sin (φ/) os (φ/) tan (φ/) os (φ/) os (φ/) tan (φ/) os (φ/) + sin (φ/) tan (φ/) + tan (φ/) x + x + z dz I z + z dz z log z + ( ( )) φ log tan +, (.6) rotation RS(γ) generated by the urve γ x(u i ) {r(u ) os u, r(u ) sin u, h(u )} ((u, u ) I (0, π)). (.7) Proposition.3 The pseudo sphere generated by the tratrix given by (.) has a parametri representation { } x(u i ) e u os u, e u sin u, d e u du ((u, u ) (log (/d), ) (0, π)) (Figure 3). (.8) Figure 3 A Pseudo sphere where is a onstant of integration. We put d( + ). Then it follows from (.3), (.4), (.5) (.6) that ( ( ( )) ) φ y(φ) d log tan + os φ +. If we hoose suh that lim φ π/ y(φ) 0, then we have 0, (.) is an immediate onsequene. We reall that a surfae of rotation is generated by the rotation of a urve γ in the x x 3 plane about the x 3 -axis. If γ is given by a parametri representation x(t) {r(t), 0, h(t)} for t in some open interval I, where r h are ontinuously differentiable on I with r(t) > 0 on I, then we write u t u for the angle of rotation, obtain the following parametri representation for the surfae of Proof. Writing t t(t ) e t > 0 y y(t(t )), we obtain from (.) with the lower sign d y dt d y dt dt dt d e t ( e t ) e t d e t for t > log (/d). 6

7 . The Tratrix Pseudo Spheres A STUDY OF PSEUDO SPHERES We put u t the parametri representation (.8) is an immediate onsequene of (.7). Remark.4 Pseudo spheres are surfaes of revolution of onstant negative Gaussian urvature K. If we assume that K : I IR is a given funtion write u u, for short, then the surfae of rotation that has K as its Gaussian urvature is given by (for details) r (u) + K(u)r(u) 0 h(u) ± (r (u)) du, (.9) we may hoose the upper sign for h without loss of generality. Surfaes of rotation with a given onstant Gaussian urvature are alled spherial or pseudo spherial surfaes depending on whether K > 0 or K < 0. First we assume K > 0 put K / for some onstant > 0. Then the general solution of the differential equation in (.9) is ( u ) r(u) λ os with λ > 0 we obtain h(u) ( λ u ) sin du. (.0) There three different types of spherial surfaes orresponding to the ases λ, λ > or λ <. Case. λ Then the surfae has a parametri representation ( ( ) ( ) ( )) u x(u i u ) os os u u, os sin u, sin ((u, u ) ( π/, π/) (0, π)). This is a sphere with radius entre in the origin. Case. λ > The orresponding surfaes are alled hyperboli spherial surfaes. Now the integral for h in (.0) only exists for values of u with ( u ) sin λ, that is u I k [ ( ( ] arsin + kπ, arsin + kπ λ) λ) for k 0, ±, ±,. (left in Figure 4). Every interval I k defines a region of the surfae. The radii of the irles of the u lines are minimal at the end points of the intervals I k equal to r λ, whereas the maximum radius R λ is attained in the middle of eah region (left in Figure 6). Case 3. λ < The orresponding surfaes are alled ellipti spherial surfaes (right in Figure 4). Now the integral for h in (.0) exists for all u the radii r of the irles of the u lines attain all values r λ. Figure 4 Spherial surfaes 7

8 . The Tratrix Pseudo Spheres A STUDY OF PSEUDO SPHERES Now we assume K < 0 put K / for some onstant > 0. The general solution of the differential equation in (.9) is ( u ) ( u ) r(u) C osh + C sinh with onstants C C. Case. C C λ 0 The orresponding surfaes are alled paraboli pseudo spherial surfaes (left in Figure 5). They have a parametri representation with r(u ) λ exp h(u ) ( u ) ( ) λ exp u du for u > log ( λ /). Case. C 0 C λ 0 The orresponding surfaes are alled hyperboli pseudo spherial surfaes (middle in Figure 5). They have a parametri representation with ( ) u r(u ( ) ) λ osh h(u ) λ u sinh du ( ) ( ) for u arsinh log λ λ + λ +. for all u with h(u ) λ osh ( ) u osh λ ; ( u ) du (.) (sine osh u for all u, we must have λ ) (right in Figure 5); the integral for h in (.) is ellipti. The radii r of the irles of the u lines satisfy 0 r λ. Figure 5 Pseudo spherial surfaes The radii r of the irles of the u lines satisfy λ r λ + (right in Figure 6). Case 3. C 0 C λ 0 The orresponding surfaes are alled ellipti pseudo spherial surfaes. They have a parametri representation with ( ) u r(u ) λ sinh paraboli hyperboli ellipti 8

9 .3 Aymptoti Lines on a Pseudo Sphere A STUDY OF PSEUDO SPHERES Figure 6 Hyperboli spherial pseudo spherial surfaes with minimal maximal radii of u lines.3 Aymptoti Lines on a Pseudo Sphere We reall that the seond fundamental oeffiients L ik (i, k, ) of a surfae of rotation with a parametri representation x(u i ) {r(u ) os u, r(u ) sin u, h(u )} where r(u ) > 0 r (u ) + h (u > 0 are given by L (u i ) L (u ) h (u )r (u ) h (u )r (u ) (r (u )) + (h (u )), L (u i ) L (u i ) 0 L (u i ) L (u ) h (u )r(u ) (r (u )) + (h (u )). In the ase of the pseudo sphere, we have (r ) + (h ), hene r r + h h 0 so h r r /h, the seond fundamental oeffiients are given by L h r h r (r ) r h r h r h ( (r ) + (h ) ) r h (.) L h r. (.3) Aysmptoti lines on a surfae with seond fundamental oeffiients L ik (k, ) are the real solutions of the differential equations L (u i ) ( du ) + L (u i )du du + L (u i ) ( du ) 0; (.4) they only exist in neighbourhoods of so alled hyperboli points of the surfae, that is points with L L L (L ) > 0. 9

10 .3 Aymptoti Lines on a Pseudo Sphere A STUDY OF PSEUDO SPHERES In the ase of surfaes of rotation, the differential equation (.4) for asymptoti lines redues to du du ± L (u ) L (u ) r (u ) r(u )(h (u )) for r(u )r (u ) > 0. (.5) Now we give the asymptoti lines on the pseudo sphere. Proposition.5 The asymptoti lines on the pseudo sphere with a parametri representation x(u i ) { } e u os u, e u sin u, e u du ((.8) with d ) are given by for (u, u ) (0, ) (0, π) (.6) ( ) + e t u (t) t u (t) ± log + e t for all t > 0, (.7) sine h (u) > 0 for all u > 0. Therefore we have to solve the integral du I. e u The substitution t e u yields dt I t t ( dt ). t t Now we put z /t, obtain dz dt/t dz dz ( I z z z log z + ) z ( ) ( ) + t + e u log log. t e u Figure 7 Families of asymptoti lines on a speudo sphere where IR is a onstant of intgration (Figure 7). Proof. Again, we write u u, observe that r(u) e u, h (u) e u r (u) r(u). Hene the differential equation (.5) for the asymptoti lines redues to du du ± h (u) ± h (u), 0

11 .3 Aymptoti Lines on a Pseudo Sphere A STUDY OF PSEUDO SPHERES We hoose the upper sign 0 in (.7), that is, we onsider the aysmptoti line given by ( ) + e t u (t) t u (t) log for all t > 0, (.8) It is useful to obtain a parametri representation for the asymptoti line with respet to its ar length s. Proposition.6 The asymptoti line given by (.8) has a parametri representation x (s) e t { } osh s os s, sin s, s tanh s osh s for all s > 0. (.9) Proof. We reall that the first fundamental oeffiients for surfaes of rotation are given by g (u i ) g (u ) (r (u ) + (h (u )), g (u i ) g (u i ) 0 g (u i ) g (u ) (r(u )). Therefore, we obtain for the pseudo sphere with a parametri representation (.6) g (u ) g (u ) e u, hene for the aymptoti line given by (.8) ( ) du x (t) g (u ( (t) du (t)) + g (u (t) (t)) dt dt + e t e t e. t Now it follows for the ar length of the asymptoti line by (.7) dt s(t) x (t) dt e t ( ) + e t log u (t). t This implies e s e t e t e t ( e s + ) e t e s 0, that is, sine e t > 0 Furthermore, we have e t h(s) h(t(s)) ( ) es e s + osh s. (.0) dt(s) e t(s) ds ds osh (s) osh (s) ds osh (s) ) ds s tanh (s). This (.0) together yield the parametri representation (.9).

12 .4 The Vetors of the Trihedra the Curvature along an Asymptoti Line A STUDY OF PSEUDO SPHERES Remark.7 Another, simpler proof is to hek if x (s) where x (s) d x (s). ds Writing φ(s) / osh s, we obtain { } x (s) φ(s) os s, φ(s) sin s, φ (s) + Sine we obtain { φ(s) sin s, φ(s) os s, 0} x (s) ( φ(s)) + ( φ (s)) + φ (s) ( φ(s)) ( φ(s)) + φ (s) + φ 4 (s). ( ( )) d sinh (s) ds osh s osh 4 (s) osh (s) osh 4 (s) φ (s) φ 4 (s), x (s) φ (s) φ 4 (s) + φ (s) + φ 4 (s) for all s > 0. Figure 8 The vetors of a trihedron.4 The Vetors of the Trihedra the Curvature along an Asymptoti Line Now we ompute the vetors of the trihedra the urvature along the aymptoti line given by (.8). Proposition.8 The vetors v k (s) (k,,, 3) of the trihedra (Figure 8), the urvature κ(s) torsion τ(s) along the asymptoti line given by (.9) are v (s) sinh s osh {os s, sin s, sinh s} (s) + { sin s, os s, 0}, (.) osh s v (s) osh {os s, sin s, sinh s} (s) sinh s { sin s, os s, 0}, (.) osh s v 3 (s) {sinh s os s, sinh s sin s, } (.3) osh s κ(s) osh s. (.4) τ(s). (.5) Proof. We omit the arguement s in the funtion φ. We already know from Remark.7 that the tangent vetor is v (s) x (s) { φ os s, φ sin s, φ } + { φ sin s, φ os s, 0} sinh s osh {os s, sin s, sinh s} + { sin s, os s, 0} (s) osh s

13 .4 The Vetors of the Trihedra the Curvature along an Asymptoti Line A STUDY OF PSEUDO SPHERES whih is (.). This yields x (s) {( φ φ) os s, ( φ φ) sin s, φφ}+ It follows from that Therefore, we have φ sinh s osh (s) φ sinh s { φ sin s, φ os s, 0}. φ φ osh s φφ sinh s φ + φ 3 sinh s φ + φ φ 3 φ φ 3 φ φ φ 3. x (s) osh 3 {os s, sin s, sinh s} (s) sinh s osh { sin s, os s, 0}. (s) Now we obtain for the urvature κ(s) along the asymptoti line hene ( + sinh ) κ (s) x (s) (s) 4 osh 6 + sinh (s) (s) osh 4 (s) 4 osh 4 (s) ( + 4 sinh (s)) osh (s), κ(s) osh s whih is (.4). Now the prinipal normal vetors v (s) along the asymptoti line are given by v (s) κ(s) x (s) sinh s osh {os s, sin s, sinh s} { sin s, os s, 0} (s) osh s whih is (.). Furthermore, sine b(s) {os s, sin s, sinh s} { sin s, os s, 0} {sinh s os s, sinh s sin s, }, the binormal vetors v 3 (s) along the asymptoti line are given by ( sinh ) (s) v 3 (s) v (s) v (s) osh 3 (s) + b osh 3 (s) {sinh s os s, sinh s sin s, } osh s whih is (.3). Finally, we obtain from (.3) sinh s v 3 (s) osh {os s, sin s, sinh s} + { sin s, os s, 0} s osh s (.6) so by (.3) τ(s) v (s) v 3 (s) osh 4 (s) ( + sinh (s)) + sinh s osh s osh s ( + sinh s). 3

14 .5 The Osulating Cirles Spheres along an Asymptoti Line on a Pseudo Sphere A STUDY OF PSEUDO SPHERES Remark.9 (a) We an easily hek the result (.3) for the binormal vetor of the asymptoti line by using the well known fat, that the osulating planes the tangent planes of a surfae along an asymptoti line with non vanishing urvature oinide, that means that v 3 (s) ± N(u i (s)) where N(u i ) x (u i ) x (u i ) x (u i ) x (u i ) are the surfae normal vetors of the surfae with a parametri representation x(u i ). We note that the surfae normal vetors of a surfae of rotation are N(u i ) (r (u )) + (h (u )) { h (u ) os u, h(u ) sin u, r (u )}. We already know that (r (u )) +(h (u )) for the pseudo sphere, for the asymptoti line h (u (s)) sinh s osh s osh s sinh s osh s, sine s > 0, r (u (s)) r(u (s)) osh s u (s) s. Thus the surfae normal vetors of the pseudo sphere along the asymptoti line are given by N(u i (s)) osh s { sinh s os s, sinh s sin s, } v 3(s). (b) The pseudo sphere has onstant Gaussian urvature K(u i ). Therefore we have by the Beltrami-Enneper theorem for the torsion along the asymptoti line τ(s) K(u i (s))..5 The Osulating Cirles Spheres along an Asymptoti Line on a Pseudo Sphere Now we onsider the osulating irles spheres along the asymptoti line on our pseudo sphere. Proposition.0 Let γ be the asymptoti line with a parametri representation (.9). (a) Then the entres of urvature along γ are given by x m (s) { os s osh s, sin s osh s, s } tanh s sinh s{ sin s, os s, 0}, (.7) hene the osulating irle of γ at s has a parametri representation y m,s (t) x m (s) + osh s (os t v (s) + sin t v (s)) for t (0, π). s (.8) (b) The entre radius of the osulating sphere of γ at s are m(s) { osh s os s, } osh s sinh s, s sinh s{ sin s, os s, 0} (.9) r m (s) osh s + sinh s. (.30) Proof. (a) The radius of urvature of γ at s is by (.4) ρ(s) κ(s) osh s. 4

15 .5 The Osulating Cirles Spheres along an Asymptoti Line on a Pseudo Sphere A STUDY OF PSEUDO SPHERES Therefore we obtain from (.9) (.) for the entre of urvature of γ at s x m (s) x (s) + ρ(s) v (s) { os s osh s, sin s osh s, s sinh s } osh s { os s osh s, sin s osh s, sinh s } osh s sinh s { sin s, os s, 0} } { os s osh s, sin s osh s, s tanh s sinh s { sin s, os s, 0} whih is (.7). Now (.8) follows immediately from the definition of the osulating irle of a urve at s. Figure 9 Osulating plane irle { os s osh s, sin s osh s, s } tanh s sinh s{ sin s, os s}+ { sinh s os s +, sinh s sin s osh s osh s, tanh s } s { osh s os s, } osh s sinh s, s sinh s{ sin s, os s, 0} whih is (.9). Finally, we obtain for the radius of the osulating sphere ( ) ρ(s) rm(s) ρ (s) + ( osh s + sinh s ), τ(s) 4 (.30) is an immediate onsequene. Now we onsider the urve of the entres of the osulating spheres of the asymptoti line. Proposition. Let γ be the asymptoti line with a parametri representation (.9) γ m be the urve of the entres of the osulating spheres along γ. Then the ar length s along γ m is given by s (s) sinh s, (.3) γ m has the following parametri representation with respet to s (b) Sine ρ(s) sinh(s)/ τ(s) by (.5), we obtain the entre of the osulating sphere of γ from (.3) (.7) m(s) x (s) + ρ(s) v (s) + ρ(s) τ(s) v 3(s) x m (s) + sinh s v 3 (s) m (s ) { } + (s ) sin (ψ (s )), ψ (s ) + (s ) os (ψ (s )), s { sin (ψ (s )), os (ψ (s )), 0}where ψ (s ) Arsinh(s ). (.3) 5

16 .5 The Osulating Cirles Spheres along an Asymptoti Line on a Pseudo Sphere A STUDY OF PSEUDO SPHERES The vetors v (s ) of the trihedra, the urvature κ (s ) of γ m at s are v (s ) { s + (s ) os (ψ (s )), v (s ) v 3 (s ) s + (s ) sin (ψ (s )), }, + (s ) (.33) ( + (s ) + (s ) {os (ψ (s )), sin (ψ (s )), s } + { s sin (ψ (s )), s os (ψ (s )), 0}). (.34) + (s ) { sin (ψ (s )), os (ψ (s )), 0} s + (s ) {os (ψ (s )), sin (ψ (s )), s } (.35) κ (s ) Figure 0 Osulating plane sphere + (s ) (.36) Figure Osulating sphere Proof. Omitting the argument s, using Frenet s formulae τ, we obtain d m ds d ( x + ρ v + ρ ) v + ρ v + ρ ds τ v + ρ v 3 + ρ v v + ρ v + ρ ( ρ ) v + v 3 + ρ v 3 ρ v (ρ + ρ) v 3. Sine ρ(s) ρ(s) osh s/, it follows from (.4) that d m(s) ds osh s v 3 (s) {sinh s os s, sinh s sin s, } (.37) d m(s) ds sinh s + osh s osh s, hene the ar length s along γ m is given by s d m(s) (s) ds ds osh s ds sinh s, whih is (.3). This implies s Arsinh(s ) log ( s + ) (s ) + ψ (s ). Substituting this in (.9) observing that osh s + (s ), we obtain (.3). Sine m (s ) m(s(s )), an appliation of the hain rule (.37) (.3) together yield 6

17 .5 The Osulating Cirles Spheres along an Asymptoti Line on a Pseudo Sphere A STUDY OF PSEUDO SPHERES v (s ) d m (s ) d m(s(s )) ds(s ) ds ds ds osh (s(s )) v 3 (s(s )) osh (s(s )) { } s + (s ) os (ψ (s s )), + (s ) sin (ψ (s )), + (s ) whih is (.33). Sine v (s ) v 3 (s(s )), appliation of the hain rule (.6) together yield v (s ) d v 3(s(s )) ds(s ) ds ds ( osh (s(s )) osh (s(s )) {os (ψ (s )), sin (ψ (s )), s } + { s sin (ψ (s )), s os (ψ (s )), 0}) ( + (s ) + (s ) {os (ψ (s )), sin (ψ (s )), s } + { s sin (ψ (s )), s os (ψ (s )), 0}). Now we obtain the urvature κ (s ) of γ m at s from (κ (s )) v (s ) ( + (s ) ) ( ( + (s ) ) ( + s ) whih yields (.36). Furthermore, we obtain (.34) from ) + (s ) ( + s ) + (s ) v (s ) v (s ) κ (s ). + (s ), Sine a(s ) {s os (ψ (s )), s sin (ψ (s )), } {os (ψ (s )), sin (ψ (s )), s } { ( + (s ) ) sin (ψ (s )), ( + (s ) ) os (ψ (s )), 0 } ( + (s ) ){ sin (ψ (s )), os (ψ (s )), 0}, b(s ) {s os (ψ (s )), s os (ψ (s )), } we obtain (.35). { sin (ψ(s )), os (ψ(s )), 0} { os (ψ (s )), sin (ψ (s )), s } v 3 (s ) Figure Surfae generated by the osulating planes s ( ) 3 a + + (s ) + (s ) b, Figure 3 Envelope of the osulating spheres 7

18 .6 Geodesi Lines on the Pseudo-Sphere A STUDY OF PSEUDO SPHERES.6 Geodesi Lines on the Pseudo-Sphere Here we determine the geodesi lines on the pseudo sphere with a parametri representation (.6), whih are given by the differential equations { } i ü i + u i u k 0 for i,, (.38) jk { } i where (i, j, k, ) denote the seond Christoffel symbols, jk summation is arried out in (.38) with respet to j, k. Proposition. Let (u 0, u 0) (0, ) (0, π) be given. Then the u line orresponding to u 0 is a geodesi line. Furthermore, if Θ 0 [0, π) \ {π/, 3π/} then the geodesi line through (u 0, u 0) with an angle of Θ 0 to the u line through (u 0, u 0) is given by u (s) log ( δ osh (s + s 0 )) u (s) δ tanh (s + s 0) + (.39) where ( ) δ os Θ 0 e u 0, u 0 + e u 0 sin Θ0 tan Θ0 s 0 log. + sin Θ 0 (.40) Proof. are Sine the first fundamental oeffiients for the pseudo sphere g (u i ) g (u ), g (u i ) 0 g (u i ) g (u ) e u, we obtain the seond Christoffel symbols { } g (u ) { } { } 0, { } g (u ) g (u ) du { } 0, { } { } { } 0, dg (u ) du 0, e u, dg (u ) g (u ) du Figure 4 A geodesi line Figure 5 Families of geodesi lines on a speudo sphere the differential equations (.38) redue to ü + e u ( u ) 0 (.4) ü u u 0. (.4) If u 0 then we obtain a u line orresponding to u u 0, that is, for a suitable orientation of the ar length u s + u 0 u (s) u 0 for s > u 0. 8

19 .6 Geodesi Lines on the Pseudo-Sphere A STUDY OF PSEUDO SPHERES Now we assume u 0. Then (.4) yields ü u u, that is u δe u Substituting this in (.4), we obtain for some onstant δ 0. (.43) ü + δ e u 0, that is d ds ( ( u ) + δ e u) 0, hene ( ) u d δ e u for some onstant d 0 u < log d δ. This yields I u ± d δ e u for u log d δ. (.44) du d δ e u ±(s + s 0) for some onstant s 0. To evaluate the integral I(u ), we substitute x δ e u obtain dx I x d x ( ) d + d log d x (d d x log + ) d δ e u, δ e u hene ( ) δ e u e ( d(s+s0)) d d δ e u e u de ( d(s+s 0)) δ ( + e ( d(s+s 0)) ) d δ osh (d(s + s 0 )). (.45) Therefore we have ( ) u d (s) log for all s IR, δ osh (d(s + s 0 )) sine osh (d(s + s 0 )) for all s implies u (s) log d/δ for all s. Now it follows from the identity on the right h side in (.43) (.45) that u d δ osh (d(s + s 0 )), so u d δ tanh (d(s + s 0)) + for some onstant. Furthermore, sine s is the ar length along the geodesi line, it follows from (.44) the identity on the right h side in (.43) that g (u )( u ) + g (u )( u ) d δ e u + δ e u d, hene d ±. Therefore, the general solution of the differential equations (.4) (.4) is u (s) log δ osh (s + s 0 )) u (s) ± δ tanh (±(s + s 0)) + δ tanh (s + s 0) +. Thus we have shown the identities in (.39). Now we determine the onstants δ, s 0 suh that the initial onditions are satisfied. Let x (s) be a parametri representation of the geodesi line. Then it follows from x (0) x (u 0) x (u 0) g (u 0) du (0) ds tanh s 0 sin Θ 0, 9

20 REFERENCES REFERENCES that s 0 Artanh( sin Θ 0 ) ( ) sin log Θ0, + sin Θ 0 whih is the third identity in (.40). Furthermore, x (0) x (u 0) x (u 0) g (u 0) du (0) ds e u 0 δe u 0 os Θ0 implies δ e u 0 os Θ0, whih is the first identity in (.40). Finally Referenes [] E. Malkowsky, A software for the visualisation of differential geometry, Visual Mathematis 4(): Eletroni publiation, 00 index.htm 3 yields u 0 δ tanh s 0 + u 0 + sin Θ 0 e u 0 os Θ 0 u 0 + e u 0 tan Θ0, whih is the seond identity in (.40). [] E. Malkowsky, Visualisation animation in mathematis physis, Proeedings of the Institute of Mathematis of NAS of Ukraine 50(3): 45 4, Proeedings003.html 3 [3] E.Malkowsky, W. Nikel, Computergrafik und Differentialgeometrie, Vieweg Verlag, Braunshweig, [4] E. Malkowsky, V. Veličković, Analyti transformations between surfaes with animations, Proeedings of the Institute of Mathematis of NAS of Ukraine 50(3): , Proeedings003.html 3 0