Two-parametric Motions in the Lobatchevski Plane

Transcription

1 Journal for Geometry and Graphics Volume 6 (), No., Two-parametric Motions in the Lobatchevski Plane Marta Hlavová Department of Technical Mathematics, Faculty of Mechanical Engineering CTU Karlovo nám. 3, Praha, Czech Republic hlavova@marian.fsik.cvut.cz Abstract. In this paper we classify two-parametric motions in the Lobatchevski plane L. These motions are surfaces on the Lie group SO(, ). In the first part the basic properties of motions in L are derived and it turns out that the kinematical space belonging to these motions is locally the space SO(, )/SO(, ) realized as the unit quadric with signature (, ) in the vector space R. The remaining part contains explicit expressions and graphic representations of surfaces induced by motions with constant invariants. We also present some special cases developable surfaces. Key Words: two-parameter motion, hyperbolic geometry, Lobatchevski plane MSC : 53A7, 5N5. Introduction Let us consider vector space V = R 3 with scalar product (x, y) = x y + x y x 3 y 3 and vector product x y = (x y 3 x 3 y, x 3 y x y 3, x y x y ) for any x = (x, x, x 3 ) and y = (y, y, y 3 ) from V. We can choose an orthonormal base {e, e, e 3 } in V (e = e =, e 3 =, (e i, e j ) = for i j) for which e e = e 3, e e 3 = e, e 3 e = e. The Lobatchevski (hyperbolic) plane L is the projective plane P, realized as one-dimensional subspaces of V. Planar hyperbolic geometry is the geometry of L with respect to the group of projective transformations preserving the so-called absolute conic x + x x 3 = (see []). This group is isomorphic to the Lie group G = SO(, ) realized as a group of 3 3 matrices acting on L by matrix multiplication. Each element g of SO(, ) satisfies the condition J = g T Jg, where J is the matrix representation of the quadratic form x + x x 3, J = ISSN /$.5 c Heldermann Verlag.

2 8 M. Hlavová: Two-parametric Motions in the Lobatchevski Plane The 3-dimensional Lie algebra of the group SO(, ), denoted by so(, ), is interpreted as the tangent space in the unit element of the group G and every element X from so(, ) is of the form x 3 x X = x 3 x x x or X v = (x, x, x 3 ) in vector notation. The operation of Lie bracket in so(, ) is the same as the vector product introduced above, [X, Y ] = X v Y v. For the invariant Killing form we obtain K(X, Y ) = (X v, Y v ), where (X v, Y v ) means the scalar product introduced above. From this we see that the Lie algebra so(, ) can be identified with the vector space V with the vector product introduced above. We also notice that introduced operations are similar to the usual operations of vector algebra, which correspond to spherical kinematics (see []). This analogy will be also used in what follows.. Motion in a homogeneous space By a two-parametric motion in L we mean a -dimensional motion in the group G = SO(, ), which is an immersion g of a -dimensional manifold X into G. Let us choose two copies of L and in each of them a fixed orthonormal frame: R in moving plane L and R in fixed plane L. The group G = SO(, ) acts as a group of linear maps from L into L by the rule g( R ) = R g. Then for fixed elements g, g in G we consider motions g gg and g equivalent. This shows that we have to consider the homogeneous space G = G G/Diag(G G), where Diag(G G) is the group of all elements (g, g) for g from G. We consider the action of G G on G given by the rule (g, g )g = g gg and this shows that G can be identified with G because the isotropy group of the unit element e of the G is the group Diag(G G) (see [3]). This homogeneous space G is called the kinematical space of the group G. By a -dimensional motion in G we now understand an immersion of a -dimensional manifold X into G and the properties of the motion in G are properties of an immersion of X into G. The classical example is the geometry of the group SO(3) realized as the group of motions of the unit sphere S in E 3, the homogeneous space G of which is locally the space SO()/SO(3) and it is the 3-dimensional elliptic space (see [, ]). In our case, SO(, ) SO(, ) is locally isomorphic with SO(, ), which is the group of projective transformations in P 3 preserving the quadric with signature (, ). This follows that the homogeneous space G is locally the space SO(, )/SO(, ) of dimension 3. We can realize this space as the unit quadric with the signature (, ) in the projective space P 3 realized as one-dimensional subspaces of vector space R. Because the space SO(, )/SO(, ) is 3-dimensional, each two-parametric motion can be represented as a surface in this space and properties of such surfaces are the same as properties of two-parametric motions. 3. Two-parametric motions in L Let g(x) be a two-parametric motion in L. We may identify the elements of SO(, ) SO(, ) with pairs of orthonormal frames by R = R g, R = R g, so that the pair (g, g ) is identified with the pair (R, R). Let (R, R) be a lift of g, by definition a pair of orthonormal frames such that g R = R. From the following formulas dr = Rg dg, d R = Rg dg

3 M. Hlavová: Two-parametric Motions in the Lobatchevski Plane 9 we obtain the left invariant form (ϕ, ψ) of the lift (g, g ), where ϕ = g dg and ψ = g dg. Let s define new forms ω = (ϕ ψ)/ and η = (ϕ + ψ)/. Forms ϕ, ψ, ω and η have values in so(, ). Then dr = Rϕ, d R = Rψ and or, by using ω and η dϕ + ϕ ϕ =, dψ + ψ ψ =, () dω + η ω + ω η =, dη + ω ω + η η =, where () are Maurer-Cartan relations obtained by exterior differentiation of ϕ and ψ. This gives the integrability conditions in the form dω = ω η 3 + η ω 3 dη = η η 3 + ω ω 3 dω = ω 3 η + η 3 ω dη = η 3 η + ω 3 ω dω 3 = ω η + η ω dη 3 = η η + ω ω. () If we choose another lift of g, say (g h, g h), where h SO(, ), then for the new forms ω and η we get ω = h ωh η = h ηh + h dh. Due to the isomorphism between so(, ) so(, ) and so(, ) for an adapted frame R = {e, e, e, e 3 } of g(x) (an adapted frame is any orthonormal frame with respect to the invariant quadratical form x x x + x 3 such that e is the point of induced surface) we obtain dr = RM, where M = ω ω ω 3 ω η 3 η ω η 3 η ω 3 η η According to the Killing form K(ω) = ω + ω ω 3 we have to distinguish three cases: K(ω) is positive definite (has signature (,)), K(ω) is indefinite (has signature (,)) and K(ω) is singular. In this paper we leave out the last case because it is not interesting. 3.. K(ω) is positive definite We may always find a lift of g such that ω 3 =. An adapted frame is then any orthonormal frame such that e is the point of induced surface, e, e determine the tangent plane of surface and e 3 is normal to surface. Then ω ω and the corresponding isotropy group is h = cosφ sinφ sinφ cosφ.. The integrability conditions are given in (). Using Cartan s Lemma we get η = βω + γω η = αω + βω.

4 3 M. Hlavová: Two-parametric Motions in the Lobatchevski Plane From the new forms ω, η given by the change of frame h we obtain two invariants: H = (α + γ) (3) K = β αγ, () where H is the mean curvature and K is the Gauss curvature of the surface induced by two-parametric motion. There always exists a lift such that β =. α and γ are then called the main curvatures of the surface. In the case α = γ = we get a planar point and if this condition is satisfied on a neighbourhood, we get a part of a plane. Let us denote η 3 = κω + λω, dα = α ω + α ω, dγ = γ ω + γ ω, dκ = κ ω + κ ω and dλ = λ ω + λ ω. Computation gives: and γ = (γ α)λ α = (γ α)κ κ + κ λ + λ αγ =. (6) These equations (5) and (6) are analogous to Codazzi and Gauss equations from the surface theory in E Surfaces with constant main curvatures In the case α = γ we get analogical results as in the theory of two-parametric motions in O(3) (see [6]). Then d(e e 3 /α) = and computation gives the equation of the surface (5) x + y z = α, (7) which is a double-sheet hyperboloid in dehomogenized coordinates. In the following figures we present given surfaces in grey shade together with the absolute one-sheet hyperboloid, similarly as in Fig.. Let α γ be constants, then from (5) γ = α = and κ = λ =. Then η 3 = and αγ = (due to (6)). According to integrability conditions dω = dω = we get ω = du, ω = dv. The Frenet formulas (applied in the same sense as in the theory of curves) for the surface are Integration gives de = du e + dv e de = du e + α du e 3 de = dv e /α dv e 3 de 3 = α du e /α dv e. e = f e +α u + f e +α u e = f e +α v α + f 3 e +α v α, where {f, f, f, f 3 } is a fixed orthonormal base. Computation yields e = ( e + α u + e ) α v and the equation of the surface is α x = yz. (8)

5 M. Hlavová: Two-parametric Motions in the Lobatchevski Plane 3 z z y x y 3 3 x Figure : The surface (7) for choice α = Figure : The surface (8) for α = / 3.. K(ω) is indefinite If this condition is satisfied, we can take such a lift for which ω =. Then ω ω 3 and the isotropy group is h = coshφ sinhφ. sinhφ coshφ Using Cartan s Lemma we get η = βω + γω 3, η 3 = αω βω 3. The mean and Gauss curvatures are H = (α + γ), K = β + αγ. Let us denote η = κω + λω 3, dα = α ω + α 3 ω 3, dγ = γ ω + γ 3 ω 3, dκ = κ ω + κ 3 ω 3 and dλ = λ ω + λ 3 ω 3. For a choosen frame such that β = (possibility α = γ = leads to the same conclusion as in 3..) we obtain and γ = (γ α)λ α 3 = (α γ)κ 3... Surfaces with constant main curvatures κ 3 κ λ + λ + αγ =. () In the case α = γ we get d(e + e /α) = and computation gives the equation of the surface in the form x y z = α, () (9)

6 3 M. Hlavová: Two-parametric Motions in the Lobatchevski Plane 8 6 z y x Figure 3: The surface () for α = /3 which is a one-sheet hyperboloid (Fig. 3). Let α γ be constants. Then κ = λ = and γ = /α. From () we get ω = du, ω 3 = dv and from the Frenet formulas we obtain the equation of the following surface: already mentioned in Section 3.., Fig... Developable surfaces α x = yz, () A surface is called developable, if the Gauss curvature K =. For the first type of motion (Section 3.) we obtain αγ =. Let us suppose α and γ =. Then λ =, α = ακ and κ + κ = from (5) and (6). The corresponding matrix is M = ω ω ω κω αω ω κω αω From the integrability conditions we get dω = and dη = d(αω ) =. So let us denote ω = du, ω = dv/α. Integration gives κ = tanh(u g(v)), α = h(v)/cosh(u g(v)), where g(u), h(u) are arbitrary functions. Computation gives. e = f (v)e u + f (v)e u, (3) where f, f are orthonormal vectors depending on one variable v.

7 M. Hlavová: Two-parametric Motions in the Lobatchevski Plane z y x - - Figure : Developable surface (3) z y x - - Figure 5: Developable surface ()

8 3 M. Hlavová: Two-parametric Motions in the Lobatchevski Plane For example we choose f = ( v, cosv v, sinv ) ( v,, f =, sinv v, cosv v The corresponding surface is displayed in Fig.. In the second case (Section 3.) the properties are similar. We obtain αγ = and then λ = and α 3 = ακ, κ 3 κ = (due to (9),()). The matrix M = ω ω 3 αω ω αω κω ω 3 κω and due to dω 3 = and dη 3 = = d(αω ) we can denote ω 3 = du, ω = dv/α. Then we get κ = tan(v g(u)) and α = cos(g(u))h(u)/cos(v g(u)), where g(u), h(u) are arbitrary function again. Integration yields, v e = f (v) cosu + f (v) sinu, () where f, f mean the same as in the previous case. For the choice ( ) ( f = v, v cosv, v sinv,, f = v, cosv sinv, sinv ) + cosv, v v we get the example of a developable surface () presented in Fig Conclusion The results we have presented show that the theory of two-parametric motion in the hyperbolic plane is similar to the theory of the same kind of motion of unit sphere S in the Euclidean space (see []), closely connected with the elliptic surface theory. It yields a natural interpretation of the group of projective space transformations preserving a one-sheet hyperboloid. All computations and figures were obtained by using MAPLE software. References [] W. Blaschke: Nicht-Euklidische Geometrie und Mechanik. Hamb. Math. Einzelschriften, 3. Heft, 9. [] W. Blaschke: Kinematik und Quaternionen. Deutscher Verlag der Wiss., Berlin 96. [3] S. Helgason: Differential Geometry and Symmetric Spaces. Academic Press, New York 96. [] V. Hlavatý: Úvod do neeuklidovské geometrie. JČSMF 99. [5] M. Husty, P. Nagy: Dreidimensionale Lie Gruppen und ebene Kinematik. Math. Pannonica, 5 (99). [6] A. Karger: Two-Parametric Motions in E 3. Aplikace Matematiky /3, 96 9 (987). ).

9 M. Hlavová: Two-parametric Motions in the Lobatchevski Plane 35 [7] A. Karger: Kinematic geometry of regular motions in homogeneous space. Czech. Math. J. 8 (3), (978). [8] A. Karger, J. Novák: Prostorová kinematika a Lieovy grupy. SNTL 978. English translation: Space Kinematics and Lie Groups. Gordon & Breach Science Publishers, Montreux 985. [9] J.-P. Serre: Lie algebras and Lie groups. Springer-Verlag 99. Received May 7, ; final form July 3,