Möbius Geometry DMV- Jahrestagung

Köln 2011

by

Rolf Sulanke

The Examples

This Chapter contains some examples describing applications of interesting new functions introduced in the

packages presented in Chapter 1.

ü Example 1. Pseudo-Euclidean Orthogonalization

ü The Package neuvec.m

The package neuvec.m contains enhancements of Mathematica for applications to pseudo-Euclidean geometry.

?neuvec`*

neuvec`

ch ide orthonorm pscross psgram

chsort indexorder orthopair psCross pssp

dual normalize pr psfilter

The most important and interesting function in this section is

? orthonorm

Info3524719575-2877488

orthonorm: If b is a list of vectors in the n-dimensional

pseudo-Euclidean space of index k, then orthonorm@b,optsD is an orthogonal basis

of span@bD. orthonorm can be applied to finite dimensional vector spaces and

symmetric bilinear forms prod using the option innerprod -> prod;

the default is prod = pssp.

Options @orthonorm D

8innerprod → pssp, normed → True, print → False, neglect → −10<We show its action on a random sequence of vectors under the assumptions

dim = 7; ind = 3;

The Gram matrix of the pseudo - Euclidean scalarproduct:

Table @pssp @stb @i D, stb @j DD, 8i, dim <, 8j, dim <D êê MatrixForm

−1 0 0 0 0 0 0

0 −1 0 0 0 0 0

0 0 −1 0 0 0 0

0 0 0 1 0 0 0

0 0 0 0 1 0 0

0 0 0 0 0 1 0

0 0 0 0 0 0 1

We generate 10 random vectors; its pseudo-orthonormal coordinates are the rows of the random matrix

rm = randommatrix @10D

88−0.528316, −0.954825, 0.272704, −0.408792, −0.222726, 0.546233, 0.714865<,80.170297, −0.953355, 0.712824, −0.539402, 0.469621, 0.916763, 0.347088<,80.105655, 0.465732, 0.245054, 0.118592, −0.822119, 0.695669, −0.874039<,80.0537307, 0.0065073, 0.528303, 0.654277, 0.00855584, 0.733803, −0.062905<,8−0.122997, 0.462322, −0.981062, 0.766798, −0.169642, 0.749498, 0.55834<,8−0.702823, −0.0864053, −0.59759, −0.547316, −0.168555, 0.668541, 0.283818<,8−0.725197, 0.135776, 0.54258, −0.769913, 0.268296, 0.607473, 0.888303<,80.221531, 0.534493, −0.329622, 0.0112996, 0.759209, 0.515555, −0.0964204<,8−0.819058, −0.990289, 0.957216, −0.393598, 0.267347, 0.607301, 0.504531<,80.774958, 0.598806, −0.676517, 0.229728, −0.360818, −0.943774, −0.906604<<Their scalarproducts form the 10x10-matrix

Table @pssp @%@@i DD, %@@j DDD, 8i, Length @%D<, 8j, Length @%D<D êê MatrixForm

−0.239055 −0.149913 0.323491 −0.0229802 1.17686 0.538502 0.820362 0.756246

−0.149913 0.026418 0.135677 −0.0775437 1.54865 1.39076 1.2727 1.49642

0.323491 0.135677 1.64972 0.497853 0.301884 0.551621 −0.785263 −0.37145

−0.0229802 −0.0775437 0.497853 0.688542 1.53701 0.467219 −0.360119 0.557029

1.17686 1.54865 0.301884 1.53701 0.298895 −0.364321 0.695727 −0.330797

0.538502 1.39076 0.551621 0.467219 −0.364321 −0.00307509 0.860686 0.188052

0.820362 1.2727 −0.785263 −0.360119 0.695727 0.860686 0.984115 0.689457

0.756246 1.49642 −0.37145 0.557029 −0.330797 0.188052 0.689457 0.40821

−0.845556 −0.417213 0.0282104 −0.29658 1.68589 0.630364 0.212972 1.48924

−0.0115015 −0.552113 0.264753 −0.176425 −1.82141 −0.761059 −0.804574 −1.38522

We meaure the time needed for their orthogonalization in seconds:

Timing @om = orthonorm @rmDD

80.023997, 88−1.08055, −1.95288, 0.557755, −0.836091, −0.455535, 1.1172, 1.46209<,81.44543, −1.02175, 1.56128, −0.815625, 1.75574, 1.65466, −0.291641<,8−0.230082, −0.715304, 0.640006, −0.413818, −0.54729, 1.22588, 0.0257437<,80.580207, 0.170022, 0.78795, 0.889816, 0.683157, 0.814003, −0.255608<,8−2.94264, −0.428341, −2.50138, −0.114565, −2.47845, −1.69613, 2.25095<,82.52647, 1.92712, 1.72898, 0.768467, 1.89014, 0.47266, −3.11442<,80.187564, 1.51695, 0.943119, 0.71178, 0.266984, −0.0415539, −1.28303<<<

The process lasted ca. 0.03 s. By numerical reasons, the result contains numerical small errors:

2 DMV2011sh.nb

Table @pssp @om@@i DD, om@@j DDD, 8i, Length @omD<, 8j, Length @omD<D

99−1., 2.22045 × 10−16, −4.85723 × 10−16, −5.55112 × 10−17, 0.,

−8.88178 × 10−16, −2.44249 × 10−15=, 92.22045 × 10−16, 1., 1.31839 × 10−16,

1.52656 × 10−16, −1.33227 × 10−15, 2.55351 × 10−15, 7.77156 × 10−16=,9−4.85723 × 10−16, 1.31839 × 10−16, 1., 4.85723 × 10−17, −1.38778 × 10−16,

−1.41553 × 10−15, −1.07553 × 10−15=, 9−5.55112 × 10−17, 1.52656 × 10−16, 4.85723 × 10−17,

1., −2.22045 × 10−16, 1.11022 × 10−15, 7.77156 × 10−16=, 90., −1.33227 × 10−15,

−1.38778 × 10−16, −2.22045 × 10−16, −1., −8.88178 × 10−16, −1.33227 × 10−15=,9−8.88178 × 10−16, 2.55351 × 10−15, −1.41553 × 10−15, 1.11022 × 10−15, −8.88178 × 10−16, 1., 0.=,9−2.44249 × 10−15, 7.77156 × 10−16, −1.07553 × 10−15,

7.77156 × 10−16, −1.33227 × 10−15, 0., −1.==The built-in function Chop nullifies too small numbers

? Chop

Info3524719708-2877488

Chop@exprD replaces approximate real numbers in expr that are close to zero by the exact integer 0. à

Chop@%%D êê MatrixForm

−1. 0 0 0 0 0 0

0 1. 0 0 0 0 0

0 0 1. 0 0 0 0

0 0 0 1. 0 0 0

0 0 0 0 −1. 0 0

0 0 0 0 0 1. 0

0 0 0 0 0 0 −1.

Evaluating the next cell we obtain an orthonormal basis in its standard order :

som = indexorder @omD

88−1.08055, −1.95288, 0.557755, −0.836091, −0.455535, 1.1172, 1.46209<,8−2.94264, −0.428341, −2.50138, −0.114565, −2.47845, −1.69613, 2.25095<,80.187564, 1.51695, 0.943119, 0.71178, 0.266984, −0.0415539, −1.28303<,81.44543, −1.02175, 1.56128, −0.815625, 1.75574, 1.65466, −0.291641<,8−0.230082, −0.715304, 0.640006, −0.413818, −0.54729, 1.22588, 0.0257437<,80.580207, 0.170022, 0.78795, 0.889816, 0.683157, 0.814003, −0.255608<,82.52647, 1.92712, 1.72898, 0.768467, 1.89014, 0.47266, −3.11442<<

Table @Chop@pssp @%@@i DD, %@@j DDDD, 8i, Length @%D<, 8j, Length @%D<D êê MatrixForm

−1. 0 0 0 0 0 0

0 −1. 0 0 0 0 0

0 0 −1. 0 0 0 0

0 0 0 1. 0 0 0

0 0 0 0 1. 0 0

0 0 0 0 0 1. 0

0 0 0 0 0 0 1.

ü Example 2. Spheres and the Coxeter Invariant

ü The Package mspher.m

The package mspher.m contains functions relating Möbius and Euclidean geometry of spheres in the 3-dimen-

sional Möbius space.

DMV2011sh.nb 3

? mspher`*

mspher`

coxeterinv invstepro spt

euklidsphere plane spunit

euklidsphereplot3D planevec stepro

gencos pstg5frame vcenter

hplanevec sph4ptsvec vradius

hspherevector spherevec

The subspace orthogonal to a spacelike unit vector is a 4-dimensional pseudo-Euclidean space defining a sphere

in the 3-dimensional Möbius space, and vice versa. The space of all oriented spheres(including the planes as

spheres of radius 0) is isomorphic as transformation group under the action of the Möbius group G = OH1, 4L+ to

the hyper-hyperboloid of spacelike unit vectors vec[u] in the Minkowski space:

pssp @vec @uD, vec @uDD � 1

−u@1D2 + u@2D2 + u@3D2 + u@4D2 + u@5D2 � 1

The vector corresponding to the sphere with center {x,y,z} and radius r is

spv = spherevec @x, y, z, r D

:1 − r2 + x2 + y2 + z2

2 r,x

r,y

r,z

r,

−1 − r2 + x2 + y2 + z2

2 r>

Simplify @pssp @%, %DD

1

Consider the Hesse normal form of the equation of a plane

8a, b, c <.vec @x, 3 D � p

a x@1D + b x@2D + c x@3D � p

with distance p from the origin and the normal unit vector

nv = 8a, b, c <

nnv = nv.nv −> 1

a2 + b2 + c2 → 1

In[61]:= npv = planevec @ a, b, c, p D

Out[61]= 8p, a, b, c, p<stepro @81, 0, 0, 0, 1 <D

8∞, ∞, ∞<The infinite point belongs to each “sphere” whose spacelike unit vector has the shape npv:

In[62]:= pssp @npv, 81, 0, 0, 0, 1 <D

Out[62]= 0

Conversely, to any spacelike unit vector corresponds a uniquely defined sphere or plane with the parameter

representation

A sphere:

euklidsphere @spv D@s, t D

:x + r Cos@π sD CosBπ t

2F, y + r CosBπ t

2F Sin@π sD, z + r SinBπ t

2F>

A plane:

4 DMV2011sh.nb

In[63]:= euklidsphere @npv D@s, t D

Out[63]= : a p

a2 + b2 + c2+

b2 + c2

a2 + b2 + c2s,

b p

a2 + b2 + c2−

a b s

Ib2 + c2M Ia2 + b2 + c2M+

c2 t

b2 + c2 Abs@cD,

c p

a2 + b2 + c2−

a c s

Ib2 + c2M Ia2 + b2 + c2M−

b c t

b2 + c2 Abs@cD>

The standard unit sphere:

euklidsphereplot3D @spherevec @0, 0, 0, 1 DD

The plane at distance 1 from the origin parallel to the x, y - plane :

DMV2011sh.nb 5

euklidsphereplot3D @planevec @0, 0, 1, 1 DD

To enter options one has to enter all parameters of

? euklidsphereplot3D

euklidsphereplot3D @−stb @5D, −1, 1, −1, 1, PlotStyle → Opacity @.4 D, Mesh → NoneD

sph1 = %;

6 DMV2011sh.nb

ü Sphere through 4 Random Points

? sph4ptsvec

Info3524720195-2877488

sph4ptsvec@p1, p2, p3, p4D yields a spacelike vector of the

pseudo-Euclidean 5-space corresponding to the sphere through the four points

p1, p2, p3, p4 Hin general positionL of the Euclidean 3-space.

pa = randomv@3D; pb = -randomv@3D; pc = randomv@3D; pd = -2*randomv@3D;

Print@8pa, pb, pc, pd<D;

vv = sph4ptsvec@pa, pb, pc, pdD;

Print@vvD;

gr1 = euklidsphereplot3D@vv, -1, 1, -1, 1, PlotStyle Æ Opacity@.2D, Mesh Æ NoneD;

gr2 = Graphics3D@8PointSize@0.015D, Point@paD, Point@pbD, Point@pcD, Point@pdD<D;

gr3 = Graphics3D@8PointSize@0.02D, Blue, Point@vcenter@vvDD<D;

Show@8gr1, gr2, gr3<D

880.961433, 0.0317091, −0.832076<, 80.128135, 0.57306, 0.638669<,80.156624, 0.112656, −0.0886155<, 8−0.915503, 0.0486348, −0.20589<<

80.5542, −0.223947, 2.06904, −1.05049, 0.00316966<

ü The Coxeter Invariant (Inversive Distance of two spheres)

? coxeterinv

Info3524720351-2877488

coxeterinv@r1, r2, dD is the conformal invariant of two

hyperspheres

with radii r1, r2 and distance d of their centers.

coxeterinv @r, R, d D

−d2 + r2 + R2

2 r R

DMV2011sh.nb 7

We show that the Coxeter invariant coincides with the single Möbius geometric invariant of two hyperspheres. It

follows that it is a conformal invariant.

mr = spherevec @x, y, z, r D; mR = spherevec @X, Y, Z, R D; Simplify @pssp @mr, mRDD

−1

2 r RI−r2 − R2 + x2 − 2 x X + X2 + y2 − 2 y Y + Y2 + z2 − 2 z Z + Z2M

Simplify Ax2 − 2 x X + X2 + y2 − 2 y Y + Y2 + z2 − 2 z Z + Z2 − 8X− x, Y − y, Z − z<. 8X− x, Y − y, Z − z<E

0

%% ê. x 2 − 2 x X + X2 + y2 − 2 y Y + Y2 + z2 − 2 z Z + Z2 → d^2

−d2 − r2 − R2

2 r R

Simplify @% == coxeterinv @r, R, d DD

True

This proves the Möbius invariance of the Coxeter inversive distance. Similarly, we obtained Euclidean expres-

sions for the Möbius invariants of pairs of circles in the notebook mcircles.nb, Example 6 below, and for other

pairs of subspheres in the notebook pairs,nb, see [13].

Clear @mr, mRD

ü Example 3. Circle through Three Random Points

The package mcirc.m containes functions specific for the geometry of circles in the 3-dimensional Euclidean

space and the Möbius space. For details of the geometry of the 6-dimensional circle space see [10].

? mcirc`*

mcirc`

adaptsplframe control ppdet

center gencircle pptr

center3pts plotcircle2spv pscomplement

circle2spv plotcircle3D radius

circle3D plotcircle3pts radius3pts

circle3pts plotgencircle threepoints

circlefamily posvec tube

circleplane posvec3pts unitvec

circlespacevectors pp vspace

A circle is defined by three of its points in the Eucldean n-space. On the other hand, in projective or Möbius

geometry, it is an elliptic quadric in a 2-dimensional subspace, corresponding to a 3-dimensional pseudo-

Euclidean subspace of the (n+2)-dimensional Lorentz space. Such a subspace is obtained as a function of the

points by

? vspace

Info3524728355-7085145

If p is a List of points of the Euclidean n-space, then vspace@pD yields an

orthonormal basis of the Euclidean subspace

in the Hn+2L-dimensional Lorentz space being the orthogonal complement

of the pseudo-orthogonal subspace corresponding to the maximal subsphere through the points of the List p.

vspace@p1,p2,p3D yields a basis of the Euclidean subspace corresponding to the circle through the points p1,p2,p3.

? randommatrix

8 DMV2011sh.nb

rm3 = randommatrix @3, 3 D

88−0.0458312, −0.148651, 0.708336<,80.495644, 0.449637, 0.0763914<, 8−0.89047, 0.17216, −0.780091<<

vspace @rm3D

880.184636, −0.406552, 0.806978, 0.413907, 0.215114<,8−0.188408, 0.365205, −0.158207, 0.0991218, 0.931272<<

Table @Chop@pssp @%@@i DD, %@@j DDDD, 8i, Length @%D<, 8j, Length @%D<D êê MatrixForm

K 1. 0

0 1.O

Table @Chop@pssp @invstepro @rm3@@i DDD, %%@@j DDDD, 8i, 3 <, 8j, 2 <D

880, 0<, 80, 0<, 80, 0<<These orthogonality relations show tha the three points lie on the circle corresponding to vspace[rm3]- Example:

pa = randomv@3D; pb = -randomv@3D; pc = 2*randomv@3D;

Print@8pa, pb, pc<D;

gr1 = plotcircle3pts@pa, pb, pcD;

gr2 = Graphics3D@8PointSize@0.015`D, Point@paD, Point@pbD, Point@pcD<D;

gr3 = Graphics3D@8PointSize@0.02D, Blue, Point@center3pts@pa, pb, pcDD<D;

Show@8gr1, gr2, gr3<D

88−0.562791, −0.946697, 0.0787639<,80.741523, −0.4055, −0.88538<, 8−1.5862, 1.66307, 0.0883381<<

-1

0

1

-1

0

1

2

-1.0

-0.5

0.0

0.5

DMV2011sh.nb 9

pa = randomv@3D; pb = 5* randomv@3D; pc = Hpa + pbL ê2;

Print@8pa, pb, pc<D;

gr1 = plotcircle3pts@pa, pb, pcD;

gr2 = Graphics3D@8PointSize@0.015`D, Point@paD, Point@pbD, Point@pcD<D;

Show@8gr1, gr2<D

88−0.271244, 0.094243, −0.0798477<,82.93209, 3.76536, 4.95649<, 81.33042, 1.9298, 2.43832<<

-10

-5

0

5

10

-10

-5

0

5

10

-10

0

10

ü Example 4. Tubes of Torus Knots

? tube

Info3524720782-2877488

tube@x,a,b,r,optD plots the tube of the curve x HEnter only

the name, not the parameter!L with tube radius r for the parameter s,

asb, with options opt of ParametricPlot3D.

The general circular torus is the surface of revolution

Clear@torussfD; torussf@a_, b_D@u_, v_D := 8Ha + b*Cos@uDL*Cos@vD, Ha + b*Cos@uDL*Sin@vD, b*Sin@uD<

10 DMV2011sh.nb

In[66]:= ParametricPlot3D@torussf@1, 1 ê3D@u, vD, 8u, 0, 2*Pi<,

8v, 0, 2*Pi<, MeshÆ None, PlotStyleÆ Opacity@.6D, PlotRange Æ AllD

Out[66]=

In[67]:= torus1 =%;

A torus knot is a curve on the torus with the parameter representation

torkn@a_, b_D@p_, q_D@t_D := 8Ha + b*Cos@p* tDL*Cos@q* tD, Ha + b*Cos@p* tDL*Sin@q* tD, b*Sin@p* tD<

ParametricPlot3D @torkn @1, 1 ê 3D@4, 3 D@t D, 8t, 0, 2 ∗ Pi <, PlotStyle → 8Red, Thick <D

-1

0

1

-1

0

1

-0.2

0.0

0.2

tc1 = %;

DMV2011sh.nb 11

Show@8tc1, torus1 <, Axes → None, Boxed → False D

A tube of radius r of such a curve with parameter p and circleparameter s can be defined as

torusknottube @a_, b_ D@p_, q_, r_, opts___ D : = tube @torkn @a, b D@p, q D, 0, 2 ∗ Pi, r, opts D

Here a,b are the parameters defining the torus, p,q are integers defining the knot, and r is the tube radius

In[64]:= torusknottube @1, 1 ê 3D@4, 3, 0.05, PlotPoints → 880, 20 <D

Out[64]=

12 DMV2011sh.nb

Show@8%, torus1 <, Axes → None, Boxed → False D

Out[69]=

ü Example 5. Euclidean and Möbius Invariants of a Circle Pair

ü The Double Projection and the Möbius Invariants of Circle Pairs

With the notatioons and assumptions of the last subsection using the orthogonal decompositions of the pseudo-

Euclidean vector space defined by the subspaces U0,U1, we consider the double projection

? pp

Info3524721352-2877488

pp: If V and W are the Euclidean vector subspaces spanned by the

orthonormal bases 8v1, v2<, 8w1, w2< respectively, then pp@v1, v2, w1,w2D

is the matrix of the composition p = p2 p1 of the orthogonal projections

p1: V -> W, and p2: W -> V, with respect to the base 8v1, v2<.

An elementary calculation yields the following symmetric matrix

Clear @v1, v2, w1, w2 D; pp @v1, v2, w1, w2 D êê MatrixForm

pssp@v1, w1D2 + pssp@v1, w2D2 pssp@v1, w1D pssp@v2, w1D + pssp@v1, w2

pssp@v1, w1D pssp@v2, w1D + pssp@v1, w2D pssp@v2, w2D pssp@v2, w1D2 + pssp@v2, w2

The same matrix is obtained if one considers the extremal problem for the function pssp[v,w] under the side

conditions vœ U0, wœU1, with pssp[v,v] = 1, pssp[w,w]= 1. The eigenvalues of the matrix correspond to the

extrema of the function under the side conditions.

A complete system of Euclidean invariants of a pair (C0,C1) of circles is obtained by the

following entities:

The radii r of C0, R of C1,

the distance d of the centers,

the angle a between the position vectors nc0, nc1,

the angle b between the position vector nc0, and the line connecting the centers,

the angle c between the position vector nc1, and the line connecting the centers.

Proposition. The double projection pp is a selfadjoint operator Möbius equivariantly associated

to the circle pair. Its Eigenvalues or, equivalently, its trace and its determinant, are a complete

invariant system in the manifold of all circle pairs. they can be expressed by the Eucliden

invariants mentioned above by the formulas

DMV2011sh.nb 13

euforminvdet@r, R, d, a, b, cD

II−d2 + r2 + R2M Cos@aD + 2 d2 Cos@bD Cos@cDM24 r2 R2

euforminvtr@r, R, d, a, b, cD

1

4 r2 R2

Id4 + r4 + 4 r2 R2 + R4 + 2 r2 R2 Cos@2 aD + 2 d2 Ir2 Cos@2 bD + R2 Cos@2 cDMMThe proof of the proposition can be found in [6] or [4]. The formulas are derived in the Mathematica notebook

[10]; we repeat the calculations in the next subsection.

3. Curves of Constant Möbius Curvatures and Dupin Cyclides

The curves of constant curvatures in any Klein geometry coincide with the orbits of 1-parametric subgroups of the

transitive transformation group defining the geometry. In [11] the classification of the 1-parametric subgroups of

the Möbius group O[1,4] with respect to conjugation and equivalently the classification of the curves of constant

Möbius curvatures in the 3-dimensional Möbius space has been carried out in this general context. Here we show

some aspects of this classification.

ü Direct Calculation of the Curves of Constant Curvatures

We consider the underlying pseudo-Euclidean vector space in isotropic-orthogonal coordinates, since the

points,represented by isotropic vectors, are the basic objects now. The scalar products of the standard basis

vectors satisfy

Table @stb @i D.io @D.stb @j D, 8i, dim <, 8j, dim <D êê MatrixForm

0 0 0 0 −1

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

−1 0 0 0 0

A generally curved curve in the Möbius space has two curvatures k, h being functions of a natural parameter t,

and a canonical defined isotropic-orthogonal moving frame (vec[b[i][t]]), i = 1,...,5, satisfying the Frenet formu-

las with the Frenet matrix

Clear @cc D; cc @k_, h_ D : = mfre3D @k, h D

cc @k, h D êê MatrixForm

0 k 1 0 0

1 0 0 0 k

0 0 0 −h 1

0 0 h 0 0

0 1 0 0 0

The solution of the Frenet equation with constant curvatures k,h and the start condition b[0][i] = stb[i] is

gr @k_, h_ D@t_ D : = MatrixExp @cc @k, h D ∗ t D

FullSimplify @D@gr @k, h D@t D, t D − gr @k, h D@t D.cc @k, h DD

880, 0, 0, 0, 0<, 80, 0, 0, 0, 0<, 80, 0, 0, 0, 0<, 80, 0, 0, 0, 0<, 80, 0, 0, 0, 0<<The matrix cc[k,h] represents an element of the Lie algebra of the Möbius group in isotropic-orthogonal

coordinates:

14 DMV2011sh.nb

Transpose @cc @k, h DD.io @D + io @D.cc @k, h D

880, 0, 0, 0, 0<, 80, 0, 0, 0, 0<, 80, 0, 0, 0, 0<, 80, 0, 0, 0, 0<, 80, 0, 0, 0, 0<<Therefore gr[k,h][t] is a 1-parametric subgroup of the Möbius group, with group parameter t, whose orbits are

curves of constant curvatures, or eventually fixed points. The stereographic projection with the North pole

{0,0,0,1} of the unit 3-sphere as center applied to the orbit yields an Euclidean image of the orbit. Its parameter

representation is

cccurve @k_, h_, x_, y_, z_ D@t_ D : =

stepro @transoi @D.gr @k, h D@t D.transio @D.invstepro @8x, y, z <DD

transio @D.invstepro @80, 0, 0 <D

: 2 , 0, 0, 0, 0>The orbit of the origin is a curve with constant curvatures k,h:

cco @k_, h_ D@t_ D : = stepro @transoi @D.Transpose @gr @k, h D@t DD@@1DDD

Simplify @cco @k, h D@t D − cccurve @k, h, 0, 0, 0 D@t DD

80, 0, 0<ParametricPlot3D @cco @−2, 1 ê 2D@t D, 8t, −16 ∗ Pi, 16 ∗ Pi <, PlotRange → All D

-10

1

-1

0

1

0.0

0.2

0.4

0.6

cco1 = %;

In[70]:= Limit @cco @−2, 1 ê 2D@t D, t → Infinity, Direction → 1D

Mathematica can’t calculate these Limits, but an elementary calculation shows that the Limit equals the point

ppt correspopnding to the fixed vector vv (see below).

vv = fv @−2, 1 ê 2D

:1, 0, 0, 1,1

2>

vv.io @D.vv

0

ppt = stepro @transoi @D.vv D

:0, 0,1

2>

DMV2011sh.nb 15

lppt = Graphics3D @8PointSize @.015 D, Point @ppt D<D

Show@8cco1, lppt <, PlotRange → All D

-1

0

1

-1

0

1

0.0

0.2

0.4

0.6

16 DMV2011sh.nb

ParametricPlot3D @sphericalreflection @cco @−2, 1 ê 2D@t D, ppt, 1 D,

8t, −10 ∗ Pi, 10 ∗ Pi <, PlotRange → All D

-2

0

2-10

-5

0

5

10

-0.50.00.5

ü A Raw Classification

The geometry of the curve depends on the invariant defining the character of the fixed vector corrresponding to

the eigenvalue 0:

ses = Simplify @Eigensystem @cc @k, h DDD;

eva@k_, h_ D = ses @@1DD; eve @k_, h_ D = ses @@2DD; Clear @ses D

Here are the Eigenvalues of cc[k,h]

eva @k, h D

:0, −

� h2 − 2 k + 4 + h4 + 4 h2 k + 4 k2

2, −

h2

2+ k −

1

24 + h4 + 4 h2 k + 4 k2 ,

− −h2

2+ k +

1

24 + h4 + 4 h2 k + 4 k2 , −

h2

2+ k +

1

24 + h4 + 4 h2 k + 4 k2 >

Thus the eigenvalues eva[[2]],eva[[3]] are purely imaginary and conjugated.

And here are the corresponding Eigenvectors in isotropic-orthonormal coordinates:

DMV2011sh.nb 17

eve @k, h D

::−k, 0, 0,1

h, 1>, :1

2−h2 − 4 + h4 + 4 h2 k + 4 k2 , −

� h2 − 2 k + 4 + h4 + 4 h2 k + 4 k2

2,

� h2 − 2 k + 4 + h4 + 4 h2 k + 4 k2 h2 + 2 k + 4 + h4 + 4 h2 k + 4 k2

2 2,

−1

2h h2 + 2 k + 4 + h4 + 4 h2 k + 4 k2 , 1>,

:12

−h2 − 4 + h4 + 4 h2 k + 4 k2 , −h2

2+ k −

1

24 + h4 + 4 h2 k + 4 k2 ,

−

� h2 − 2 k + 4 + h4 + 4 h2 k + 4 k2 h2 + 2 k + 4 + h4 + 4 h2 k + 4 k2

2 2,

−1

2h h2 + 2 k + 4 + h4 + 4 h2 k + 4 k2 , 1>,

:12

−h2 + 4 + h4 + 4 h2 k + 4 k2 , − −h2

2+ k +

1

24 + h4 + 4 h2 k + 4 k2 ,

h2 + 2 k − 4 + h4 + 4 h2 k + 4 k2 −h2 + 2 k + 4 + h4 + 4 h2 k + 4 k2

2 2,

−1

2h h2 + 2 k − 4 + h4 + 4 h2 k + 4 k2 , 1>,

:12

−h2 + 4 + h4 + 4 h2 k + 4 k2 , −h2

2+ k +

1

24 + h4 + 4 h2 k + 4 k2 ,

−h2 − 2 k + 4 + h4 + 4 h2 k + 4 k2 −h2 + 2 k + 4 + h4 + 4 h2 k + 4 k2

2 2,

−1

2h h2 + 2 k − 4 + h4 + 4 h2 k + 4 k2 , 1>>

The first eigenvector corresponds to the eigenvalue 0 and is a fixed vector of the 1-parametric subgroup gr[k,h[[t]

eve @k, h D@@1DD

:−k, 0, 0,1

h, 1>

To avoid infinite expressions, we consider the proportional eigenvector

fv @k_, h_ D : = 8H−hL ∗ k, 0, 0, 1, h <

cc @k, h D.fv @k, h D

80, 0, 0, 0, 0<and therefore a fixed vector of the 1-parametric subgroup gr[k,h][t]:

18 DMV2011sh.nb

Simplify @gr @k, h D@t D.fv @k, h DD

8−h k, 0, 0, 1, h<Deciding for the geometry of the orbits is the character of this fixed vector:

fv @k, h D.io @D.fv @k, h D

1 + 2 h2 k

chB:: usage =

"chB @k,h D defines the character of the curve of constant curvatures k, h in

the 3 −dimensional Moebius space; it equals the invariant B.";

The invariant B appears in [12].

chB@k_, h_ D : = 1 + 2 ∗ k ∗ h^2

We distinguish three cases:

chB[k,h] = 0, the Euclidean case: the fixed vector is isotropic. The 1-parametric subgroup is contained

in the isometry group of the Euclidean space.

chB[k,h] < 0, the elliptic case: the fixed vector is timelike. The 1-parametric subgroup is contained in

the isometry group of the elliptic space, or the 3-sphere.

chB[k,h] > 0, the hyperbolic case: the fixed vector is spacelike. The 1-parametric subgroup is contained

in the group O[1,3] of the hyperbolic space.

One may show:

Lemma. To any “generic” infinitesimal transformationcc[k,h] there exists a uniquely defined

maximal abelian Lie subalgebra, which is generically two-dimensional.

The two-dimensional orbits of the corresponding two-dimensional abelian subgroups of the Moebius group are

named Dupin cyclides. It follows:

Corollary. To any generic curve of constant curvatures there exists a uniquely defined Dupin

cyclide containing it.

In the next subsection we consider the three cases in particular.

ü The Euclidean Case chB[k,h] = 0

The generally curved orbits are the helices.

DMV2011sh.nb 19

r = 3; b = 2; ParametricPlot3D @euccc @b, r, 0, 0 D@t D,

8t, −4 ∗ Pi, 4 ∗ Pi <, PlotStyle → 8 Red, Thick <D

-20

2

-2

0

2

-10

0

10

helb2r3 = %;

The corresponding Dupin cyclide is the circular cylinder

eudupinsf @b_, x_, y_, z_ D@u_, v_ D : =

stepro @transoi @D.eudupingr @1, b D@u, v D.transio @D.invstepro @8x, y, z <DD

20 DMV2011sh.nb

ParametricPlot3D @eudupinsf @2, 3, 0, 0 D@u, v D, 8u, 0, 2 ∗ Pi <, 8v, −13, 13 <,

PlotRange → All, PlotPoints → 100, PlotStyle → 8Opacity @.6 D<, Mesh → NoneD

dupinsfb2r3 = %;

Show@8dupinsfb2r3, helb2r3 <, Boxed → False, Axes → NoneD

DMV2011sh.nb 21

r = 3; b = 2; ParametricPlot3D @sphericalreflection @euccc @b, r, 0, 0 D@t D, 85, 0, 0 <, 1 D,

8t, −40 ∗ Pi, 40 ∗ Pi <, PlotStyle → 8 Red, Thick <, PlotRange → All D

4.6

4.8

5.0

-0.1

0.0

0.1

-0.10

-0.05

0.00

0.05

0.10

refleucc = %;

ParametricPlot3D @sphericalreflection @eudupinsf @2, 3, 0, 0 D@u, v D, 85, 0, 0 <, 1 D,

8u, 0, 2 ∗ Pi <, 8v, −50, 50 <, PlotRange → All, PlotPoints → 100,

PlotStyle → 8Opacity @.6 D<, Mesh → None, PlotRange → All D

reflcyl = %;

22 DMV2011sh.nb

Show@8reflcyl, refleucc <, Boxed → False, Axes → NoneD

ü The Elliptic Case chB[k,h] <0

Since now chB < 0 a timelike unit vector, say stb[1], is invariant, and the 1-parametric subgroup lies in the

isometry group of the elliptic space, modelled in the 4-dimensional Euclidean space by the unit sphere. There-

fore the curves of constant Möbius curvatures coincide withe the curves of constantmetric curvatures on the 3-

sphere. These are the torus knots

The stereographic projection shows a conformal image in the Euclidean 3-space:

ParametricPlot3D @sterproj @grell @3, 5 ê 3, t D. 81, 0, 1, 0 < ê Sqrt @2DD,

8t, 0, 6 ∗ Pi <, PlotRange → All, PlotStyle → 8Red, Thick <D

-2

-1

0

1

2-2

-1

0

1

2

-1.0

-0.5

0.0

0.5

1.0

elcc = %;

The torus containing this curve is the orbit of the toral group

DMV2011sh.nb 23

ParametricPlot3D @sterproj @tor @3, 5 ê 3, u, v D. 81, 0, 1, 0 < ê Sqrt @2DD,

8u, 0, 2 ∗ Pi <, 8v, 0, 2 ∗ Pi <, PlotStyle → Opacity @.2 D, Mesh → NoneD

torus2 = %;

Show@8torus2, elcc <, Boxed → False, Axes → NoneD

Proposition. The elliptic curves of constant conformal curvatures grell[a, b, t].ptell[u, v, w] are

isogonal trajectories of the meridians of the toral orbits tor[1,1, s,t].ptell[u, v, w]. They are

curves of constant Riemannian curvatures nin the standard 3-sphere.

ü The Hyperbolic Case chB[k,h] >0

In this case the fixed vector is spacelike. Its orthogonal complement is a four-dimenional pseudo-Euclidean

subspace. Its isotropie group is the pseudo-orthogonal group O[1,3] (the Lorentz group) The hyper-hyperboloids

pssp @vec @x, 4 D, vec @x, 4 DD � − r^2

−x@1D2 + x@2D2 + x@3D2 + x@4D2 � −r2

are orbits of the Lorentz group. With the induced Riemannian metric they are isometric models of the 2-dimen-

24 DMV2011sh.nb

are orbits of the Lorentz group. With the induced Riemannian metric they are isometric models of the 2-dimen-

sional hyperbolic space. It followws

Proposition. A curve of constant conformal curvatures of hyperbolic type is also a curve of

constant Riemannian curvatures; the invariant sphere of the generating 1-parametric subgroup

may serve as the Infinity of the hyperbolic space being the set of inner points of the ball

bounded by the sphere. The fixed points of the generating subgroup belong to the sphere. Any

non-flat curve of constant curvatures runs from Infinity to Infinity

ü A General Example

Here is a line as annon generic orbit

ParametricPlot3D @orbit @2, 8, 0, 0, 0 D@t D,

8t, −20, 20 <, PlotStyle → 8Red, Thickness @0.005 D<D

-1.0

-0.5

0.0

0.5

1.0

-1.0

-0.5

0.0

0.5

1.0

-1.0

-0.5

0.0

0.5

1.0

xax = %;

A more interesting orbit of the sanme subgroup is

orbit @2, 8, Cos @.3 D, Sin @.3 D, 0 D@t D

: 1. H0. + 0.955336 Cosh@2 tD + Sinh@2 tDL0. + Cosh@2 tD + 1.91067 Cosh@tD Sinh@tD,

0. +0.29552 Cos@8 tD

0. + Cosh@2 tD + 1.91067 Cosh@tD Sinh@tD, 0. +0.29552 Sin@8 tD

0. + Cosh@2 tD + 1.91067 Cosh@tD Sinh@tD>

DMV2011sh.nb 25

ParametricPlot3D @orbit @2, 8, Cos @.3 D ê 2, Sin @.3 D, 0 D@t D,

8t, −10, 10 <, PlotStyle → 8Green, Thickness @0.005 D<, PlotRange → All D

-1.0

-0.5

0.0

0.5

1.0

-0.2

0.0

0.2

-0.2

0.0

0.2

a2b3cc = %;

Analogously tom the elliptic case the hyperbolic Dupin cyclides are defined as the orbits of the maximal abelian

subgroups

dupin @a_, b_, x_, y_, z_, u_, v_ D : = stepro @grpso @a ∗ u, b ∗ v, 1 D.invstepro @8x, y, z <DD

ParametricPlot3D @dupin @2, 8, Cos @.3 D ê 2, Sin @.3 D, 0, u, v D, 8u, −5, 5 <,

8v, 0, 2 ∗ Pi ê 8<, PlotRange → All, PlotStyle → 8Opacity @.2 D<, Mesh → NoneD

dup1 = %;

26 DMV2011sh.nb

Show@8dup1, a2b3cc, xax, sph1 <, Boxed → False, Axes → NoneD

ü An Euclidean Realization of the Hyperbolic Curves of Constant Curvatures

In the hyperbolic case we have two real isotropic eigenvectors to opposite real non zero eigenvalues. The corre-

sponding points of the Möbius space are fixed points of the 1-parametric transformation group gr[k,h]. Therefore

the complement of such a fixed point is invariant under these transformations too. Taking it as the conformal

model of the Euclidean space the orbits of the non fixed points of this complement are Euclidean realizations of

the hyperbolic curves of constant curvatures. In general they are not helices and don't have constant Euclidean

curvatures. An example of this kind are the isogonal trajectories of the generators of a circular conus which is a

Dupin cyclide, too.

The spiral groups of the Euclidean space are 1-parametric groups of conformal transformations depending on one

parameter a, defined by

gd@a_, t_ D : = 88Cos@t D, −Sin @t D, 0 <, 8Sin @t D, Cos @t D, 0 <, 80, 0, 1 << ∗ Exp@a ∗ t D

FullSimplify @gd@a, t D.gd @a, s D � gd@a, t + sDD

True

The 3D-spirals are defined as their orbits for b∫0:

gdcc @a_, b_ D@t_ D : = gd@a, t D. 8b, 0, 1 <

gdcc @a, b D@t D

9b a t Cos@tD, b a t Sin@tD, a t=

DMV2011sh.nb 27

ParametricPlot3D @gdcc @.1, .5 D@t D, 8t, −30, 3 <, Axes → False,

Boxed → False, PlotRange → All, PlotStyle → 8Thick <D

consp = %;

Here is the parameter representation of a circular cone depending on the parameter b

conus @b_, u_, v_ D : = 8b ∗ v ∗ Cos@uD, b ∗ v ∗ Sin @uD, v <

28 DMV2011sh.nb

ParametricPlot3D @conus @.5, u, v D, 8u, 0, 2 ∗ Pi <, 8v, 0, 1.4 <, Mesh → None,

Axes → False, Boxed → False, PlotStyle → 8Opacity @.2 D, Blue <, PlotRange → All D

con1 = %;

Here is a parameter representation of the spiral cylinder:

gsp @a_, b_ D@t_, v_ D : = 9b a t Cos@t D, b a t Sin @t D, v ∗ a t =

ParametricPlot3D @gsp @.1, .5 D@t, s D, 8t, −30, 3 <, 8s, 0, 1 <, Mesh → None,

Axes → False, Boxed → False, PlotPoints → 100, PlotStyle → 8Opacity @.4 D, Red <D

spsf = %;

Show@8con1, consp, spsf <D

DMV2011sh.nb 29

sp3D = %;

It follows:

Proposition. The 3D-spiral gdcc[a,b] is the intersection of two homogeneous surfaces of the Möbius space, a

spiral cylinder and a circular cone.

4. The Homogeneous Tori Problem

In the papers Moebius Geometry V - VII (see items [29], [31], [32] in the Publication List on my homepage) we

classified the homogeneous surfaces in the 3-dimensional Moebius space. The only compact homogeneous, non

umbilic surfaces are the homogeneous tori as defined in subsection 3.5. Any surface without umbilic points has

an invariant “first fundamental form”, i. e. a Riemannian metric invariant under Moebius transformations. On a

homogeneous surface the Gauss curvature K of this metric must be constant, and by the Gauss- Bonnet theorem

we conclude K = 0 for homogeneous tori. The problem I want to remember is:

Do there exist immersions of the torus manifold M2 = S1x S1 into the Moebius space S3 without umbilic

points with Gauss curvature K = 0 not being Moebius equivalent to a homogeneous torus?

As far as I know the problem is unsolved yet. The proof of the non-existence of such immersion would be an

interesting global characterization of the homogeneous tori.

References

Homepage

Home

http://www-irm.mathematik.hu-berlin.de/~sulanke

30 DMV2011sh.nb