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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 [email protected], Mesh Æ NoneD;
gr2 = Graphics3D@[email protected], Point@paD, Point@pbD, Point@pcD, Point@pdD<D;
gr3 = Graphics3D@[email protected], 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@[email protected]`D, Point@paD, Point@pbD, Point@pcD<D;
gr3 = Graphics3D@[email protected], 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@[email protected]`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 [email protected], 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