# Möbius Geometry

## Full text

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### 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,optsDis 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@orthonormD

8innerprodpssp, normedTrue, printFalse, 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:

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Table@pssp@stb@iD, stb@jDD,8i, dim<,8j, dim<D êêMatrixForm

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

880.528316,0.954825, 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<, 80.702823,0.0864053,0.59759,0.547316,0.168555, 0.668541, 0.283818<, 80.725197, 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<, 80.819058,0.990289, 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@%@@iDD,%@@jDDD,8i, Length@%D<,8j, Length@%D<D êêMatrixForm

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

80.023997,881.08055,1.95288, 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<, 82.94264,0.428341,2.50138,0.114565,2.47845,1.69613, 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:

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Table@pssp@om@@iDD, om@@jDDD,8i, Length@omD<,8j, Length@omD<D 991., 2.22045×1016,−4.85723×1016,5.55112×1017, 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=, 94.85723×10−16, 1.31839×10−16, 1., 4.85723×10−17,1.38778×10−16,

1.41553×1015,1.07553×1015=,95.55112×1017, 1.52656×1016, 4.85723×1017, 1.,−2.22045×1016, 1.11022×1015, 7.77156×1016=,90.,−1.33227×1015,

−1.38778×10−16,−2.22045×10−16,−1.,−8.88178×10−16,1.33227×10−15=,

98.88178×10−16, 2.55351×10−15,1.41553×10−15, 1.11022×10−15,8.88178×10−16, 1., 0.=, 92.44249×10−15, 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@exprDreplaces approximate real numbers inexprthat are close to zero by the exact integer 0. à Chop@%%D êêMatrixForm

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

881.08055,1.95288, 0.557755,0.836091,0.455535, 1.1172, 1.46209<, 82.94264,0.428341,2.50138,0.114565,2.47845,1.69613, 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<, 80.230082,0.715304, 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@%@@iDD,%@@jDDDD,8i, Length@%D<,8j, Length@%D<D êêMatrixForm

−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.

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? mspher`*

mspher`

coxeterinv invstepro spt

euklidsphere plane spunit

euklidsphereplot3D planevec stepro

gencos pstg5frame vcenter

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@5D21

The vector corresponding to the sphere with center {x,y,z} and radius r is spv = spherevec@x, y, z, rD

:1r2+x2+y2+z2 2 r

, x r ,

y r ,

z r ,

1r2+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, 3D p

a x@1D+b x@2D+c x@3Dp

with distance p from the origin and the normal unit vector nv =8a, b, c<

nnv = nv.nv −> 1 a2+b2+c21

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

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@spvD@s, tD :x+r Cos@πsDCosBπt

2 F, y+r CosBπt

2 FSin@πsD, z+r SinBπt 2 F>

A plane:

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In[63]:= euklidsphere@npvD@s, tD

Out[63]= : a p

a2+b2+c2

+ b2+c2 a2+b2+c2

s, b p a2+b2+c2

a b s

Ib2+c2M Ia2+b2+c2M

+ c2t 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, 1DD

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

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

To enter options one has to enter all parameters of

? euklidsphereplot3D

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

sph1 = %;

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ü Sphere through 4 Random Points

? sph4ptsvec

Info3524720195-2877488

sph4ptsvec@p1, p2, p3, p4Dyields a spacelike vector of the

pseudo-Euclidean 5-space corresponding to the sphere through the four points p1, p2, p3, p4Hin general positionLof 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, PlotStyleÆ Opacity@.2D, Mesh Æ NoneD;

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<,80.915503, 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, dDis the conformal invariant of two hyperspheres

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

coxeterinv@r, R, dD

−d2+r2+R2 2 r R

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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, rD; mR = spherevec@X, Y, Z, RD; Simplify@pssp@mr, mRDD

1

2 r RIr2R2+x22 x X+X2+y22 y Y+Y2+z22 z Z+Z2M

SimplifyAx22 x X+X2+y22 y Y+Y2+z22 z Z+Z2 8Xx, Yy, Zz<.8Xx, Yy, Zz<E 0

%%ê. x22 x X+X2+y22 y Y+Y2+z22 z Z+Z2 d ^ 2

d2r2R2 2 r R

Simplify@% == coxeterinv@r, R, dDD 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`

center gencircle pptr

center3pts plotcircle2spv pscomplement

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@pDyields an orthonormal basis of the Euclidean subspace

in theHn+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,p3Dyields a basis of the Euclidean subspace corresponding to the circle through the points p1,p2,p3.

? randommatrix

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rm3 = randommatrix@3, 3D

880.0458312,0.148651, 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<, 80.188408, 0.365205,0.158207, 0.0991218, 0.931272<<

Table@Chop@pssp@%@@iDD,%@@jDDDD,8i, Length@%D<, 8j, Length@%D<D êêMatrixForm

K1. 0

0 1.O

Table@Chop@pssp@invstepro@rm3@@iDDD,%%@@jDDDD,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;

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

Show@8gr1, gr2, gr3<D

880.562791,−0.946697, 0.0787639<,

80.741523,0.4055,0.88538<,81.5862, 1.66307, 0.0883381<<

-1

0

1 -1

0 1

2

-1.0 -0.5 0.0 0.5

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

Print@8pa, pb, pc<D;

gr1=plotcircle3pts@pa, pb, pcD;

Show@8gr1, gr2<D

880.271244, 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,optDplots the tube of the curve xHEnter 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<

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

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

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, 1ê3D@4, 3D@tD,8t, 0, 2Pi<, PlotStyle 8Red, Thick<D

-1

0

1

-1 0

1 -0.2

0.0 0.2

tc1 = %;

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Show@8tc1, torus1<, Axes None, Boxed FalseD

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, bD@p, qD, 0, 2Pi, r, optsD 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, 1ê3D@4, 3, 0.05, PlotPoints 880, 20<D

Out[64]=

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Show@8%, torus1<, Axes None, Boxed FalseD

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 bases8v1, 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 base8v1, v2<.

An elementary calculation yields the following symmetric matrix Clear@v1, v2, w1, w2D; pp@v1, v2, w1, w2D êêMatrixForm

pssp@v1, w1D2+pssp@v1, w2D2 pssp@v1, w1Dpssp@v2, w1D+pssp@v1, w2 pssp@v1, w1Dpssp@v2, w1D+pssp@v1, w2Dpssp@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.

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2

2

2

2

2

2

2

2

2

4

4

2

2

4

2

2

2

2

2

### Cos @ 2 c DMM

The 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@iD.io@D.stb@jD,8i, dim<,8j, dim<D êêMatrixForm 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@ccD; cc@k_, h_D := mfre3D@k, hD cc@k, hD êêMatrixForm

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, hDtD

FullSimplify@D@gr@k, hD@tD, tD gr@k, hD@tD.cc@k, hDD

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:

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Transpose@cc@k, hDD.io@D + io@D.cc@k, hD

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, hD@tD.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, hD@tDD@@1DDD Simplify@cco@k, hD@tD cccurve@k, h, 0, 0, 0D@tDD

80, 0, 0<

ParametricPlot3D@cco@2, 1ê2D@tD,8t,−16Pi, 16Pi<, PlotRange AllD

-1 0 1

-1

0

1

0.0 0.2 0.4 0.6

cco1 = %;

In[70]:= Limit@cco@−2, 1ê2D@tD, 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, 1ê2D :1, 0, 0, 1,

1 2>

vv.io@D.vv 0

ppt = stepro@transoi@D.vvD

:0, 0, 1

2

>

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

Show@8cco1, lppt<, PlotRange AllD

-1

0

1

-1 0

1

0.0 0.2 0.4 0.6

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ParametricPlot3D@sphericalreflection@cco@−2, 1ê2D@tD, ppt, 1D, 8t,−10Pi, 10Pi<, PlotRange AllD

-2 0

2 -10

-5 0

5 10

-0.00.5 0.5

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, hDDD;

eva@k_, h_D =ses@@1DD; eve@k_, h_D= ses@@2DD; Clear@sesD Here are the Eigenvalues of cc[k,h]

eva@k, hD

:0,

h22 k+ 4+h4+4 h2k+4 k2 2

, h2 2

+k1 2

4+h4+4 h2k+4 k2 ,

h2 2

+k+1 2

4+h4+4 h2k+4 k2 , h2 2

+k+ 1 2

4+h4+4 h2k+4 k2 >

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

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

Updating...

## References

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