File: spheres.txt

Title: Spheres

(Enhancements/Geometry/0210-092. Version 2)

Date: June 1, 1999

Author: Rolf Sulanke

The packages and notebooks collected under the title "Spheres" consist of a system

of Mathematica notebooks and accompanying packages yielding tools for working

in classical geometric fields: Vector calculus, euclidean geometry, Moebius

Geometry. We emphasize applications to Moebius geometry.

The stuff composed here in a unified manner originated in a longer period

of working with Mathematica.I express my sincere gratitude to Alfred Gray,

who introduced me in Mathematica and discussed with me working problems very

intensively, and to the Humboldt University for the continuous support.

Especially I thank Dr. Spitzer and Mrs. Schnabel from the Computer Center,

and Dr. Hubert Gollek from the Institute of Mathematics of the University for

numerous hints and practical help.

MAIN CHANGES

In version 2 the new files pairs.nb, spiralsf.m, liealg.m appear.

In eusphere.nb a section about spiral surfaces has been added.

The considerations about orthogonality in pseuvec.m and pseuklid.nb are generalized

and unified. This file: spheres.txt, has been adapted and completed.

GENERAL HINTS

1. Put all the submitted files into the same directory, and

you will not suffer from Path problems.

2. All private (not delivered from the Mathematica system) symbols defined in

the submitted files start with small letters.

3. For using the concepts defined in the packages euvec.m, pseuvec.m,

mspher.m, mcirc.m, and spiralsf.m it suffices to import the file init.m. This

proceeds automatically if one activates Kernel/Evaluation/Evaluate Initialization

from the frontends menu

4. Not all concepts and constructs created in the notebooks are collected in

the packages mentioned in 3. The new file liealg.m is needed for the notebook

pairs.nb; it is not initialized by init.m. Loading it, some constructs of

linear Lie algebra, in particular the Killing forms of some Lie algebras, are

introduced within the Global Context.

5. The notebooks and packages in the collection "Spheres" are free software.

Any user may change and adapt them to his aims. If publishing such adaptions or

applications, please cite the sources and sign with your name as the author.

I am grateful for copies of such applications, for your hints,

corrections, comments etc. Please, e-mail them to

rolf.sulanke@t-online.de

6. Please, excuse posible errors and bad, too simplified use of anglo-american

English in my texts. I never learnt the English systematically.

CONTENTS

SPHERES IN EUCLIDEAN SPACES

Notebook: eusphere.nb

Needs: euvec.m, spiralsf.m

In euvec.m defined and protected Symbols:

"cross", "hyperplane", "invsterproj", "normalized", "nul", "ortho1",

"orthoerg", "orthoframe", "outzero", "rank", "smoothing", "sphere1",

"sphere3", "spherrefl", "stb", "sterproj", "tg4frame", "unit".

In spiralsf.m defined and protected Symbols:

"spiral", "spiralsurface", "plotspiralsurface".

Summary:

This notebook contains basic definitions needed to describe spheres as

objects of euclidean and spherical Riemannian geometry. As an application

the construction and plotting of a sphere through four points in the

euclidean 3-space is given. Furthermore, it contains a recursive definition of

the generalised geographical parameter representations of n-spheres in the

(n+1)-dimensional euclidean space. Reflections at hyperspheres (also called

inversions) are introduced. Spiral surfaces, in particular the spiral cylinder,

is treated.

Keywords:

vector objects, orthoframes, unitvectors, norming, cross product (general),

outzero, nullvector, standard base , hyperplanes, stereographical projection,

invers stereographical projection, 3-sphere, spheres, spheres through four points,

inversion at hyperspheres, flat torus, geodesics on the flat torus,

orthogonal complement in the euclidean 4-space, Parameter representation for

n-spheres, spiral transformations, spiral surfaces, MathGl3d, ThreeScript.

PSEUDO-EUCLIDEAN VECTOR SPACES

Notebook: pseuklid.nb

Needs: pseuvec.m, {euvec.m}

In pseuvec.m defined and protected Symbols:

"ch", "dual", "ide", "normed", "ortho", "orthonorm", "orthopair", "pr", "pscross",

"psfilter", "pssp"

Summary:

The notebook contains the basic definition of vector operations in the

n-dimensional pseudo-Euclidean space, including the Euclidean case. The

dimension dim and the index ind - the number of diagonal elements equal -1 in

an orthogonalized basis- are the characteristing constants for the

pseudo-euclidean vector spaces, which are vector spaces over the real numbers.

In special relativity theory the "world" = "space-time" is based on the

pseudo-euclidean vector space of dimension dim = 4 and index ind = 1,

corresponding to the three-dimensionality of the physical space and the

one-dimensionality of time. For the N-dimensional Moebius space we have to

set dim = N +2 and ind = 1 . In the n-dimensional euclidean case we have, of

course, dim = n and ind = 0. In the Lie geometry of spheres one has dim = 6

and ind = 2. Among others we introduce an orthogonalization procedure which

works for pseudo-euclidean vector spaces.

Keywords:

Dimension dim, index ind, pseudo-euclidean scalar product, generalized cross

product, spacelike, timelike, isotropic vectors, norming, orthogonalization,

orthopairing.

MÖBIUS GEOMETRY OF SPHERES

Notebook: mspheres.nb

Needs: mspher.m {pseuvec.m, euvec.m}

In mspher.m defined and protected Symbols:

"confinv", "gencos", "invstepro", "planev", "plotsphere",

"psphere", "pstg5frame", "paramsphere", "showsphere", "spacevec",

"sphvec", "spt","spunit", "stepro".

Summary:

This notebook treates 2-spheres within the 3-sphere, or the euclidean 3-space,

as objects of Möbius Geometry. First we introduce some basic objects of

Möbius gemetry, which is the conformal geometry of the n-sphere. We emphasize

the case n = 3, in which Möbius geometry can be visualized in the Euclidean

3-space by stereographical projection from the north pole. For this purpose we

construct a version of the stereographic projection, and their inversion,

which relates isotropic vectors of the pseudo-euclidean 5-space and points of

the 3-sphere. Then commands building spheres which correspond to a given

spacelike vector are constructed. Conversely, a command yielding a spacelike

vector of the sphere through four points (in general position) in the

3-dimensional Euclidean space is formulated. Using it we find a function

giving a spacelike unit vector for the sphere with center {x,y,z} and radius

r. Finally we scetch the geodesics of the sphere space.

Keywords:

Generalized angles between spheres. Conformal invariant. Spheres defined by

spacelike vectors. Spacelike vectors corresponding to spheres and planes.

Spheres through four points. Stereographical projection of isotropic vectors.

Space of all spheres. Geodesics in the sphere space; spacelike, timelike, and

isotropic geodesics.

GEOMETRY OF CIRCLES

Notebook: mcircles.nb

Needs mcirc.m {mspher.m, pseuvec.m, euvec.m}

In mcirc.m defined and protected Symbols:

"circle3D", "circlefamily", "control", "gencircle", "mcircle",

"paramcircle", "pcirc3D", "pgencirc", "pp", "ppdet", "pptr",

"tube", "unitvec", "vspace".

Summary:

This notebook describes the Möbius invariants for circles in the 3-sphere,

or, by stereographical projection, in the euclidean 3-space . Section 2

contains the needed concepts for the euclidean geometry of circles in the

3-space (or in the 3-sphere). On the level of the euclidean 3-space we give a

parametrization of the 6-dimensional space of circles. We construct two plot

commands which plot the circles corresponding to these parameters.

Furthermore, we define functions giving the euclidean invariants radius,

center and position vector of a circle through three points on the euclidean

level. As an application, a plot command for tubes of general space curves is

developed. Section 3 gives the basic concepts for circles in the

3-dimensional Möbius space. The circles are represented by 2-dimensional

euclidean subspaces of the 5-dimensional pseudo-euclidean vector space of

index 1, which are defined by orthonormal pairs of vectors. By stereographical

projection of the 3-sphere onto the euclidean 3-space the circles become

usual euclidean 3D-graphics (important for considering the mutual position of

circles in space). We construct a function which for three given points in the

euclidean 3-space yields the circle through them, more exactly, gives the

corresponding euclidean 2-subspace of the 5-space; the mcircle command then

plots the corresponding circle. In Section 4 we define a complete system of

Möbius-geometric invariants for pairs of circles in the 3-sphere, and try to

find out their geometric meaning. Section 5 gives the normal forms of the

circle pairs in relation to this system of invariants. Geodesics in the circle

space are studied as circle orbits and plotted in section 6; here the file liealg.m

is imported within the Global Context. This file yields some tools useful for

calculations in linear Lie algebras; in particular certain Killing forms are

introduced. The last section imports ThreeScript and MathGL3d, which can be applied

for animations of the created 3D-graphics.

Keywords:

pseudo-euclidean geometry, Moebius geometry, 3D-circles, circles defined by

subspaces, orthogonal circles, isospherical circles, stationary angles,

eigenspheres, invariants of pairs of circles, normal forms of pairs of circles,

geodesics in the circle space, MathGL3d.

PAIRS OF SUBSPHERES IN THE 3-SPHERE

Notebook: pairs.nb

Needs init.m (and the packages described above, declared in init.m)

For section 6 also the package liealg.m is needed.

Summary:

This notebook describes the Moebius invariants for pairs of subspheres in the

3-sphere, or, by stereographical projection, in the euclidean 3-space . The most

important cases, pairs of spheres and pairs of circles, are treated in the notebooks

mspheres.nb and mcircles.nb, some procedures of which are needed in the present

notebook. In section 2 pairs consisting of a sphere and a circle are considered;

they are characterised up to Möbius equivalence by a single invariant invsc.

Certain expressions of invsc by euclidean invariants of the pairs are deduced.

Section 3 treats pairs consisting of a sphere and a point pair; remember that

point pairs are 0-spheres. Their mutual position is described again by a single

invariant named invspp. In section 4 pairs of 1-spheres and 0-spheres, i.e. circles

and point pairs, are considered; their mutual position depends on two invariants:

the eigenvalues of a double projection. Finally, section 5 treats point quadrupels

as pairs of 0-spheres, and in section 6 the geodesics in the space of 0-spheres

(= point pairs) are considered. The last section imports ThreeScript and MathGL3d,

which can be applied for animations of the created 3D-graphics.

Keywords:

pseudo-euclidean geometry, Moebius geometry, 3D-circles,

invariants of pairs of subspheres (arbitrary dimensions), double projection,

stationary angles, eigenspheres, point pairs as 0-spheres, point quadrupels,

geodesics in the space of 0-spheres, Killing form, MathGL3d.