Clifford Algebras: Geometric Modelling and Chain Geometries with Application in Kinematics

228 Pages·2015·2.28 MB·English

by  Daniel Klawitter (auth.)

Cliff ord Algebras
Daniel Klawitter
Cliff ord Algebras
Geometric Modelling
and Chain Geometries
with Application in Kinematics
Foreword by Prof. Dr. Gunter Weiss
Dr. Daniel Klawitter
Dresden, Germany
Dissertation TU Dresden, 2014
ISBN 978-3-658-07617-7 ISBN 978-3-658-07618-4 (eBook)
DOI 10.1007/978-3-658-07618-4
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To my parents,
Manuela and Burghard
Foreword
What is the right mathematical model to a real phenomenon of our
world? Do there exist criteria whether a model can be called eleg-
ant as well as practically efficient? These questions will surely be
answered differently by e.g. a pure mathematician on the one hand
and an engineer on the other. But both will have to start with ab-
stracting real world phenomena to objects of a more or less platonic
world. Inthisidealizedworldoneconstructsthe”structuredgeomet-
ricimage”ofrealworldprocessesandobjectsinconsideration. Thus,
on the way to constructing explicit mathematical models, geometric
representations,andgeometricreasoningwillalwaysplayanessential
partandprovidesinsightintoperhapsnotthatobviousinterrelations
and structures.
The idea for this book took its origins from space kinematics and
robotics and the standardized way. Its aim is to be understood and
handledpractically. Therearetwoessentiallydifferentwaystomodel
space kinematics, although both ways are based on an Euclidean mo-
tion as the core element of the manifold of motions.
Loosely spoken, a seemingly clear model will interpret such a basic
element as a point and the manifold of motions as a point manifold.
(It turns out, that, via a clever transfer mapping, this point mani-
fold becomes a hyper-quadric in a seven-dimensional real projective
space. This hyper-quadric is referred to as Study’s quadric.) The
second model interprets a motion as a number rather than a point on
a line: In order to describe the position of it on that line, one needs a
pair of Hamilton quaternions as coordinates, which in addition fulfill
conditions. Both models relate to well understood families of models.
Thefirstonecanbeseene.g. asahigherdimensionalanaloguetothe
geometry of lines and screws. Indeed, screws are closely connected
to motions. The second model relates to so-called circle geometries.
VIII Foreword
Their structures are those of a line with coordinates of a ring. (A
standard example of such a circle geometry is the so-called (planar)
Mo¨bius geometry; here it turns out that the coordinate ring of the
corresponding line is even a field, namely the field of complex num-
bers.)
Both standard models for space kinematics have their merits and can
be considered elegant. The first model aims at a visualization of the
structured set of motions, where coordinates become inessential. In
spite of the transfer between the original set of motions and Study’s
quadric, the seven-dimensional real projective space is described by
coordinates. Now the model induces objects of further research: For
example, planar intersections of Study’s quadric lead to interesting
subsets of motions. The second model aims at a clearly perform-
able calculus and the analytic treatment of forced motions. Here as
well, one feels the need for visualization. Such transfer from calculus
to geometry establishes a third model and can be derived as follows:
Pairsofquaternionsareinterpretedaselementsofafour-dimensional
module over the ring of so-called dual numbers, which again are in-
terpreted as points. Points originating from motions are contained
by a hypersphere in this four-dimensional module space and, again,
planar intersections of this sphere become a matter of interest. But
these planar intersections stem from helical motions in the original
world.
In this setting, space kinematics and its models are used to point
out an obvious fact: The models and model spaces, which are struc-
turedfortheirown,influencethetopicsofresearchoftheoriginalreal
world objects and ” re-structure ” these objects in a specific manner.
Therefore, the answer to the introductory question, which model is
best, will be rather a matter of taste and pre-knowledge of related
models than being based on objectified facts. But still the origin of
different models is the object of consideration! So the question about
relations between different models and their structures arises. Is is
possible to unify these view points?
This last question is the core topic of this book: The author in-
troduces the seemingly universal tool of Clifford Algebras and their
Foreword IX
groupstructurestomodelnotonlyEuclideankinematicsandlinegeo-
metry, but also circle geometries and their generalizations, so-called
” chain geometries ” and, to take it even further, also non-Euclidean
kinematics and line geometry. Projective and chain geometric prop-
erties alternately relate to algebraic properties and concepts. Again,
the concept ” cross ratio ” becomes such an interface between Geo-
metry and Algebra, being on the one hand visualized as quadruplets
of points and on the other being a number resp. a ring element as
well as responsible for structuring e.g. specific point sets of a line
to chains. The presented method allows to connect disciplines of
(Applied) Mathematics, that, up to now, are treated independently.
Furthermore, it provides an effective tool for solving practical prob-
lems of Mechanical Engineering and Robotics, too.
Is this book a Maths book on theoretical and practical applications
of Clifford Algebras? Yes, but in its rich details and examples it aims
at visualizing the more or less hidden Geometries of the Clifford Al-
gebra calculus, and therefore, it could rightly be called a Geometry
book. By its wide range of examples it is profitable to a broad span
of readers, be it mechanical engineers or mathematicians. Like me,
they surely consider its publication as an absolute stroke of luck.
Wien Gunter Weiss
Preface
Usually, dual unit quaternions are used to describe Euclidean dis-
placements in three-dimensional kinematics. Engineers widely use
this elegant calculus next to a real projective geometric model called
Study’s quadric. Moreover, there exists a ” sphere-model ” in a four-
dimensional module space over the ring of dual numbers. However,
there are no investigations connecting these three models up to now.
After recalling Study’s quadric, the ” sphere-model ”, and other mod-
els,wepresentasurvey-likeintroductiontoCliffordalgebrasandtheir
Spin and Pin groups. The advantage of Clifford algebras is that geo-
metric objects and transformation may be represented as algebra ele-
ments. Using the so-called sandwich operator, it is possible to apply
transformations that are represented by algebra elements directly to
geometric objects, also described by algebra elements. We introduce
the homogeneous and conformal Clifford algebra model. Further-
more, we show how to model special Cayley-Klein geometries and
their isometry groups in a homogeneous Clifford algebra model. As
anexample, wefocusonthehomogeneousCliffordalgebramodelcor-
responding to line geometry and derive the correspondence between
projective transformations of three-dimensional projective space and
the Pin group Pin . Therefore, we introduce the Clifford algebra
(3,3,0)
C(cid:2) , constructedoverthequadraticspaceR(3,3), anddescribehow
(3,3,0)
points on Klein’s quadric are embedded as null vectors. We discuss
how geometric entities that are known from Klein’s model, i.e., lin-
ear line manifolds can be transferred to this homogeneous Clifford
algebra model. All entities known from line geometry occur natur-
ally in this model and can be transformed projectively by the ap-
plication of the sandwich operator. The action of grade-1 elements
correspondstothe actionof null polarities onP3(R), i.e., correlations
that are involutions as the basic elements building up the group of
regular projective transformations. It is proven that every regular
projective transformation of P3(R) can be expressed as the product
XII Preface
of six null polarities, i.e., skew-symmetric 4×4 matrices at the most.
The results achieved for Klein’s quadric may be transferred to any
quadric, hence, we present the homogeneous Clifford algebra model
corresponding to Lie sphere geometry as an example. Additionally, a
new geometric algebra allowing the description of inversions with re-
specttoquadricsinprincipalpositionasPingroupispresented. This
model serves as a generalization for the conformal geometric algebra
and is constructed for dimension two and three in detail. Further-
more, the generalization to arbitrary dimension is shown.
A further focus of the thesis is applying chain geometry to Clifford
algebras in order to examine the cross ratio in Clifford algebras. It
is well known that the cross ratio of four complex numbers is real
if, and only if, they all lie on a Mo¨bius circle, i.e., a circle or a line
augmented by a point at infinity. A generalization of the so called
M¨obius geometry is obtained by using different algebras instead of
complex numbers. This leads to a branch of geometry called chain
geometry. Chains are subsets of the projective line over a ring which
can be parametrised with the cross ratio. Therefore, it is natural to
apply this theory to dual quaternions and to examine the kinematic
and geometric interpretation. A more general point of view can be
achieved by the use of Clifford algebras and Spin groups instead of
dual quaternions and dual unit quaternions. After recalling the fun-
damental chain geometric background, we define the cross ratio for
CliffordalgebrasandtheirPinandSpingroups. Wepresentaquadric
model corresponding to the dual unit quaternions and homogeneous
Clifford algebra models of Klein’s, Study’s and Lie’s quadric where
chainsthatarecontainedinthegrade-1subspacecorrespondtoconic
sections. Moreover,wederiveanalgebraicbiarcconstructionwiththe
help of contact spaces. Chains of the grade-1 subspace that are in
contact at a certain point are parametrized with the cross ratio and
correspond to conic sections. Moreover, it has been proven that the
connected components of the Pin- and Spin groups define subspaces
of chain spaces. Every element contained in a chain defined by three
elements of the same connected component of the Pin- or the Spin
group is contained in the same connected component of the Pin- or
the Spin group. The question for the cross ratio of dual unit qua-
ternions has been answered in detail.