1,980 research outputs found
Using geometric algebra to interactively model the geometry of Euclidean and non-Euclidean spaces.
This research interprets and develops the 'conformal model of space' in a way appropriate for a graphics developer interested in the design of interactive software for exploring 2-dimensional non-Euclidean spaces.
The conformal model of space extends the standard projective model – instead of adding just one extra dimension to standard Euclidean space, a second one is added that results in a Minkowski space similar to that of relativistic spacetime. Also, standard matrix algebra is replaced by geometric ( i.e. Clifford) algebra.
The key advantage of the conformal model is that both Euclidean and non- Euclidean spaces are accommodated within it. Transformations in conformal space are generated by bivectors which are special elements of the geometric algebra. These induce geometric transformations in the embedded non Euclidean spaces. However, the relationship between the bivector generated transformations of the Minkowski modelling space and the geometric transformations they induce is extremely obscure.
This thesis provides new analytical tools for determining the nature of this relationship. Their derivation was motivated by the need to successfully solve key implementation problems relating to navigation and in-scene mouse interaction.
The analytic approaches developed not only successfully solved these problems but pointed the way to implementing other unplanned features. These include facilities for dynamically altering on-screen geometry as well as using multiple viewports to allow the user to interact with the same objects embedded in different geometries. These new analytical approaches could be powerful tools for solving future and as yet unforeseen implementation problems
Spacetime deployments parametrized by gravitational and electromagnetic fields
On the basis of a "Punctual" Equivalence Principle of the general relativity
context, we consider spacetimes with measurements of conformally invariant
physical properties. Then, applying the Pfaff theory for PDE to a particular
conformally equivariant system of differential equations, we make explicit the
dependence of any kind of function describing a "spacetime deployment", on
n(n+1) parametrizing functions, denoting by n the spacetime dimension. These
functions, appearing in a linear differential Spencer sequence and determining
gauge fields of spacetime deformations relatively to a "substrat spacetime",
can be consistently ascribed to unified electromagnetic and gravitational
fields, at any spacetime dimensions n greater or equal to 4.Comment: 26 pages, LaTeX2e, file macro "suppl.sty", correction in the
definition of germs and local ring
Navigation of Spacetime Ships in Unified Gravitational and Electromagnetic Waves
On the basis of a "local" principle of equivalence of general relativity, we
consider a navigation in a kind of "4D-ocean" involving measurements of
conformally invariant physical properties only. Then, applying the Pfaff theory
for PDE to a particular conformally equivariant system of differential
equations, we show the dependency of any kind of function describing "spacetime
waves", with respect to 20 parametrizing functions. These latter, appearing in
a linear differential Spencer sequence and determining gauge fields of
deformations relatively to "ship-metrics" or to "flat spacetime ocean metrics",
may be ascribed to unified electromagnetic and gravitational waves. The present
model is based neither on a classical gauge theory of gravitation or a
gravitation theory with torsion, nor on any Kaluza-Klein or Weyl type
unifications, but rather on a post-Newtonian approach of gravitation in a four
dimensional conformal Cosserat spacetime.Comment: 28 pages. Relative to the second version some changes in the
mathematical results have been corrected without consequences in the physical
model. The conformally flatness of the substratum spacetime which is an
assumption used throughout in the mathematical developements from chapter 2,
has been well precised in the first chapter. Clearer explanations at the very
end of chapter 3 about accelerating frames are given. New references are
indicated and some of them correcte
Twistor geometry of a pair of second order ODEs
We discuss the twistor correspondence between path geometries in three
dimensions with vanishing Wilczynski invariants and anti-self-dual conformal
structures of signature . We show how to reconstruct a system of ODEs
with vanishing invariants for a given conformal structure, highlighting the
Ricci-flat case in particular. Using this framework, we give a new derivation
of the Wilczynski invariants for a system of ODEs whose solution space is
endowed with a conformal structure. We explain how to reconstruct the conformal
structure directly from the integral curves, and present new examples of
systems of ODEs with point symmetry algebra of dimension four and greater which
give rise to anti--self--dual structures with conformal symmetry algebra of the
same dimension. Some of these examples are analogues of plane wave
space--times in General Relativity. Finally we discuss a variational principle
for twistor curves arising from the Finsler structures with scalar flag
curvature.Comment: Final version to appear in the Communications in Mathematical
Physics. The procedure of recovering a system of torsion-fee ODEs from the
heavenly equation has been clarified. The proof of Prop 7.1 has been
expanded. Dedicated to Mike Eastwood on the occasion of his 60th birthda
Conformal Parametrisation of Loxodromes by Triples of Circles
We provide a parametrisation of a loxodrome by three specially arranged
cycles. The parametrisation is covariant under fractional linear
transformations of the complex plane and naturally encodes conformal properties
of loxodromes. Selected geometrical examples illustrate the usage of
parametrisation. Our work extends the set of objects in Lie sphere
geometry---circle, lines and points---to the natural maximal
conformally-invariant family, which also includes loxodromes.Comment: 14 pages. 9 PDF in four figures, AMS-LaTe
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