Distance geometry and geometric algebra

@article{Dress1993DistanceGA,
  title={Distance geometry and geometric algebra},
  author={Andreas W. M. Dress and Timothy F. Havel},
  journal={Foundations of Physics},
  year={1993},
  volume={23},
  pages={1357-1374}
}
As part of his program to unify linear algebra and geometry using the language of Clifford algebra, David Hestenes has constructed a (well-known) isomorphism between the conformal group and the orthogonal group of a space two dimensions higher, thus obtaining homogeneous coordinates for conformal geometry.(1) In this paper we show that this construction is the Clifford algebra analogue of a hyperbolic model of Euclidean geometry that has actually been known since Bolyai, Lobachevsky, and Gauss… 
Geometric Algebra in Linear Algebra and Geometry
This article explores the use of geometric algebra in linear and multilinear algebra, and in affine, projective and conformal geometries. Our principal objective is to show how the rich algebraic
Computational Synthetic Geometry with Clifford Algebra
TLDR
A MAPLE package is implemented, called Gibbs, for the elementary expansion and simplification of expressions in Gibbs' abstract vector algebra, and it is shown how to translate any origin-independent scalar-valued expression in the algebra into an element of the corresponding invariant ring, which is christened the Cayley-Menger ring.
Geometric Algebra and Möbius Sphere Geometry as a Basis for Euclidean Invariant Theory
Physicists have traditionally described their systems by means of explicit parametrizations of all their possible individual configurations. This makes a local description of the motion of the system
Hyperbolic Conformal Geometry with Clifford Algebra
AbstractIn this paper, we study hyperbolic conformal geometry following a Clifford algebraic approach. Similar to embedding an affine space into a one-dimensional higher linear space, we embed the
Surface Evolution and Representation using Geometric Algebra
TLDR
By moving from a projective to a conformal representation (5d representation of 3d space), one is able to extend the range of geometrical operations that can be carried out in an efficient and elegant way.
A covariant approach to geometry using geometric algebra
TLDR
Using the mathematical framework of conformal geometric algebra – a 5-dimensional representation of 3-dimensional space – is shown to provide an elegant covariant approach to geometry, thus enabling us to deal simply with the projective and non-Euclidean cases.
Geometry of complex data
  • K. J. Sangston
  • Mathematics
    IEEE Aerospace and Electronic Systems Magazine
  • 2016
TLDR
This tutorial provides a basic introduction to geometric algebra and presents formulations of known electrical engineering and signal processing concepts to illustrate some inherent advantages of geometric algebra for formulating and solving problems involving vectors.
Generalized homogeneous coordinates for computational geometry
The standard algebraic model for Euclidean space E n is an n-dimensional real vector space ℝ n or, equivalently, a set of real coordinates. One trouble with this model is that, algebraically, the
Some Applications of Clifford Algebra to Geometries
  • Hongbo Li
  • Mathematics
    Automated Deduction in Geometry
  • 1998
TLDR
A Clifford algebra model is focused on that integrates symbolic representation of geometric entities with that of geometric constraints such as angles and distances, and is appropriate for both symbolic and numeric computations.
Conformal Geometric Algebra
The geometric algebra of a 3D Euclidean space \(G_{3,0,0}\) has a point basis and the motor algebra \(G_{3,0,1}\) a line basis. In the latter geometric algebra, the lines expressed in terms of
...
...

References

SHOWING 1-10 OF 33 REFERENCES
UNIVERSAL GEOMETRIC ALGEBRA
The claim that Clifiord algebra should be regarded as a universal geometric algebra is strengthened by showing that the algebra is applicable to nonmetrical as well as metrical geometry. Clifiord
The design of linear algebra and geometry
Conventional formulations of linear algebra do not do justice to the fundamental concepts of meet, join, and duality in projective geometry. This defect is corrected by introducing Clifford algebra
Möbius groups over general fields using clifford algebras associated with spheres
The space of 2-by-2 Hermitian matrices is isometric to Minkowski space. This is commonly used to exhibit the groupSL(2, ℂ) as a twofold cover of the identity component of the Lorentz group. That
Projective geometry with Clifford algebra
Projective geometry is formulated in the language of geometric algebra, a unified mathematical language based on Clifford algebra. This closes the gap between algebraic and synthetic approaches to
Multilinear Cayley Factorization
Möbius Transformations and Clifford Numbers
TLDR
An approach is advocated which works directly in ℝ n and uses formulas strikingly analogous to those in the complex case and is therefore advocated.
Möbius Tramsformations and Clifford Algebras of Euclidean and Anti-Euclidean Spaces
L. Ahlfors studied Mobius transformations employing Clifford algebras of anti-euclidean spaces with negative definite quadratic forms. This paper gives a passage from the euclidean space (positive
New Foundation of Euclidean Geometry
My second paper on metrical geometry * contains a characterisation of the n-dimensional euclidean space among general semi-metrical spaces in terms of relations between the distances of its points.
...
...