# 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…

## 53 Citations

Geometric Algebra in Linear Algebra and Geometry

- Mathematics
- 2002

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

- MathematicsAutomated Deduction in Geometry
- 1996

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

- Mathematics
- 1995

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

- Mathematics
- 2001

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

- MathematicsIMA Conference on the Mathematics of Surfaces
- 2000

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

- Mathematics, Computer Science
- 2004

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

- MathematicsIEEE Aerospace and Electronic Systems Magazine
- 2016

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

- Mathematics
- 2001

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

- MathematicsAutomated Deduction in Geometry
- 1998

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

- MathematicsGeometric Algebra Applications Vol. II
- 2018

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

- Mathematics
- 1988

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

- Mathematics
- 1991

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…

Some Examples of the Use of Distances as Coordinates for Euclidean Geometry

- MathematicsJ. Symb. Comput.
- 1991

Möbius groups over general fields using clifford algebras associated with spheres

- Mathematics
- 1990

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

- Mathematics
- 1991

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…

Möbius Transformations and Clifford Numbers

- Computer Science
- 1985

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

- Mathematics
- 1989

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

- Mathematics
- 1931

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.…