• Corpus ID: 124816889

A covariant approach to geometry using geometric algebra

@inproceedings{Lasenny2004ACA,
  title={A covariant approach to geometry using geometric algebra},
  author={A Lasenny and Joan Lasenby and Rj Wareham},
  year={2004}
}
This report aims to show that using the mathematical framework of conformal geometric algebra – a 5-dimensional representation of 3-dimensional space – we are able to provide an elegant covariant approach to geometry. In this language, objects such as spheres, circles, lines and planes are simply elements of the algebra and can be transformed and intersected with ease. In addition, rotations, translation, dilations and inversions all become rotations in our 5-dimensional space; we will show how… 
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31 Citations
Highly Influential Citations
2
Background Citations
15
Methods Citations
8
Results Citations
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31 Citations

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References

SHOWING 1-10 OF 17 REFERENCES
Distance geometry and geometric algebra
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…
New Geometric Methods for Computer Vision: An Application to Structure and Motion Estimation
TLDR
A coordinate-free approach to the geometry of computer vision problems is discussed, believing the present formulation to be the only one in which least-squares estimates of the motion and structure are derived simultaneously using analytic derivatives.
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.
Geometric Algebra for Physicists
Geometric algebra is a powerful mathematical language with applications across a range of subjects in physics and engineering. This book is a complete guide to the current state of the subject with…
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…
PHYSICAL APPLICATIONS OF GEOMETRIC ALGEBRA
TLDR
This course introduces Geometric Algebra as a new mathematical technique to add to your existing base as a theoretician or experimentalist and develops applications of this new technique in the fields of classical mechanics, engineering, relativistic physics and gravitation.
The making of GABLE: a geometric algebra learning environment in Matlab
Geometric algebra extends Clifford algebra with geometrically meaningful operators with the purpose of facilitating geometrical computations. Present textbooks and implementation do not always convey…
Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics
1 / Geometric Algebra.- 1-1. Axioms, Definitions and Identities.- 1-2. Vector Spaces, Pseudoscalars and Projections.- 1-3. Frames and Matrices.- 1-4. Alternating Forms and Determinants.- 1-5.…
Old Wine in New Bottles: A New Algebraic Framework for Computational Geometry
My purpose in this chapter is to introduce you to a powerful new algebraic model for Euclidean space with all sorts of applications to computer-aided geometry, robotics, computer vision and the like.…
Hyperbolic Geometry
We introduce the third of the classical geometries, hyperbolic geometry.
...
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