# 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…

## 31 Citations

Calculating the Rotor Between Conformal Objects

- MathematicsAdvances in Applied Clifford Algebras
- 2019

In this paper we will address the problem of recovering covariant transformations between objects—specifically; lines, planes, circles, spheres and point pairs. Using the covariant language of…

Conformal Geometric Algebra

- Mathematics
- 2018

This chapter is devoted to a presentation of conformal geometric algebra (CGA) targeted to the sort of applications dealt with in chapters 4 and 5, and is designed so that it can encode all conformal transformations of E3 in spinorial form.

Articulating Space: Geometric Algebra for Parametric Design - Symmetry, Kinematics, and Curvature

- Mathematics
- 2015

Author(s): Colapinto, Pablo | Advisor(s): Peljhan, Marko | Abstract: To advance the use of geometric algebra in practice, we develop computational methods for parameterizing spatial structures with…

Some Applications of Clifford Algebra in Geometry

- MathematicsStructure Topology and Symplectic Geometry
- 2020

In this chapter, we provide some enlightening examples of the application of Clifford algebra in geometry, which show the concise representation, simple calculation, and profound insight of this…

A Clifford Algebraic Framework for Coxeter Group Theoretic Computations

- Mathematics
- 2014

Real physical systems with reflective and rotational symmetries such as viruses, fullerenes and quasicrystals have recently been modeled successfully in terms of three-dimensional (affine) Coxeter…

Oriented Conformal Geometric Algebra

- Mathematics
- 2008

Abstract.In [16] Stolfi developed a complete theory of Oriented Projective Geometry. He showed that assigning meaning to the sign of an otherwise homogeneous representation of geometry could provide…

Exploring Novel Surface Representations via an Experimental Ray-Tracer in CGA

- MathematicsAdvances in Applied Clifford Algebras
- 2021

Conformal Geometric Algebra (CGA) provides a unified representation of both geometric primitives and conformal transformations, and as such holds significant promise in the field of computer…

Applications of Conformal Geometric Algebra in Computer Vision and Graphics

- Mathematics, Computer ScienceIWMM/GIAE
- 2004

A new method for pose and position interpolation based on CGA is discussed which firstly allows for existing interpolation methods to be cleanly extended to pose andposition interpolation, but also allows for this to be extended to higher-dimension spaces and all conformal transforms (including dilations).

Expressing Discrete Geometry using the Conformal Model

- Mathematics
- 2012

Primitives and transformations in discrete geometry, such as lines, circles, hyperspheres, hyperplanes, have been defined with classical linear algebra in dimension 2 and 3, leading to different…

Inverse Kinematics Solutions Using Conformal Geometric Algebra

- MathematicsGuide to Geometric Algebra in Practice
- 2011

A novel iterative Inverse Kinematics (IK) solver that is implemented using Conformal Geometric Algebra (CGA), FABRIK, that is real-time implementable and exploits the advantages of CGA for applications in computer vision, graphics and robotics.

## References

SHOWING 1-10 OF 17 REFERENCES

Distance geometry and geometric algebra

- Mathematics
- 1993

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

- MathematicsInternational Journal of Computer Vision
- 2004

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

- 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.

Geometric Algebra for Physicists

- Mathematics, Physics
- 2003

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

- 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…

PHYSICAL APPLICATIONS OF GEOMETRIC ALGEBRA

- Mathematics
- 2006

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

- Mathematics
- 2001

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

- Mathematics
- 1984

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

- Computer Science
- 2001

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

- Mathematics
- 1996

We introduce the third of the classical geometries, hyperbolic geometry.