Old Wine in New Bottles: A New Algebraic Framework for Computational Geometry

  title={Old Wine in New Bottles: A New Algebraic Framework for Computational Geometry},
  author={David Hestenes},
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. A detailed description and analysis of the model is soon to be published elsewhere, so I can concentrate on highlights here, although with a slightly different formulation that I find more convenient for applications. Also, I can assume that this audience is familiar with Geometric Algebra, so we… 

Applications of Conformal Geometric Algebra in Computer Vision and Graphics

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).

Some Considerations about Geometric Algebras in relation with Visibility in Computer Graphics

The use of conformal geometric algebra is emphasized since, among other reasons, it allows us to study easily the visibility for flat varieties and, due to the same algebraic expression of hyper-spheres and linear varieties, the results might be generalized to non-flat objects.

A 1d Up Approach to Conformal Geometric Algebra: Applications in Line Fitting and Quantum Mechanics

We discuss an alternative approach to the conformal geometric algebra (CGA) in which just a single extra dimension is necessary, as compared to the two normally used. This is made possible by working

Conformal Geometric Algebra for Robotic Vision

The authors believe that the framework of conformal geometric algebra can be, in general, of great advantage for applications using stereo vision, range data, laser, omnidirectional and odometry based systems.

Advanced Geometric Approach for Graphics and Visual Guided Robot Object Manipulation

Conformal Geometric Algebra appears to be a promising mathematical tool for building intelligent man-machine interfaces and is shown to be very well suitable for applications of all kind of robot manipulator kinematics, representation and visualization and object robot manipulation.

Surface Evolution and Representation using Geometric Algebra

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.

Geometry of complex data

  • K. J. Sangston
  • Mathematics
    IEEE 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.

New Tools for Computational Geometry and Rejuvenation of Screw Theory

This paper is a comprehensive introduction to a CGA tool kit and designs for a complete system of powerful tools for the mechanics of linked rigid bodies are presented.

Geometric computing in computer graphics and robotics using conformal geometric algebra

The main contribution of this thesis is the geometrically intuitive and - nevertheless - efficient algorithm for a computer animation application, namely an inverse kinematicsgorithm for a virtual character based on an embedding of quaternions in Conformal Geometric Algebra.



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

Geometrical Methods in Robotics

This book provides an introduction to the geometrical concepts that are important to applications in robotics and shows how these concepts may be used to formulate and solve complex problems encountered in the design and construction of robots.

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

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

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.

Lie-groups as Spin groups.

It is shown that every Lie algebra can be represented as a bivector algebra; hence every Lie group can be represented as a spin group. Thus, the computational power of geometric algebra is available

Invariant body kinematics: I. Saccadic and compensatory eye movements

Invariant body kinematics: II. Reaching and neurogeometry

Introduction to Theoretical Mechanics

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