Surface Evolution and Representation using Geometric Algebra

  title={Surface Evolution and Representation using Geometric Algebra},
  author={Anthony N. Lasenby and Joan Lasenby},
  booktitle={IMA Conference on the Mathematics of Surfaces},
Recent developments in geometric algebra have shown that 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. For example, while in projective space one is able to intersect lines and planes in a simple fashion, in conformal space one is able to intersect and represent spheres, lines, circles and planes. In addition, all the operations of… 
Conformal Geometry, Euclidean Space and Geometric Algebra
A solution to the problem of handling the Euclidean distance between points based on conformai geometry is discussed, including a compact formula for reflecting a line off a general spherical surface.
Circle and sphere blending with conformal geometric algebra
A simple, robust set of algorithms for performingcircle blends for performing circle blends for a range of cases is demonstrated and the basic idea is extended to the case of sphere blending to interpolate over a surface.
A covariant approach to geometry using geometric algebra
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.
The Twist Representation of Shape
We give a contribution to the representation problem of free-form curves and surfaces. Our proposal is an operational or kinematic approach based on the Lie group SE(3). While in Euclidean space the
Twists - An Operational Representation of Shape
An operational or kinematic approach based on the Lie group SE(3) that makes possible the robust and fast estimation of the pose of 3D objects from incomplete and noisy image data and the equivalence of the proposed shape model to that of the Fourier representations.
Monocular Pose Estimation of Kinematic Chains
In this paper conformal geometric algebra is used to formalize an algebraic embedding for the problem of monocular pose estimation of kinematic chains. The problem is modeled on a base of several
Applications of Geometric Algebra in Electromagnetism, Quantum Theory and Gravity
We review the applications of geometric algebra in electromagnetism, gravitation and multiparticle quantum systems. We discuss a gauge theory formulation of gravity and its implementation in
Estimating rigid motions via the conformal model of Euclidean space
  • D. Tweed
  • Computer Science, Mathematics
    Proceedings of the 17th International Conference on Pattern Recognition, 2004. ICPR 2004.
  • 2004
An origin independent semi-closed form is derived for the optimal transformation of rigid motions from 3-D point correspondences and image projections, solving it using an SVD-based technique.
6-DOF haptic rendering using geometric algebra
  • D. Garroway, F. Bogsanyi
  • Computer Science
    IEEE International Workshop HAVE Haptic Virtual Environments and Their
  • 2002
This paper presents an approach that uses an evenly distributed collection of points to represent the tool and shows that there is a limit to this concentration where it yields a continuous model of the volume of the actual tool.


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
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
Analysis and Computation of Projective Invariants from Multiple Views in the Geometric Algebra Framework
This paper looks at the formation of 3D projective invariants from multiple images, show how they can be formed from image coordinates and estimated tensors and gives results on simulated and real data.
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.
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
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
Objects in contact: boundary collisions as geometric wave propagation.
The motivation behind this work is to make the computation of collision-free motions of robots efficiently computable. For translational motions, the boundary of permissible translations of a
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.
Reconciling Distance Functions and Level Sets
This paper proposes an alternative to the use of Hamilton-Jacobi equations which eliminates this contradiction: in the method the implicit representation always remains a distance function by construction, and the implementation does not differ from the theory any more.
Geometric Computing with Clifford Algebras
  • G. Sommer
  • Mathematics, Computer Science
    Springer Berlin Heidelberg
  • 2001
A recipe for symbolic geometric computing: long geometric introduction to william kingdon clifford's geometric algebras an introduction to geometric algebra with some preliminary minkowski geometric algebra of complex sets.