Search results for " Geometric algebra"

An Introduction to Geometric Algebra and Conics

2016

This chapter introduces the conics and characterizes them from an algebraic perspective. While in depth geometrical aspects of the conics lie outside the scopes of this chapter, this chapter is an opportunity to revisit concepts studied in other chapters such as matrix and determinant and assign a new geometric characterization to them.

On the geometric structure of the class of planar quadratic differential systems

2002

In this work we are interested in the global theory of planar quadratic differential systems and more precisely in the geometry of this whole class. We want to clarify some results and methods such as the isocline method or the role of rotation parameters. To this end, we recall how to associate a pencil of isoclines to each quadratic differential equation. We discuss the parameterization of the space of regular pencils of isoclines by the space of its multiple base points and the equivariant action of the affine group on the fibration of the space of regular quadratic differential equations over the space of regular pencils of isoclines. This fibration is principal, with a projective group…

Graphic Coprocessors with Native Clifford Algebra Support

2009

A brief introduction to Clifford algebra

2010

Geometric algebra (also known as Clifford algebra) is a powerful mathematical tool that offers a natural and direct way to model geometric objects and their transformations. It is gaining growing attention in different research fields as physics, robotics, CAD/CAM and computer graphics. Clifford algebra makes geometric objects (points, lines and planes) into basic elements of computation and defines few universal operators that are applicable to all types of geometric elements. This paper provides an introduction to Clifford algebra elements and operators.

4D Clifford algebra based on fixed-size representation

2008

Geometric algebra (also known as Clifford algebra) is a powerful mathematical tool that offers a natural and direct way to model geometric objects and their transformations. It is gaining growing attention in different research fields as physics, robotics, CAD/CAM and computer graphics. In particular, 4D geometric algebra implements homogeneous coordinates, which are used to model 3D scenery in most computer graphics applications. The research work on Clifford algebra is actually aimed at finding efficient implementations of the algebra. This paper wants to give a contribution to this research effort by proposing a direct hardware support for geometric algebra operators. The paper introduce…

ConformalALU: A Conformal Geometric Algebra Coprocessor for Medical Image Processing

2015

Medical imaging involves important computational geometric problems, such as image segmentation and analysis, shape approximation, three-dimensional (3D) modeling, and registration of volumetric data. In the last few years, Conformal Geometric Algebra (CGA), based on five-dimensional (5D) Clifford Algebra, is emerging as a new paradigm that offers simple and universal operators for the representation and solution of complex geometric problems. However, the widespread use of CGA has been so far hindered by its high dimensionality and computational complexity. This paper proposes a simplified formulation of the conformal geometric operations (reflections, rotations, translations, and uniform …

GAPPCO: An Easy to Configure Geometric Algebra Coprocessor Based on GAPP Programs

2017

Because of the high numeric complexity of Geometric Algebra, its use in engineering applications relies heavily on tools and devices for efficient implementations. In this article, we present a novel hardware design for a Geometric Algebra coprocessor, called GAPPCO, which is based on Geometric Algebra Parallelism Programs (GAPP). GAPPCO is a design for a coprocessor combining the advantages of optimizing software with a configurable hardware able to implement arbitrary Geometric Algebra algorithms. The idea is to have a fixed hardware easily and fast to be configured for different algorithms. We describe the new hardware design together with the complete tool chain for its configuration.

Fixed-size Quadruples for a New, Hardware-Oriented Representation of the 4D Clifford Algebra

2010

Clifford algebra (geometric algebra) offers a natural and intuitive way to model geometry in fields as robotics, machine vision and computer graphics. This paper proposes a new representation based on fixed-size elements (quadruples) of 4D Clifford algebra and demonstrates that this choice leads to an algorithmic simplification which in turn leads to a simpler and more compact hardware implementation of the algebraic operations. In order to prove the advantages of the new, quadruple-based representation over the classical representation based on homogeneous elements, a coprocessing core supporting the new fixed-size Clifford operands, namely Quad-CliffoSor (Quadruple-based Clifford coproces…

Calibration of the Norwegian motion laboratory using conformal geometric algebra

2017

This paper applies Conformal Geometric Algebra (CGA) as a tool for calibrating the robotic equipment found in the Norwegian Motion Laboratory. By using the inner product of CGA to measure the distance between a point and the surface of a plane/sphere, the least-squares method can be used to solve for the unknown parameters describing the plane/sphere in an efficient and intuitive way given n measured points. Positional data samples were acquired from using a high precision Laser tracker (FARO Xi), and the overall calibration error was found to be no more than 4.90mm, and the maximum standard deviation 3.25mm. In addition, the applied least-squares algorithm using CGA was twice as fast, when…

The non-degenerate Dupin cyclides in the space of spheres using Geometric Algebra

2012

International audience; Dupin cyclides are algebraic surfaces of degree 4 discovered by the French mathematician Pierre-Charles Dupin early in the 19th century and \textcolor{black}{were} introduced in CAD by R. Martin in 1982. A Dupin cyclide can be defined, in two different ways, as the envelope of a one-parameter family of oriented spheres. So, it is very interesting to model the Dupin cyclides in the space of spheres, space wherein each family of spheres can be seen as a conic curve. In this paper, we model the non-degenerate Dupin cyclides and the space of spheres using Conformal Geometric Algebra. This new approach permits us to benefit from the advantages of the use of Geometric Alge…