Three-dimensional quadrics in extended conformal geometric algebras of higher dimensions from control points, implicit equations and axis alignment

International audienceWe introduce the quadric conformal geometric algebra (QCGA) inside the algebra of R 9,6. In particular, this paper presents how three-dimensional quadratic surfaces can be defined by the outer product of conformal geometric algebra points in higher dimensions, or alternatively by a linear combination of basis vectors with coefficients straight from the implicit quadratic equation. These multivector expressions code all types of quadratic surfaces in arbitrary scale, location, and orientation. Furthermore, we investigate two types of definitions of axis aligned quadric surfaces, from contact points and dually from linear combinations of R 9,6 basis vectors

Garamon: A Geometric Algebra Library Generator

International audienceThis paper presents both a recursive scheme to perform Geometric Algebra operations over a prefix tree, and Garamon, a C++ library generator implementing these recursive operations. While for low dimension vector spaces, precomputing all the Geometric Algebra products is an efficient strategy, it fails for higher dimensions where the operation should be computed at run time. This paper describes how a prefix tree can be a support for a recursive formulation of Geometric Algebra operations. This recursive approach presents a much better complexity than the usual run time methods. This paper also details how a prefix tree can represent efficiently the dual of a multivector. These results constitute the foundations for Garamon, a C++ library generator synthesizing efficient C++ / Python libraries implementing Geometric Algebra in both low and higher dimensions, with any arbitrary metric. Garamon takes advantage of the prefix tree formulation to implement Geometric Algebra operations on high dimensions hardly accessible with state-of-the-art software implementations. Garamon is designed to produce easy to install , easy to use, effective and numerically stable libraries. The design of the libraries is based on a data structure using precomputed functions for low dimensions and a smooth transition to the new recursive products for higher dimensions. Mathematics Subject Classification (2010). Primary 99Z99; Secondary 00A00

Visiting bijective digitized reflections and rotations using geometric algebra

International audienceGeometric algebra has become popularly used in applications dealing with geometry. This framework allows us to reformulate and redefine problems involving geometric transformations in a more intuitive and general way. In this paper, we focus on 2D bijective digitized reflections and rotations. After defining the digitization through geometric algebra, we characterize the set of bijective digitized reflections in the plane. We derive new bijective digitized rotations as compositions of bijective digitized reflections since any rotation is represented as the composition of two reflections. We also compare them with those obtained through geometric transformations by computing their distributions

A Geometric Algebra Implementation using Binary Tree

International audienceThis paper presents an efficient implementation of geometric algebra, based on a recursive representation of the algebra elements using binary trees. The proposed approach consists in restructuring a state of the art recursive algorithm to handle parallel optimizations. The resulting algorithm is described for the outer product and the geometric product. The proposed implementation is usable for any dimensions, including high dimension (e.g. algebra of dimension 15). The method is compared with the main state of the art geometric algebra implementations , with a time complexity study as well as a practical benchmark. The tests show that our implementation is at least as fast as the main geometric algebra implementations

Implantation des opérateurs de l'algèbre géométrique par arbres binaires récursifs

International audienceL'algèbre géométrique est un outil permettant de représenter et manipuler les objets géométriques de manière générique, efficace et intuitive. Différentes approches ont été prospectées afin d'obtenir des implantations efficaces, mais restent toutefois perfectibles notamment sur leur attractivité face à un usage industriel. Cet article présente une implantation de ces algèbres basée sur une représentation de ses éléments sous forme d'arbres binaires. Cette représentation aboutit à des définitions récursives des produits permettant, une fois déroulés, la mise en oeuvre de nouveaux algorithmes itératifs. Ces algorithmes consistent à assimiler les produits à des constructions itératives de listes. L'approche obtenue est valable en toutes dimensions. Les algorithmes proposés ont été implantés en C++ et comparés avec les méthodes de l'état de l'art

Computational Aspects of Geometric Algebra Products of Two Homogeneous Multivectors

International audienceThis paper addresses the study of the complexity of products in geometric algebra. More specifically, this paper focuses on both the number of operations required to compute a product, in a dedicated program for example, and the complexity to enumerate these operations. In practice, studies on time and memory costs of products in geometric algebra have been limited to the complexity in the worst case, where all the components of the multivector are considered. Standard usage of Geometric Algebra is far from this situation since multivectors are likely to be sparse and usually full homogeneous, i.e., having their non-zero terms over a single grade. We provide a complete computational study on the main Geometric Algebra products of two full homogeneous multivectors, that are outer, inner, and geometric products. We show tight bounds on the number of the arithmetic operations required for these products. We also show that some algorithms reach this number of arithmetic operations

Computational Aspects of Geometric Algebra Products of Two Homogeneous Multivectors

Studies on time and memory costs of products in geometric algebra have been limited to cases where multivectors with multiple grades have only non-zero elements. This allows to design efficient algorithms for a generic purpose; however, it does not reflect the practical usage of geometric algebra. Indeed, in applications related to geometry, multivectors are likely to be full homogeneous, having their non-zero elements over a single grade. In this paper, we provide a complete computational study on geometric algebra products of two full homogeneous multivectors, that is, the outer, inner, and geometric products of two full homogeneous multivectors. We show tight bounds on the number of the arithmetic operations required for these products. We also show that algorithms exist that achieve this number of arithmetic operations

Computational Aspects of Geometric Algebra Products of Two Homogeneous Multivectors

International audienceThis paper addresses the study of the complexity of products in geometric algebra. More specifically, this paper focuses on both the number of operations required to compute a product, in a dedicated program for example, and the complexity to enumerate these operations. In practice, studies on time and memory costs of products in geometric algebra have been limited to the complexity in the worst case, where all the components of the multivector are considered. Standard usage of Geometric Algebra is far from this situation since multivectors are likely to be sparse and usually full homogeneous, i.e., having their non-zero terms over a single grade. We provide a complete computational study on the main Geometric Algebra products of two full homogeneous multivectors, that are outer, inner, and geometric products. We show tight bounds on the number of the arithmetic operations required for these products. We also show that some algorithms reach this number of arithmetic operations

Surfaces quadratiques à base d'Algèbres Géométriques Conforme

International audienceLes Algèbres Géométriques constituent un outil intuitif et performant pour représenter et manipuler des objets géométriques ainsi que pour calculer les intersections entre plusieurs objets géométriques. Parmi les Algèbres Géométriques les plus courantes figure l'Algèbre Géométrique Conforme qui permet de représenter des objets ronds (cercles et sphères) et plats (droites et plans) de façon compacte et élégante. Dans l'Algèbre Géométrique Conforme, les objets plats se construisent comme un objet rond dont on a remplacé un point de construction par un point à l'infini. Cela a pour effet de voir les objets plats comme des objets ronds avec une courbure nulle. La structure de ces objets plats n'est pas différentes de celles des objets ronds (une droite et un cercle de courbure nulle). Tous ces objets se construisent avec le nombre minimum de points de contrôle nécessaires à leur définition, ou bien par leur formule implicite communément utilisée en algèbre linéaire. Mais, contrairement à l'algèbre linéaire, ces objets se manipulent (transformations rigides) tous avec exactement les mêmes opérateurs, indifféremment de leur type. Enfin, l'intersection entre ces objets se calcule elle aussi avec un unique opérateur. En revanche, la diversité des objets exprimés dans cette algèbre se limite à quelques objets (sphères, plans, cercles, droites, ...). Dans cet article, nous nous intéressons à une extension de l'Algèbre Géométrique Conforme qui permet de représenter et manipuler les surfaces quadriques, l'Algèbre Géométrique Conforme Quadrique. Les surfaces quadratiques suscitent de plus en plus d'attention dans la communauté des Algèbres Géométriques et plusieurs algèbres ont été proposées pour représenter et manipuler ces surfaces. En nous plaçant dans l'Algèbre Géométrique Conforme Quadrique, nous étudions les objets ronds et plats de cette algèbre. L'objectif est, dans un premier temps, de retrouver les objets ronds et plats de l'Algèbre Géométrique Conforme et ensuite d'explorer quels sont les nouveaux objets ronds et plats que l'on peut définir et manipuler avec l'Algèbre Géométrique Conforme Quadrique