scholarly journals Conformal Geometry, Euclidean Space and Geometric Algebra

2002 ◽  
pp. 41-58 ◽  
Author(s):  
Chris Doran ◽  
Anthony Lasenby ◽  
Joan Lasenby
Keyword(s):  
Euclidean Space ◽  
Geometric Algebra ◽  
Conformal Geometry

Modeling Spatial Displacements Using Clifford Algebra

2003 ◽  
Vol 126 (3) ◽  
pp. 420-424 ◽  
Author(s):  
Glen Mullineux
Keyword(s):  
Euclidean Space ◽  
Clifford Algebra ◽  
Geometric Algebra ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Real Numbers ◽  
B Spline ◽  
Spatial Displacements ◽  
Basis Vectors ◽  
Three Dimensional Space

This paper looks at the use of a Clifford (or geometric) algebra for handling both rotations and translations in Euclidean space. The algebra is constructed over the real numbers using four basis vectors. Three of these generate a subalgebra which models three-dimensional space; the fourth acts as a projective coordinate. Spatial displacements are represented by bivectors of a certain form. The application to the generation of smooth motions using Be´zier and B-spline techniques is illustrated.

Conformal Geometry and the Cyclides of Dupin

1980 ◽  
Vol 32 (4) ◽  
pp. 767-782 ◽  
Author(s):  
Thomas E. Cecil ◽  
Patrick J. Ryan
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Conformal Geometry ◽  
Conformally Flat ◽  
The Past ◽  
Open Set

A Riemannian manifold (M, g) is said to be conformally flat if every point has a neighborhood conformai to an open set in Euclidean space. Over the past thirty years, many papers have appeared attacking, with varying degrees of success, the problem of classifying the conformally flat spaces which occur as hypersurfaces in Euclidean space. Most of these start from the following pointwise result of Schouten.

scholarly journals Spheroidal Domains and Geometric Analysis in Euclidean Space

2021 ◽  
pp. 189-209
Author(s):  
Garret Sobczyk
Keyword(s):  
Artificial Intelligence ◽  
Data Analysis ◽  
Euclidean Space ◽  
Geometric Algebra ◽  
Geometric Analysis ◽  
Natural Generalization ◽  
Cauchy Kernel ◽  
Vector Algebra ◽  
Computer Scientists ◽  
Clifford’S Geometric Algebra

Clifford's geometric algebra has enjoyed phenomenal development over the last 60 years by mathematicians, theoretical physicists, engineers, and computer scientists in robotics, artificial intelligence and data analysis, introducing a myriad of different and often confusing notations. The geometric algebra of Euclidean 3-space, the natural generalization of both the well-known Gibbs-Heaviside vector algebra and Hamilton's quaternions, is used here to study spheroidal domains, spheroidal-graphic projections, the Laplace equation, and its Lie algebra of symmetries. The Cauchy-Kovalevska extension and the Cauchy kernel function are treated in a unified way. The concept of a quasi-monogenic family of functions is introduced and studied. 

scholarly journals Calculation for Multidimensional Topological Relations in 3D Cadastre Based on Geometric Algebra

2019 ◽  
Vol 8 (11) ◽  
pp. 469
Author(s):  
Shi ◽  
Hu ◽  
Yin ◽  
Wang ◽  
Chen ◽  
...  
Keyword(s):  
Euclidean Space ◽  
Geometric Algebra ◽  
Boundary Point ◽  
Topological Analysis ◽  
Data Models ◽  
Boundary Line ◽  
Topological Relations ◽  
Geometric Information ◽  
Outer Product ◽  
Topological Models

Maintaining topological consistency is a crucial issue for 3D cadastral modeling as this helps to represent the cadastral boundary clearly and accurately. As a result, 3D cadastral data models are mainly built on the basis of topological models that allow topology to be expressed clearly. However, topological models in Euclidean space cannot directly represent objects’ geometric information. As geometric information is important in 3D spatial analysis, 3D cadastral data models based on topological models cannot realize topological calculation and analysis. Previous research has proved that geometric and topological information for cadastral objects can be integrated and represented by conformal geometric algebra (CGA) expressions. This paper aims to realize 3D topological analysis in the cadastral field using CGA’s advantages in geometric relations computation. A calculation framework is designed on the basis of the outer product to achieve the purpose of multidimensional unity for 3D cadastral topological analysis in this paper. A calculation framework of topological relations between a boundary point (or a boundary line) and a cadastral parcel is developed. A total of 13 types of topological relations between a boundary point and a cadastral parcel and 48 types of topological relations between a boundary line and a cadastral parcel are obtained. The study indicates that the advantages of CGA in multidimensional unified representation and calculation can be used to solve problems encountered by topological models in Euclidean space.

scholarly journals Spherical Conformal Geometry with Geometric Algebra

2001 ◽  
pp. 61-75 ◽  
Author(s):  
Hongbo Li ◽  
David Hestenes ◽  
Alyn Rockwood
Keyword(s):  
Geometric Algebra ◽  
Conformal Geometry

Symplectic capacities

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Euclidean Space ◽  
Generating Functions ◽  
Weinstein Conjecture ◽  
Finite Dimensional ◽  
Hofer Metric

This chapter returns to the problems which were formulated in Chapter 1, namely the Weinstein conjecture, the nonsqueezing theorem, and symplectic rigidity. These questions are all related to the existence and properties of symplectic capacities. The chapter begins by discussing some of the consequences which follow from the existence of capacities. In particular, it establishes symplectic rigidity and discusses the relation between capacities and the Hofer metric on the group of Hamiltonian symplectomorphisms. The chapter then introduces the Hofer–Zehnder capacity, and shows that its existence gives rise to a proof of the Weinstein conjecture for hypersurfaces of Euclidean space. The last section contains a proof that the Hofer–Zehnder capacity satisfies the required axioms. This proof translates the Hofer–Zehnder variational argument into the setting of (finite-dimensional) generating functions.

ANTI-SELF-DUAL SOLUTIONS OF THE YANG-MILLS EQUATIONS IN 4n DIMENSIONS

1992 ◽  
Vol 07 (23) ◽  
pp. 2077-2085 ◽  
Author(s):  
A. D. POPOV
Keyword(s):  
Scalar Field ◽  
Euclidean Space ◽  
Lie Algebra ◽  
Linear Equations ◽  
Dual Solutions ◽  
Gauge Fields ◽  
System Of Equations ◽  
Semisimple Lie Algebra ◽  
Yang Mills ◽  
Self Duality

The anti-self-duality equations for gauge fields in d = 4 and a generalization of these equations to dimension d = 4n are considered. For gauge fields with values in an arbitrary semisimple Lie algebra [Formula: see text] we introduce the ansatz which reduces the anti-self-duality equations in the Euclidean space ℝ4n to a system of equations breaking up into the well known Nahm's equations and some linear equations for scalar field φ.

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