相关论文: Geometric Algebras
This is the first paper in a series (of four) designed to show how to use geometric algebras of multivectors and extensors to a novel presentation of some topics of differential geometry which are important for a deeper understanding of…
Geometric algebra is the natural outgrowth of the concept of a vector and the addition of vectors. After reviewing the properties of the addition of vectors, a multiplication of vectors is introduced in such a way that it encodes the famous…
The earlier approach is used for description of qubits and geometric phase parameters, the things critical in the area of topological quantum computing. The used tool, Geometric (Clifford) Algebra is the most convenient formalism for that…
In this Master of Science Thesis I introduce geometric algebra both from the traditional geometric setting of vector spaces, and also from a more combinatorial view which simplifies common relations and operations. This view enables us to…
The objective of the present paper (the second in a series of four) is to give a theory of multivector and extensor fields on a smooth manifold M of arbitrary topology based on the powerful geometric algebra of multivectors and extensors.…
Geometric (Clifford) algebra provides an efficient mathematical language for describing physical problems. We formulate general relativity in this language. The resulting formalism combines the efficiency of differential forms with the…
This is an introduction to geometric algebra, an alternative to traditional vector algebra that expands on it in two ways: 1. In addition to scalars and vectors, it defines new objects representing subspaces of any dimension. 2. It defines…
In this paper, the second in a series of eight we continue our development of the basic tools of the multivector and extensor calculus which are used in our formulation of the differential geometry of smooth manifolds of arbitrary topology…
This article provides a pedagogically oriented introduction to geometric (Clifford) calculus on pseudo-Riemannian manifolds. Unlike usual approaches to the topic, which rely on embedding the geometric algebra either within a tensor algebra…
The article presents a new approach to euclidean plane geometry based on projective geometric algebra (PGA). It is designed for anyone with an interest in plane geometry, or who wishes to familiarize themselves with PGA. After a brief…
A tutorial introduction to projective geometric algebra (PGA), a modern, coordinate-free framework for doing euclidean geometry. PGA features: uniform representation of points, lines, and planes; robust, parallel-safe join and meet…
Some notions of algebraic geometry can be defined for arbitrary varieties of algebras. This leads to universal algebraic geometry. The main idea of the presented theory is to consider interactions between algebra, logic and geometry in…
Geometric algebra is an optimal frame work for calculating with vectors. The geometric algebra of a space includes elements that represent all the its subspaces (lines, planes, volumes, ...). Conformal geometric algebra expands this…
The book is designed for a semester-long course in Foundations of Geometry and meant to be rigorous, conservative, elementary and minimalist. List of topics: Euclidean geometry: The Axioms / Half-planes / Congruent triangles / Perpendicular…
Universal geometric calculus simplifies and unifies the structure and notation of mathematics for all of science and engineering, and for technological applications. This paper treats the fundamentals of the multivector differential…
We present alternative postulates for Euclidean geometry whose merit is that they lead to a new class of invariants and associated geometries for real finite-dimensional unital associative algebras.
This book is expository and is in Russian. It is shown how in the course of solution of interesting geometric problems (close to applications) naturally appear main notions of algebraic topology (homology groups, obstructions and…
We survey the development of Clifford's geometric algebra and some of its engineering applications during the last 15 years. Several recently developed applications and their merits are discussed in some detail. We thus hope to clearly…
An introductory overview of vector spaces, algebras, and linear geometries over an arbitrary commutative field is given. Quotient spaces are emphasized and used in constructing the exterior and the symmetric algebras of a vector space.…
Projective geometry provides the preferred framework for most implementations of Euclidean space in graphics applications. Translations and rotations are both linear transformations in projective geometry, which helps when it comes to…