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Geometric Algebra

时间: 2024-08-29 09:02:53 浏览: 45

Geometric Algebra Applications Vol. I: Computer Vision, Graphics and Neurocomput

The goal of the Volume I Geometric Algebra for Computer Vision, Graphics and Neural Computing is to present a unified mathematical treatment of diverse problems in the general domain of artificial intelligence and associated fields using Clifford, or geometric, algebra. Geometric algebra provides a rich and general mathematical framework for Geometric Cybernetics in order to develop solutions, concepts and computer algorithms without losing geometric insight of the problem in question. Current mathematical subjects can be treated in an unified manner without abandoning the mathematical system of geometric algebra for instance: multilinear algebra, projective and affine geometry, calculus on manifolds, Riemann geometry, the representation of Lie algebras and Lie groups using bivector algebras and conformal geometry. By treating a wide spectrum of problems in a common language, this Volume I offers both new insights and new solutions that should be useful to scientists, and engineers working in different areas related with the development and building of intelligent machines. Each chapter is written in accessible terms accompanied by numerous examples, figures and a complementary appendix on Clifford algebras, all to clarify the theory and the crucial aspects of the application of geometric algebra to problems in graphics engineering, image processing, pattern recognition, computer vision, machine learning, neural computing and cognitive systems.

几何代数（Geometric Algebra, GA）是一种数学工具，它将向量和标量合并成统一的代数结构。GA通过结合代数和几何的概念，提供了一种简洁的方式来处理空间中的线性变换、方向和长度等信息。它扩展了传统向量分析中的加法和乘法操作，引入了外积（geometric product），这使得对旋转、反射等基本变换的操作变得直观。 GA的主要特点包括： 1. 合并旋转和平移：在GA中，旋转和位移不再是独立的概念，它们都可以由单个操作表达，简化了许多计算。 2. 方程的简洁性：相比于传统的矩阵方程，GA中的表达通常更短，更容易理解。 3. 内积和外积合而为一：内积对应于点积，而外积则包含了叉积和平行四边形积等多种运算。

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