Euclidean Geometry的几何代数
Geometric Algebras for Euclidean Geometry
摘要：
讨论如何应用几何代数欧几里德$n$ -空间一直笼罩着一些概念上的误解，我们首先确定和解决，基于彻底审查的关键，但很大程度上被遗忘的主题，从$19^{th}$世纪的数学。然后，我们介绍了对偶投影Clifford代数$\mathbf{P}(\mathbb{R}^*_{n,0,1})$（欧几里德PGA）作为最有前途的齐次（1-up）候选欧几里德几何。我们比较欧几里德PGA和流行的2-up模型CGA（共形几何代数），限制注意平面几何图元，并表明，在这个域上，他们表现出相同的形式特征集。因此，我们建立欧几里德PGA是最小的结构保持欧几里德GA。我们比较两个代数更详细，就一些实际的标准，包括实施运动学和刚体力学。然后，我们将比较扩展到包括欧几里得球体基元。我们的结论是，欧几里德PGA提供了一个自然的过渡，科学和教学，向量空间模型和更复杂和强大的CGA之间。
英文摘要：
The discussion of how to apply geometric algebra to euclidean $n$-space has been clouded by a number of conceptual misunderstandings which we first identify and resolve, based on a thorough review of crucial but largely forgotten themes from $19^{th}$ century mathematics. We then introduce the dual projectivized Clifford algebra $\mathbf{P}(\mathbb{R}^*_{n,0,1})$ (euclidean PGA) as the most promising homogeneous (1-up) candidate for euclidean geometry. We compare euclidean PGA and the popular 2-up model CGA (conformal geometric algebra), restricting attention to flat geometric primitives, and show that on this domain they exhibit the same formal feature set. We thereby establish that euclidean PGA is the smallest structure-preserving euclidean GA. We compare the two algebras in more detail, with respect to a number of practical criteria, including implementation of kinematics and rigid body mechanics. We then extend the comparison to include euclidean sphere primitives. We conclude that euclidean PGA provides a natural transition, both scientifically and pedagogically, between vector space models and the more complex and powerful CGA.
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