更好一点：具有退化度量的几何代数中对偶的变体
A bit better: Variants of duality in geometric algebras with degenerate metrics
摘要：
与伪标量$\mathbf{I}$相乘传统上用于几何代数中以执行非度量运算，例如计算坐标和回归积。在具有退化度量的代数中，如欧几里得PGA $P(\mathbb{R}^*_{3,0,1})$，这种方法失效了，导致寻找非度量形式的对偶。本文比较了对偶坐标映射$J: G \rightarrow G^*$（一种双代数对偶）和Hodge对偶$H: G \rightarrow G $（一种单代数对偶）的性质.虽然这两个映射在计算上是相同的，但只有$J$是无坐标的，并提供了对几何对偶的直接支持，即每个几何图元出现两次，一次是基于点的形式，一次是基于平面的形式，这不仅是射影几何的基本特征，也是欧几里得运动学和动力学的基本特征。我们的分析得出结论，提出了一个双重中立的软件实现，每个多向量需要一个单一的位字段。
英文摘要：
Multiplication by the pseudoscalar $\mathbf{I}$ has been traditionally used in geometric algebra to perform non-metric operations such as calculating coordinates and the regressive product. In algebras with degenerate metrics, such as euclidean PGA $P(\mathbb{R}^*_{3,0,1})$, this approach breaks down, leading to a search for non-metric forms of duality. The article compares the dual coordinate map $J: G \rightarrow G^*$, a double algebra duality, and Hodge duality $H: G \rightarrow G $, a single algebra duality for this purpose. While the two maps are computationally identical, only $J$ is coordinate-free and provides direct support for geometric duality, whereby every geometric primitive appears twice, once as a point-based and once as a plane-based form, an essential feature not only of projective geometry but also of euclidean kinematics and dynamics. Our analysis concludes with a proposed duality-neutral software implementation, requiring a single bit field per multi-vector.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
