论文标题
好一点：与退化指标的几何代数的二元性变体
A bit better: Variants of duality in geometric algebras with degenerate metrics
论文作者
论文摘要
传统上，在几何代数中使用伪级$ \ mathbf {i} $乘法来执行非金属操作，例如计算坐标和回归产品。在具有退化指标的代数中，例如Euclidean PGA $ P（\ Mathbb {r}^*_ {3,0,1}）$，此方法分解，从而搜索非金属形式的偶性。本文比较了双坐标地图$ j：g \ rightarrow g^*$，double代数双重性和霍奇二元性$ h：g \ rightarrow g $，为此目的是单个代数二元性。尽管这两个地图在计算上是相同的，但只有$ j $是无坐标的，并为几何二元性提供了直接的支持，因此每个几何原始词都会出现两次，一次是基于点的基于平面的形式，不仅是投影性几何形式的基本特征，而且是欧几里得动力学和动力学的基本特征。我们的分析以提出的二元性中立软件实现结束，每个多向量需要一个位字段。
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.
