We revisit the geometric foundations of mesh representation through the lens of Plane-based Geometric Algebra (PGA), questioning its efficiency and expressiveness for discrete geometry. We find how $k$-simplices (vertices, edges, faces, ...) and $k$-complexes (point clouds, line complexes, meshes, ...) can be written compactly as joins of vertices and their sums, respectively. We show how a single formula for their $k$-magnitudes (amount, length, area, ...) follows naturally from PGA's Euclidean and Ideal norms. This idea is then extended to produce unified coordinate-free formulas for classical results such as volume, centre of mass, and moments of inertia for simplices and complexes of arbitrary dimensionality. Finally we demonstrate the practical use of these ideas on some real-world examples.

翻译：我们通过基于平面的几何代数（PGA）的视角重新审视网格表示的几何基础，质疑其在离散几何中的效率与表达能力。我们发现$k$-单纯形（顶点、边、面、...）和$k$-复形（点云、线复形、网格、...）可以分别紧凑地表示为顶点的连接及其和。我们展示了如何从PGA的欧几里得范数和理想范数自然地推导出它们$k$-度量（数量、长度、面积、...）的统一公式。随后，这一思想被推广以构建无坐标的统一公式，用于计算任意维度单纯形和复形的经典结果，如体积、质心和转动惯量。最后，我们通过若干实际案例展示了这些思想的应用价值。