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：本文通过基于平面的几何代数（PGA）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、面等）与$k$-复形（点云、线复形、网格等）可分别简洁地表示为顶点的连接及其和。通过PGA的欧几里得范数与理想范数，我们自然推导出计算其$k$-度量（数量、长度、面积等）的统一公式。这一思想进一步扩展为无坐标的统一公式，用于计算任意维度单纯形与复形的经典几何量，如体积、质心及转动惯量。最后，我们通过实际案例展示了这些理论的应用价值。