Classification of Affine Local Vertex Curvature Functions with a Constant-Sum Law on Polygon Triangulations

This paper presents a complete classification of affine local vertex curvature functions on triangulated polygonal disks, that is, on simple 2-cell plane triangulations whose outer boundary is a simple cycle, whose total vertex curvature takes one and the same value on every member of that class. We study local expressions of the form $K\left(v\right)=\text{？}A+\text{？}B\mathrm{deg}\left(v\right)+\text{？}C{\displaystyle {\sum }_{f？v}\frac{1}{\left|f\right|}}+\text{？}T{1}_{\left\{v\in ？G\right\}}$ . Summing over all vertices and applying Euler and incidence relations reduces the total to an affine function of the global counts V and L. The total is independent of V, L and the particular triangulation if and only if A + 6B + 2C = 0 and T = 2B + C, in which case the invariant is ？6B ？ C. Necessity uses the fact that V and L vary independently inside this class, which is established here by a stellar-subdivision construction; on the subclass of polygon triangulations without interior vertices one has L = V, and the two conditions are then sufficient but not separately necessary. Higuchi’s combinatorial curvature appears as a special case. A further specialization gives uniform curvature 1/n on every vertex of a convex n-gon. Randomized ear-clipping and stellar-subdivision triangulations, run as an implementation check of the counting identity, reproduce the predicted constants exactly in rational arithmetic and to within 5 × 10^？15 in double precision. The result provides a compact local-to-global classification for discrete curvature on triangulated polygonal disks.