TITLE:
Classification of Affine Local Vertex Curvature Functions with a Constant-Sum Law on Polygon Triangulations
AUTHORS:
Bekarys Tashmukhanbet
KEYWORDS:
Discrete Gauss-Bonnet, Combinatorial Curvature, Outerplanar Graphs, Polygon Triangulation, Ear Clipping
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.16 No.10,
October
9,
2026
ABSTRACT: 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(
v
)= A+ Bdeg(
v
)+ C
∑
f∋v
1
| f |
+ T
1
{
v∈∂G }
. 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.