Can anybody define what we mean by curvature of a vertex?
Curvature of a vertex in a polyhedral surface
3
$\begingroup$
polyhedra
curvature
1 Answers
2
In modern terms, the defect at a vertex [...] is precisely the curvature at that point [...], as established by the Gauss–Bonnet theorem.
-
2It is pe$r$haps wo$r$thwhile noting that this quote is referring to the *integrated* curvature "concentrated" at the vertex. The Gauss–Bonnet theorem relates the Euler characteristic to the total curvature integrated over the entire manifold; in generalizing this to polyhedral surfaces, the curvature would be given by a sum of delta distributions located at the vertices, with coefficients given by the angular defects of the vertices, so that integrating over the curvature would yield the sum of the angular defects. – 2012-09-30