# Discrete Differential-Geometry Operators for Triangulated 2-Manifolds

Today I’ve read the paper “Discrete Differential-Geometry Operators

for Triangulated 2-Manifolds” by Mark Meyer, Mathieu Desbrun, Peter Schröder, Alan H. Barr.

## Novelty of the study

Deriving first and second order operators at the vertices of a mesh using the 1-ring(star neighborhood). In other words, providing the way to extend the definition of curvature from continuous surfaces to discrete meshes.

## Definitions of the curvature in continuous surface

The normal curvature and related notions are defined as follows.

## Approximation

The average calculation is restricted to be within the immediately neighboring triangles referred as the 1-ring.

The voronoi region is used as small area around point xi.

However, the formula for the Voronoi finite-volume area does not hold in the presence of obtuse angles. So mixed area is used in the actual calculation.

The Laplace-Beltrami operator K is computed.

The Laplace-Beltrami operator (the mean curvature normal operator) is a generalization of the Laplacian from flat spaces to manifolds.