Mesh processing (discrete differential geometry)

Level AdvancedDifficulty ★★★★★Application⌖ Open in the map

What is it?

Calculus on triangle meshes: a discrete Laplacian (cotangent weights), discrete curvatures and integrals over faces. Used for smoothing scanned models, parametrizing for textures, deforming characters and computing geodesics.

Formulas

$(\Delta f)_i = \frac{1}{2A_i}\sum_{j\in N(i)}\big(\cot\alpha_{ij} + \cot\beta_{ij}\big)\big(f_j - f_i\big)$
cotangent Laplacian

The mathematics behind it

Laplacian★★★★★fundamental
The discrete (cotangent) Laplacian is the core operator of geometry processing: smoothing, parametrization, deformation.
Curvature (basic differential geometry)★★★★★fundamental
Discrete mean curvature (from the cotangent Laplacian) drives smoothing, remeshing and feature detection.
Green's theorem★★★★★frequent
Polygon areas, centroids and orientation tests in GIS and graphics use the shoelace formula.

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.