discrete mean curvature

Mean curvature on a smooth surface measures how much the surface bows in or out on average — it is the quantity a soap film makes zero. To get an honest discrete version on a triangle mesh, the trick is not to fit a smooth surface and differentiate, but to ask: in which direction, and how strongly, does the surface want to move to shrink its area fastest? That direction-and-strength is the discrete mean curvature normal, and it falls out of the cotangent formula.

The cleanest definition uses the gradient of surface area. For a vertex v_i with neighbors v_j, the discrete mean-curvature normal is H(v_i) * N = (1 / (2*A_i)) * sum over neighbors j of (cot(alpha_ij) + cot(beta_ij)) * (v_i - v_j), where alpha_ij and beta_ij are the two angles opposite the edge (i, j) in the two triangles sharing it, and A_i is the area of the dual cell around v_i (often the Voronoi or mixed area). This vector points along the surface normal, its length is the mean curvature magnitude, and — crucially — it is exactly minus the gradient of total area with respect to v_i. So mean-curvature flow, which moves each vertex in the direction that decreases area fastest, is literally d v_i / dt = -H(v_i)*N, and this same cotangent expression is the cotangent Laplacian applied to the coordinate functions.

Discrete mean curvature is the engine of mesh smoothing, fairing, and minimal-surface computation: a mesh is discretely minimal exactly when this vector vanishes at every vertex, the discrete soap-film condition. Two honesty points matter. First, mean curvature is extrinsic — unlike the intrinsic angle defect, it depends on how the surface sits in R^3, so it is not determined by edge lengths alone but by the actual vertex positions. Second, there are several competing discretizations (cotangent/area-gradient, the Steiner/edge-based normal-cycle approach, and others) that agree in the refinement limit but differ on coarse meshes; do not assume two codes' 'mean curvature' fields are the same object.