Curvature#

Discrete differential geometry tools for computing intrinsic and extrinsic curvatures on simplicial manifolds.

Two kinds of curvature are provided:

Gaussian Curvature (intrinsic)

Computed through the angle defect method: \(K_i = (\theta_{\text{full}} - \sum \theta_{ij}) / A_i^*\), where \(A_i^*\) is the dual (Voronoi) area around vertex \(i\). Gaussian curvature is an intrinsic property (Gauss’s Theorema Egregium), so it works for any codimension. Available at both vertices and cells.

Mean Curvature (extrinsic)

Computed through the cotangent Laplacian method: \(H_i = \|L \mathbf{x}\|_i / (2 A_i^*)\). Requires codimension 1 (the mesh must have well-defined normal vectors).

Both curvatures are also accessible as cached properties on the Mesh class (mesh.gaussian_curvature_vertices, mesh.mean_curvature_vertices).

from physicsnemo.mesh.primitives.surfaces import sphere_icosahedral

mesh = sphere_icosahedral.load(subdivisions=3)

# Via standalone functions
from physicsnemo.mesh.curvature import gaussian_curvature_vertices, mean_curvature_vertices
K = gaussian_curvature_vertices(mesh)
H = mean_curvature_vertices(mesh)

# Or via Mesh properties (equivalent, cached)
K = mesh.gaussian_curvature_vertices
H = mesh.mean_curvature_vertices

API Reference#

Curvature computation for simplicial meshes.

This module provides discrete differential geometry tools for computing intrinsic and extrinsic curvatures on n-dimensional simplicial manifolds.

Gaussian curvature (intrinsic):

  • Angle-defect method: \(K(v) = \Theta(v) / |{\star}v|\), where \(\Theta(v)\) is the angle defect at vertex \(v\) (full angle minus the sum of incident cell angles) and \(|{\star}v|\) is the dual 0-cell (Voronoi) volume.

  • Works for any codimension (intrinsic property).

  • Measures intrinsic geometry (Theorema Egregium).

Mean curvature (extrinsic):

  • Cotangent-Laplacian method: \(H(v) = \|L\, p\| / (2 \, |{\star}v|)\), where \(p\) is the vertex position vector and \(L\) is the cotangent Laplacian.

  • Requires codimension-1 (needs normal vectors).

  • Measures extrinsic bending.

Examples

>>> from physicsnemo.mesh.curvature import gaussian_curvature_vertices, mean_curvature_vertices
>>> from physicsnemo.mesh.primitives.surfaces import sphere_icosahedral
>>> mesh = sphere_icosahedral.load(subdivisions=2)
>>> # Compute Gaussian curvature
>>> K = gaussian_curvature_vertices(mesh)
>>> # Compute mean curvature (codimension-1 only)
>>> H = mean_curvature_vertices(mesh)
>>> # Or use Mesh properties:
>>> K = mesh.gaussian_curvature_vertices
>>> H = mesh.mean_curvature_vertices
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