Hodge Star

HodgeStar computes the three diagonal Hodge star operators , , from primal and dual volumes. The Hodge star encodes all metric information; combined with the metric-free exterior derivative it produces all geometric operators (Laplacian, divergence, etc.).

Summary

All three are diagonal: stored as DVector<f64>, applied via component_mul.

Construction

use cartan_dec::{Mesh, HodgeStar};
 
let mesh = Mesh::unit_square_grid(4);
let hodge = HodgeStar::from_mesh(&mesh);

Applying the Hodge Star

HodgeStar stores the diagonals as DVector<f64>; apply via component_mul:

use nalgebra::DVector;
 
// ⋆₀: vertex 0-form → dual 2-form (element-wise multiply by star0)
let f  = DVector::from_element(mesh.n_vertices(), 1.0_f64);
let sf = f.component_mul(&hodge.star0);
 
// ⋆₁: edge 1-form → dual 1-form
let alpha  = DVector::from_element(mesh.n_edges(), 1.0_f64);
let salpha = alpha.component_mul(&hodge.star1);
 
// ⋆₂: triangle 2-form → dual 0-form
let omega  = DVector::from_element(mesh.n_triangles(), 1.0_f64);
let somega = omega.component_mul(&hodge.star2);
 
// Inverse: divide by the diagonal
let f_back = sf.component_div(&hodge.star0);
assert!((f_back - f).norm() < 1e-12);

Formula Reference

Diagonal entries:

where is the dual vertex area (sum of of adjacent triangle areas for barycentric dual, or circumcentric area for circumcentric dual), is the dual edge length (distance between circumcenters of adjacent triangles), and is the triangle area.

Inverse (from for ):

Numerical Notes

Well-centered condition: entries are positive iff all dual edge lengths , which holds iff the triangulation is well-centered (all circumcenters inside their triangles). A negative entry indicates an obtuse triangle; the Laplacian will not be positive semi-definite.

Building operators: the Laplace-Beltrami is (using as a weight matrix). Operators::from_mesh assembles this automatically.

Applications

• Building the Laplace-Beltrami operator from and
• Hodge decomposition of discrete vector fields
• Converting between primal and dual mesh quantities