Mesh

Mesh is a 2D simplicial complex: a collection of vertices, oriented edges, and triangles, together with their primal and dual volumes needed by the Hodge star.

Summary

Construction

Uniform grid

use cartan_dec::Mesh;
 
// 4x4 grid on [0,1]²: 16 vertices, structured triangulation
let mesh = Mesh::unit_square_grid(4);
 
println!("Vertices:  {}", mesh.n_vertices());   // 16
println!("Edges:     {}", mesh.n_edges());       // 40
println!("Triangles: {}", mesh.n_triangles());   // 18

From raw geometry

use cartan_dec::Mesh;
 
let vertices:  Vec<[f64; 2]> = vec![[0.0, 0.0], [1.0, 0.0], [0.0, 1.0], [1.0, 1.0]];
let triangles: Vec<[usize; 3]> = vec![[0, 1, 2], [1, 3, 2]];
 
let mesh = Mesh::from_triangles(vertices, triangles);

Fields

// Public fields
mesh.vertices:  Vec<[f64; 2]>    // vertex coordinates
mesh.edges:     Vec<[usize; 2]>  // edge endpoint indices (oriented)
mesh.triangles: Vec<[usize; 3]>  // triangle vertex indices (CCW)
 
// Key methods
mesh.n_vertices()      // -> usize
mesh.n_edges()         // -> usize
mesh.n_triangles()     // -> usize
mesh.triangle_area(t)  // -> f64: area of triangle t
mesh.vertex(i)         // -> [f64; 2]: coordinates of vertex i

Numerical Notes

Orientation: edges are stored with the smaller index first. Triangles are stored in counter-clockwise (CCW) order to ensure positive triangle areas and correct exterior derivative signs.

Well-centered condition: the Hodge star is positive definite only when the mesh is well-centered (circumcenter inside each triangle). unit_square_grid produces a well-centered mesh; custom meshes from from_triangles are not automatically checked.

Memory layout: vertices, edges, and triangles are stored as plain Vec arrays (no adjacency lists). The ExteriorDerivative and Operators structs precompute adjacency during construction.

Applications

• Uniform grid for heat equation and Laplacian studies
• Custom triangulated domain for Q-tensor / active nematics simulation
• Benchmark meshes for DEC operator testing