API: Geometry and DEC

Geometry

`compute_geometry`

compute_geometry(mesh::CurveMesh) → CurveGeometry{T}
compute_geometry(mesh::SurfaceMesh; dual_area=:barycentric) → SurfaceGeometry{T}

Compute all intrinsic geometric quantities.

Keyword	Values	Description
`dual_area`	`:barycentric` (default)	One-third of adjacent face areas
	`:mixed` or `:voronoi`	Meyer et al. 2003 mixed Voronoi areas

DEC assembly

`build_dec`

build_dec(mesh::CurveMesh, geom::CurveGeometry) → CurveDEC{T}
build_dec(mesh::SurfaceMesh, geom::SurfaceGeometry; laplace=:dec) → SurfaceDEC{T}

Assemble the full discrete exterior calculus structure.

Keyword	Values	Description
`laplace`	`:dec` (default)	$L = \star_0^{-1} d_0^\top \star_1 d_0$
	`:cotan`	Direct cotan assembly (numerically equivalent)

`build_laplace_beltrami`

build_laplace_beltrami(mesh, geom; method=:dec) → SparseMatrixCSC

Return only the Laplacian matrix (no other DEC operators).

`laplace_beltrami`

laplace_beltrami(mesh, geom, dec, u::Vector) → Vector

Apply the Laplacian to a scalar field: returns $L u$.

Hodge stars (individual access)

hodge_star_0(mesh, geom) → SparseMatrixCSC   # diagonal: dual areas / lengths
hodge_star_1(mesh, geom) → SparseMatrixCSC   # diagonal: cotan weights (surface)
hodge_star_2(mesh, geom) → SparseMatrixCSC   # diagonal: 1/face_area

Exterior derivatives (individual access)

incidence_0(mesh) → SparseMatrixCSC   # d0: N_E × N_V
incidence_1(mesh) → SparseMatrixCSC   # d1: N_F × N_E

Codifferentials

codifferential_1(mesh, geom, dec) → SparseMatrixCSC   # δ₁ = ⋆₀⁻¹ d₀ᵀ ⋆₁
codifferential_2(mesh, geom, dec) → SparseMatrixCSC   # δ₂ = ⋆₁⁻¹ d₁ᵀ ⋆₂

DEC gradient and divergence

gradient_0_to_1(mesh, dec, u)        → Vector  # d0 * u
divergence_1_to_0(mesh, geom, dec, α) → Vector  # δ₁ α
curl_like_1_to_2(mesh, dec, α)       → Vector  # d1 * α

Topology

build_topology(mesh::SurfaceMesh) → MeshTopology

is_closed(mesh)                 → Bool
is_manifold(mesh)               → Bool
has_consistent_orientation(mesh) → Bool
euler_characteristic(mesh)      → Int    # V - E + F

Curvature

# Curves
curvature(mesh::CurveMesh, geom) → Vector{T}         # signed κ

# Surfaces
mean_curvature(mesh, geom, dec) → Vector{T}           # scalar H
mean_curvature_normal(mesh, geom, dec) → Vector{SVector{3,T}}
gaussian_curvature(mesh, geom) → Vector{T}            # angle-defect K
compute_curvature(mesh, geom, dec) → SurfaceGeometry  # updates curvature fields

Integrals

measure(mesh, geom) → T                          # total arc length or area
enclosed_measure(mesh) → T                       # enclosed area or volume
integrate_vertex_field(mesh, geom, u) → T        # Σ u_i A_i*
integrate_face_field(mesh, geom, u) → T          # Σ u_f A_f
integrated_gaussian_curvature(mesh, geom) → T    # Σ K_i A_i*

Diagnostics

check_mesh(mesh) → NamedTuple
# Fields: n_vertices, n_edges, [n_faces], closed, manifold,
#         consistent_orientation, euler_characteristic, warnings

check_dec(mesh, geom, dec; tol=1e-10) → NamedTuple
# Fields: d1_d0_zero, d1_d0_max_residual, lap_constant_nullspace,
#         star0_positive, star1_positive, warnings

gauss_bonnet_residual(mesh, geom) → T        # |∫K dA - 2πχ|
star1_sign_report(dec) → NamedTuple          # n_nonpositive, frac, min_entry
compare_laplace_methods(mesh, geom) → NamedTuple  # ‖L_dec − L_cotan‖