Tutorial 1: Getting Started

This tutorial shows how to set up FrontIntrinsicOps.jl, generate a sphere mesh, and compute basic geometric quantities — all in less than 20 lines of code.

Prerequisites

using Pkg
Pkg.add("FrontIntrinsicOps")

Step 1: Generate a mesh

using FrontIntrinsicOps

R    = 1.0
mesh = generate_icosphere(R, 3)   # level-3 icosphere: 642 vertices, 1280 faces

The level-3 icosphere provides mesh size $h \approx 0.14$. Use level 4 or 5 for convergence studies.

Step 2: Compute geometry

geom = compute_geometry(mesh)   # barycentric dual areas (default)

This fills geom.face_normals, geom.face_areas, geom.edge_lengths, geom.vertex_dual_areas, and geom.vertex_normals.

Step 3: Build DEC operators

dec = build_dec(mesh, geom)

This assembles:

• dec.d0 — vertex-to-edge coboundary $d_0$
• dec.d1 — edge-to-face coboundary $d_1$
• dec.star0 — Hodge star $\star_0$ (dual areas)
• dec.star1 — Hodge star $\star_1$ (cotan weights)
• dec.laplacian — Laplace–Beltrami $L = \star_0^{-1} d_0^\top \star_1 d_0$

Step 4: Geometric checks

println("Area           = ", measure(mesh, geom))        # ≈ 4π ≈ 12.566
println("Enclosed vol.  = ", enclosed_measure(mesh))     # ≈ (4/3)π ≈ 4.189
println("Euler χ        = ", euler_characteristic(mesh)) # = 2 for sphere
println("Gauss–Bonnet   = ", gauss_bonnet_residual(mesh, geom))  # ≈ 0

For a sphere of radius $R = 1$:

Step 5: Curvature

H = mean_curvature(mesh, geom, dec)
K = gaussian_curvature(mesh, geom)

println("Mean curvature: min=", minimum(H), "  max=", maximum(H))
# ≈ [1.0, 1.0] for sphere of radius 1

println("Gaussian curvature: min=", minimum(K), "  max=", maximum(K))
# ≈ [1.0, 1.0] for sphere of radius 1

println("∫K dA = ", integrated_gaussian_curvature(mesh, geom))
# ≈ 4π ≈ 12.566

Step 6: DEC diagnostics

report = check_dec(mesh, geom, dec; tol=1e-10)
println(report.d1_d0_zero)          # true: d₁ d₀ = 0
println(report.lap_constant_nullspace) # true: L 1 ≈ 0
println(report.star0_positive)      # true: all dual areas > 0

Step 7: Apply the Laplacian

For a smooth function $u = z$ on the sphere, $-\Delta_\Gamma z = (2/R^2) z$:

u  = [p[3] for p in mesh.points]   # z-coordinate at each vertex
Lu = dec.laplacian * u             # L u ≈ (2/R²) u

err = maximum(abs, Lu .- 2.0 .* u)
println("Eigenvalue error: $err")   # → ~1e-2 at level 3, → 0 as level → ∞

Complete script

using FrontIntrinsicOps

R    = 1.0
mesh = generate_icosphere(R, 3)
geom = compute_geometry(mesh)
dec  = build_dec(mesh, geom)

println("=== Geometric quantities ===")
println("Area       = ", measure(mesh, geom))
println("Volume     = ", enclosed_measure(mesh))
println("χ          = ", euler_characteristic(mesh))
println("GB residual = ", gauss_bonnet_residual(mesh, geom))

H = mean_curvature(mesh, geom, dec)
println("\n=== Curvature ===")
println("H range = [", minimum(H), ", ", maximum(H), "]   (exact: 1.0)")

println("\n=== DEC check ===")
rpt = check_dec(mesh, geom, dec)
println("d1∘d0 = 0 : ", rpt.d1_d0_zero)
println("L nullspace: ", rpt.lap_constant_nullspace)

Next steps

• Surface diffusion tutorial
• Scalar transport tutorial
• Math: Discrete geometry
• API: Geometry and DEC