Surface meshing

icosphere(n) -> TriangleMesh

Return a unit icosphere (a TriangleMesh of radius 1, centred at the origin), built by subdividing an icosahedron n times.

Every vertex lies exactly on the unit sphere (midpoints are renormalized), so an icosphere is an exact discretization of the sphere's radius — the flat triangles only shorten the chords, which is why its area slightly underestimates $4\pi$ while its vertex positions are exact.

Arguments

• n: the number of subdivision levels (each level splits every triangle into four).

Returns

A TriangleMesh with $20 \cdot 4^{n}$ faces.

Example

julia> using Rasmah

julia> round(surface_area(icosphere(2)); digits = 2)
12.33 m^2

surface_mesh(g; n=nothing, adaptive=false, min_depth=0, max_depth=6,
             method=:marching, curvature=false, curvature_tol=0.1) -> TriangleMesh

Triangle-mesh the surface of an implicit geometry (ImplicitGeometry) or a Feature. This is the main surface-meshing entry point.

Theory

An implicit surface is the zero level set $\{\mathbf{x} : s(\mathbf{x}) = 0\}$ of a signed distance field $s$ (see the "Geometry & modeling" chapter). A TriangleMesh approximates that surface with flat triangles. The SDF is exact, so refining the mesh only moves the approximation closer to the true surface — the surface itself is the limit as the resolution grows.

Algorithm

With the default n = nothing the standard primitives resolve to a lightweight analytic mesh — an exact icosphere for a sphere, a 12-triangle box_mesh, a ruled cylinder/cone — which is both lighter and more accurate than marching cubes. Passing an explicit n scales the sphere's icosphere level and, for shapes with no analytic resolution, falls back to marching_cubes at n. adaptive = true selects the adaptive-octree surface-nets kernel (adaptive_surface_mesh), and method = :dual selects QEF dual_contouring, which preserves sharp edges and corners.

Keyword options

• n = nothing — grid resolution. nothing picks the analytic meshes above; an integer enables the resolution-driven kernels.
• adaptive = false — use the adaptive octree (refines near the surface only).
• min_depth = 0, max_depth = 6 — octree depth bounds (adaptive only).
• method = :marching — :marching (marching cubes) or :dual (QEF dual contouring).
• curvature = false — additionally coarsen flat surface cells (adaptive only).
• curvature_tol = 0.1 — normal-turn threshold in radians (adaptive only).

Example

julia> m = surface_mesh(sphere(2.0));

julia> size(m.vertices, 2)
642

The default n = nothing resolves a sphere to an exact icosphere, so m has 642 vertices with every vertex exactly on the surface.

See also

tetrahedralize for the volume counterpart, and adaptive_surface_mesh / dual_contouring for the advanced kernels.

surface_mesh(b::BRep; n = nothing, parallel = :auto) -> TriangleMesh

Tessellate b into an orientation-consistent, watertight triangle mesh with outward normals. Adjacent faces share welded boundary vertices.

Keyword arguments

• n: mesh resolution; nothing uses the default, an integer scales it.
• parallel: threading mode (:auto, :none, :all, true, false).

Returns

A welded TriangleMesh whose signed volume equals the solid's volume.

marching_cubes(f, lo, hi, n; parallel = true) -> TriangleMesh

Watertight marching-cubes iso-surface of the SDF f at the zero level over the box [lo, hi] at resolution n.

dual_contouring(f, lo, hi, n) -> TriangleMesh

Extract an isosurface of the SDF-like callable f over the box [lo, hi] by QEF dual contouring on an n×n×n grid. One vertex per surface cell is placed at the minimizer of the cell's quadratic error function, preserving sharp edges and corners; the vertices are stitched into fans around each crossing grid edge.

adaptive_surface_mesh(f, lo, hi; min_depth=0, max_depth=6, qef=false, curvature=false, curvature_tol=0.1) -> TriangleMesh

Extract a crack-free surface TriangleMesh of the implicit surface f(x) = 0 over [lo, hi] by adaptive octree refinement and dual contouring (surface nets, or QEF dual contouring with qef = true). With curvature = true a surface cell is refined further only while the field is non-linear across it, so flat regions settle coarser than curved ones.

surface_area(x) -> Quantity

The surface area of a TriangleMesh x, as an SI Quantity of area dimension (D_AREA; see the Units chapter).

The area is the sum of the triangle areas — half the norm of each face's edge cross product. Use area for a sketch, and surface_area(surface_mesh(g)) to measure a feature or implicit geometry.

Example

julia> using Rasmah

julia> round(surface_area(icosphere(2)); digits = 2)
12.33 m^2

decimate(m::TriangleMesh, ratio; preserve_boundary=true, feature_angle=nothing)

Quadric-error-metric surface simplification (Garland & Heckbert 1997): collapse edges in increasing quadric-error cost until the face count reaches ratio of the original. preserve_boundary=true pins boundary vertices; feature_angle (a dihedral-angle threshold in degrees) protects sharp-crest edges and corners from collapse.

smooth_surface(m::TriangleMesh; iterations=5, λ=0.5, μ=-0.8) -> TriangleMesh

Taubin λ|μ smoothing: alternating umbrella (graph-Laplacian) passes with a positive and a negative scale factor, denoising the surface without the shrinkage of plain Laplacian smoothing.