Geodesics

This page documents intrinsic geodesic-distance and path tools on triangulated surfaces.

Mathematical object

Distance is computed with a heat-method pipeline on vertex 0-forms:

1. Short-time heat solve for u,
2. normalized face vector field X = -∇u / |∇u|,
3. Poisson solve for distance potential φ,
4. constant shift to pin source value.

Cochain space / degree

• Distance field: vertex 0-form (length = nV).
• Geodesic gradient: per-face tangent vectors (length = nF).
• Shortest path output: vertex index sequence.

Orientation convention

• Distances are nonnegative scalars and orientation-independent.
• Path extraction uses vertex graph adjacency from oriented mesh topology, but returned path is geometric (ordered vertices from src to dst).

Storage convention

• Distances are vectors indexed by mesh vertex order.
• Multi-source distances use pointwise minimum over sourcewise distance vectors.

API

geodesic_distance
geodesic_distance_to_vertex
geodesic_distance_to_vertices
geodesic_gradient
shortest_path_vertices
shortest_path_points
intrinsic_ball
farthest_point_sampling_geodesic

Minimal example

using FrontIntrinsicOps
using LinearAlgebra

mesh = generate_icosphere(1.0, 1)
geom = compute_geometry(mesh)
dec = build_dec(mesh, geom)

src = argmax([p[3] for p in mesh.points])
d = geodesic_distance_to_vertex(mesh, geom, dec, src)
path = shortest_path_vertices(mesh, geom, dec, src, argmin([p[3] for p in mesh.points]); distance=d)

(mind=minimum(d), maxd=maximum(d), pathlen=length(path))

Limitations and non-goals

• Surface-only geodesic tools in this page (curve geodesic distance is not included here).
• Shortest paths are approximate vertex paths (steepest-descent + fallback), not exact polyline geodesics.
• Connected-surface assumption is enforced with explicit error handling.