Plane-Constrained Geodesic Curves on Point Clouds

Curves constructed directly on point clouds are a core primitive in reverse engineering, product design, and point-based CAD; many workflows additionally require the curve to lie in a plane—e.g., as a section profile, inspection path, or design reference. This paper presents an algorithmic framework for computing free and plane-constrained geodesic curves directly on oriented point clouds, without any intermediate surface or mesh reconstruction. A geodesic-curvature-minimizing solver that combines a Newton/conjugate-gradient flow with directed projection, elliptic Gabriel neighborhoods, and Taubin smoothing forms the backbone; the plane-constrained problem is then reduced to a one-parameter pencil of planes through the endpoint chord and solved per plane by alternating projection onto the cloud and the plane, with a projection-only pre-lift and a penalized length objective that rejects sections floating off the cloud; the returned section is the best found over a sampled pencil of candidate planes. The returned sections are attached to the cloud within a small fraction of the mean sampling distance. All algorithms are given in pseudocode with convergence criteria and complexity estimates. Two parallel realizations of the plane search are developed and measured: a multithreaded CPU backend (about 3× over the serial scan) and a WebGPU backend that evaluates the whole plane pencil in a single compute dispatch. Accuracy is validated against the analytic conic sections of a cone and against cylinder and sphere benchmarks whose optimal plane is known in closed form; robustness is assessed under noise, non-uniform sampling, missing regions, outliers, and perturbed normals, and against both a slab-projection baseline and the conventional reconstruct-then-slice route. Five applications—shoe-last reverse engineering with a C2 surface reconstruction, anthropometric girth measurement, medical transverse sectioning, dimensional metrology on industrial mold scans, and cleaning-path planning for a robotic surface-treatment task—demonstrate the plane-constrained geodesic curves in practice.