Tracing cross-free polylines oriented by a N-symmetry direction field on triangulated surfaces

TL;DR

This paper introduces a novel combinatorial approach using stream-mesh data structures to trace cross-free polylines aligned with N-symmetry direction fields on triangulated surfaces, improving mesh segmentation techniques.

Contribution

It presents a new algorithm that enforces cross-free polylines on meshes by leveraging stream-mesh structures, avoiding complex numerical integrations.

Findings

Successfully traces cross-free polylines on triangulated surfaces.

Handles vector field singularities and polyline/vertex crossings consistently.

Enables improved mesh segmentation through combinatorial methods.

Abstract

We propose an algorithm for tracing polylines on a triangle mesh such that: they are aligned with a N-symmetry direction field, and two such polylines cannot cross or merge. This property is fundamental for mesh segmentation and is very difficult to enforce with numerical integration of vector fields. We propose an alternative solution based on "stream-mesh", a new combinatorial data structure that defines, for each point of a triangle edge, where the corresponding polyline leaves the triangle. It makes it possible to trace polylines by iteratively crossing triangles. Vector field singularities and polyline/vertex crossing are characterized and consistently handled. The polylines inherits the cross-free property of the stream-mesh, except inside triangles where avoiding local overlaps would require higher order polycurves.