Integrable cross-field generation based on imposed singularity configuration -- the 2D manifold case --

TL;DR

This paper develops a mathematical framework for generating integrable 2D cross-fields on manifolds based on user-defined or natural singularity configurations, enabling advanced quad mesh generation with isotropic and anisotropic options.

Contribution

It introduces a novel formulation that computes integrable cross-fields from singularity sets via two linear PDEs, supporting both isotropic and anisotropic mesh generation.

Findings

Formulation allows computing cross-fields from singularities using two linear PDEs.

Supports both isotropic and anisotropic cross-field generation.

Enables advanced quad mesh generation with user-imposed singularity patterns.

Abstract

This work presents the mathematical foundations for the generation of integrable cross-field on 2D manifolds based on user-imposed singularity configuration. In this paper, we either use singularities that appear naturally, e.g., by solving a non-linear problem, or use as an input user-defined singularity pattern, possibly with high valence singularities that typically do not appear in cross-field computations. This singularity set is under the constraint of Abel-Jacobi's conditions for valid singularity configurations. The main contribution of the paper is the development of a formulation that allows computing an integrable isotropic 2D cross-field from a given set of singularities through the resolution of only two linear PDEs. To address the issue of possible suboptimal singularities' distribution, we also present the mathematical setting for the generation of an integrable anisotropic 2D cross-field based on a user-imposed singularity pattern. The developed formulations support both an isotropic and an anisotropic block-structured quad mesh generation. Keywords: integrable 2D cross-field, valid singularity configuration, quad layout, quad meshing