Surface Quadrilateral Meshing from Integrable Odeco Fields

Abstract

We present a method for generating orthogonal quadrilateral meshes subject to user-defined feature alignment and sizing constraints. The approach relies on computing integrable orthogonal frame fields, whose symmetries are implicitly represented using orthogonally decomposable (odeco) tensors. We extend the existing 2D odeco integrability formulation to the 3D setting, and define the useful energies in a finite element approach. Our frame fields are shear-free (orthogonal) by construction, and we provide terms to minimize area and/or stretch distortion. The optimization naturally creates and places singularities to achieve integrability, obviating the need for user placement or greedy iterative methods. We validate the method on both smooth surfaces and feature-rich CAD models. Compared to previous works on integrable frame fields, we offer better performance in the presence of mesh sizing constraints and achieve lower distortion metrics.

Figures (13)

• Figure 1: Domain $\mathcal{M}$ is mapped onto a parametric plane by a seamless parametrization $\phi$, allowing to map a grid back onto $\mathcal{M}$.
• Figure 2: Equivalent gradient pairs $(\grad{u},\grad{v})$ across the charts of a global seamless parametrization.
• Figure 3: Gradients of a parametrization and corresponding integrable frame field.
• Figure 4: Even with no explicit feature alignment constraints, we still achieve natural frame alignment to sharp creases and principal curvature directions in this result on the B0 model from the MAMBO dataset mambo_dataset.
• Figure 5: Visualization of tensor polynomials $p_{\vb{T}}(\vb{x})$. Left: an isotropic odeco tensor with equal eigenvalues. Center: an anisotropic odeco tensor with distinct eigenvalues. Right: a non-odeco tensor that does not decompose into orthogonal rank-1 terms.