Approximating triangular meshes by implicit, multi-sided surfaces

TL;DR

This paper introduces a method to approximate triangular meshes using a collection of smoothly connected implicit multi-sided I-patches, optimizing their parameters for accurate surface representation and efficient distance field approximation.

Contribution

It presents a novel approach for mesh approximation with implicit I-patches, including a refinement process and a normalization method for Euclidean distance fields.

Findings

Effective mesh approximation with implicit I-patches.

Supports T-node refinement for complex geometries.

Provides an efficient normalization for Euclidean distance fields.

Abstract

The I-patch is a multi-sided surface representation, defined as a combination of implicit ribbon and bounding surfaces, whose pairwise intersections determine the natural boundaries of the patch. Our goal is to show how a collection of smoothly connected I-patches can be used to approximate triangular meshes. We start from a coarse, user-defined vertex graph which specifies an initial subdivision of the surface. Based on this, we create ribbons that tightly fit the mesh along its edges in both positional and tangential sense, then we optimize the free parameters of the patch to better approximate the interior. If the surfaces are not sufficiently accurate, the network needs to be refined; here we exploit that the I-patch construction naturally supports T-nodes. We also describe a normalization method that nicely approximates the Euclidean distance field, and can be efficiently evaluated. The capabilities and limitations of the approach are analyzed through several examples.