Approximation Bounds for Interpolation and Normals on Triangulated Surfaces and Manifolds

TL;DR

This paper establishes new bounds on the approximation errors of triangulated surfaces and manifolds, focusing on interpolation and normal errors, with implications for surface reconstruction and mesh generation.

Contribution

It provides improved and explicit bounds on interpolation and normal errors for triangulated approximations of smooth manifolds of any dimension.

Findings

Bounds depend on local feature size and medial ball measurements.

Normal error bounds are specific to triangles, while interpolation bounds apply to simplices of any dimension.

New bounds have better constants and are the first with explicit constants in higher dimensions.

Abstract

How good is a triangulation as an approximation of a smooth curved surface or manifold? We provide bounds on the {\em interpolation error}, the error in the position of the surface, and the {\em normal error}, the error in the normal vectors of the surface, as approximated by a piecewise linearly triangulated surface whose vertices lie on the original, smooth surface. The interpolation error is the distance from an arbitrary point on the triangulation to the nearest point on the original, smooth manifold, or vice versa. The normal error is the angle separating the vector (or space) normal to a triangle from the vector (or space) normal to the smooth manifold (measured at a suitable point near the triangle). We also study the {\em normal variation}, the angle separating the normal vectors (or normal spaces) at two different points on a smooth manifold. Our bounds apply to manifolds of any dimension embedded in Euclidean spaces of any dimension, and our interpolation error bounds apply to simplices of any dimension, although our normal error bounds apply only to triangles. These bounds are expressed in terms of the sizes of suitable medial balls (the {\em empty ball size} or {\em local feature size} measured at certain points on the manifold), and have applications in Delaunay triangulation-based algorithms for provably good surface reconstruction and provably good mesh generation. Our bounds have better constants than the prior bounds we know of---and for several results in higher dimensions, our bounds are the first to give explicit constants.