Estimating parametric surface areas using spline quasi-interpolants

Abstract

This article introduces a novel method for estimating the area of parametric surfaces, utilizing numerical integration and bivariate spline quasi-interpolants (QI) over a uniform subdivision of a rectangular domain. The surfaces are approximated using QIs of bi-degree (k, k), and the area is computed via an expression involving a double integral. Our method demonstrates a convergence order of k + 2 for even k and k + 1 for odd k. These theoretical results were validated through numerical examples, focusing on both quadratic and cubic cases. Furthermore, we illustrate the practical applicability of our method by estimating the area of a human head using real 3D point data.