Hermite interpolation by piecewise polynomial surfaces with polynomial area element

TL;DR

This paper introduces a method for constructing polynomial 2-surfaces with polynomial area elements, enabling Hermite interpolation of point-normal networks using piecewise polynomial patches in Euclidean and Minkowski spaces.

Contribution

It develops a systematic approach to generate polynomial surfaces with polynomial area elements from prescribed normals, extending interpolation techniques to Euclidean and Minkowski spaces.

Findings

Polynomial surfaces with polynomial area elements can be constructed from normal fields.

The degree of the surface depends on the degree and quality of the prescribed normals.

The method successfully interpolates point-normal networks with piecewise polynomial patches.

Abstract

This paper is devoted to the construction of polynomial 2-surfaces which possess a polynomial area element. In particular we study these surfaces in the Euclidean space $\mathbb R^3$ (where they are equivalent to the PN surfaces) and in the Minkowski space $\mathbb R^{3,1}$ (where they provide the MOS surfaces). We show generally in real vector spaces of any dimension and any metric that the Gram determinant of a parametric set of subspaces is a perfect square if and only if the Gram determinant of its orthogonal complement is a perfect square. Consequently the polynomial surfaces of a given degree with polynomial area element can be constructed from the prescribed normal fields solving a system of linear equations. The degree of the constructed surface depending on the degree and the quality of the prescribed normal field is investigated and discussed. We use the presented approach to interpolate a network of points and associated normals with piecewise polynomial surfaces with polynomial area element and demonstrate our method on a number of examples (constructions of quadrilateral as well as triangular patches