Hermite interpolation by piecewise polynomial surfaces with polynomial area element

摘要

This paper is devoted to the construction of polynomial 2-surfaces whichpossess a polynomial area element. In particular we study these surfaces in theEuclidean 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 MOSsurfaces). We show generally in real vector spaces of any dimension and anymetric that the Gram determinant of a parametric set of subspaces is a perfectsquare if and only if the Gram determinant of its orthogonal complement is aperfect square. Consequently the polynomial surfaces of a given degree withpolynomial area element can be constructed from the prescribed normal fieldssolving a system of linear equations. The degree of the constructed surfacedepending on the degree and the quality of the prescribed normal field isinvestigated and discussed. We use the presented approach to interpolate anetwork of points and associated normals with piecewise polynomial surfaceswith polynomial area element and demonstrate our method on a number of examples(constructions of quadrilateral as well as triangular patches