2D piecewise algebraic splines for implicit modeling

Abstract

2D splines are a powerful tool for shape modeling, either parametrically or implicitly. However, compared with regular grid-based tensor-product splines, most of the high-dimensional spline techniques based on nonregular 2D polygons, such as box spline and simplex spline, are generally very expensive to evaluate. Though they have many desirable mathematical properties and have been proved theoretically to be powerful in graphics modeling, they are not a convenient graphics modeling technique in terms of practical implementation. In shape modeling practice, we still lack a simple and practical procedure in creating a set of bivariate spline basis functions from an arbitrarily specified 2D polygonal mesh. Solving this problem is of particular importance in using 2D algebraic splines for implicit modeling, as in this situation underlying implicit equations need to be solved quickly and accurately. In this article, a new type of bivariate spline function is introduced. This newly proposed type of bivariate spline function can be created from any given set of 2D polygons that partitions the 2D plane with any required degree of smoothness. In addition, the spline basis functions created with the proposed procedure are piecewise polynomials and can be described explicitly in analytical form. As a result, they can be evaluated efficiently and accurately. Furthermore, they have all the good properties of conventional 2D tensor-product-based B-spline basis functions, such as non-negativity, partition of unit, and convex-hull property. Apart from their obvious use in designing freeform parametric geometric shapes, the proposed 2D splines have been shown a powerful tool for implicit shape modeling.