This paper presents a new triangular B-spline scheme that allows to construct piecewise polynomial surfaces over arbitrary triangulations of the parameter plane. The development of this scheme is based on the study of polar forms [79]. Polar forms have originally been a tool from classical mathematics [88]. They have first been introduced to CAGD by P. de Faget de Casteljau [24, 26] and by L. Ramshaw [65, 66, 67]. The author has subsequently extended this theory to more general surface representations and has used polar forms for the development of B-patches [77, 76, 84]. Further extensions to simplex splines have finally led to the new triangular B-spline scheme described in this paper [18, 38, 39, 49, 78, 81]. While previous approaches to the construction of B-spline like surfaces over irregular domains have been based on subdivision, interpolation, and on the use of multisided patches the new scheme is based on blending functions and control points. The resulting surfaces are defined as linear combinations of the blending functions and are parametric piecewise polynomials over an arbitrary triangulation of the parameter plane, whose shape is determined by their control points. The paper is organized as follows: After a brief introduction to bivariate polynomials and polar forms (Section 2) we discuss triangular B'ezier patches (Section 3) and introduce a new surface representation for bivariate polynomials, the B-patch (Section 4). In connection with simplex splines (Section 5) this finally leads to the construction of the new triangular B-spline scheme (Section 6). We hope that our presentation will provide a thorough unterstanding of the polar form of a polynomial surface, of triangular B'ezier patches, and of some of the main issues that are involved in the constr...

. The simplex spline recurrence is symmetric. This leads to a simple redevelopment of simplex splines from scratch and facilitates the construction of the DMS simplex spline space. The subdivision property for DMS splines is established and explicit formulas for computing the coefficients are given. x1. Introduction A new simplex spline space has recently been developed by Dahmen, Micchelli, and Seidel in [5]. The development of this space, which we refer to as DMS simplex spline space, has been based on the combination of simplex splines with polar forms, and thus the DMS space exhibits several symmetry properties that facilitate computation and have been helpful for the development of new results [13]. Up to now, these symmetry properties have generally been perceived as properties of the DMS space, not as properties of the individual basis functions. In this paper we correct this common misunderstanding and show that the standard simplex spline recurrence [3,8] is symmetric. This l...

Abstract not found

We investigate the construction of local quasi-interpolation schemes for a family of bivariate spline functions with smoothness r ≥ 1 and polynomial degree 3r−1. These splines are defined on triangulations with Powell-Sabin refinement, and they can be represented in terms of locally supported basis functions which form a convex partition of unity. Using the blossoming technique, we first derive a Marsden’s identity representing polynomials of degree 3r − 1 in such a spline form. Then we present a simple approach to construct various families of smooth quasi-interpolation schemes involving values and/or derivatives of a given function.