Abstract
In this paper, the problem of determining the dimension of the space Sn1(△), n ≥ 3 of bivariate C1 splines of degree ≤ n over a triangulation △ is considered. The piecewise polynomials are represented as blossoms, and the smoothness conditions are written as a system of linear equations. The rank of the system matrix is analysed by repeatedly reducing small subtriangulations (cells) at the boundary of a triangulation. It is shown that the dimension of the bivariate spline space Sn1(△), n ≥ 3 is equal to Schumaker’s lower bound for a large class of triangulations.
Highlights
In the last 40 years the problem of determining the dimension of the bivariate spline space has received a considerable attention
Since it is possible to reduce most of the cases by methods for boundary cells, we focus the study to the cases with collinearities
= 6545 minors of size 32 and a careful simplification of nonzero expressions reveals that all contain the same linear expression in α1 and β1. This results in a unique condition on the positions of the edges of the cell, where the matrix considered, M (△1, △), is not of full rank
Summary
In the last 40 years the problem of determining the dimension of the bivariate spline space has received a considerable attention. The main obstacle in the study of the dimension problem is the fact that the dimension depends on the topology of the triangulation △ and on its geometry It has been conjectured (see [14]) that the dimension is equal to Schumaker’s lower bound for n ≥ 2r + 1 and that the dimension jump occurs only for singular vertices. The idea is to study the smoothness conditions between polynomial patches, written as their blossoms ([10]) This is a dual approach to the well known classical approach (see [13], e.g.) and brings a new insight to the dimension problem. Sufficient conditions for an inductive approach for determining whether the dimension of Sn1(△), n ≥ 3, is equal to Schumaker’s lower bound for a large class of triangulations △ are obtained.
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