Smooth parametric surfaces and n-sided patches

The theory of 'geometric continuity' within the subject of CAGD is reviewed. In particular, we are concerned with how parametric surface patches for CAGD can be pieced together to form a smooth Ck surface. The theory is applied to the problem of filling an n-sided hole occurring within a smooth rectangular patch complex. A number of solutions to this problem are surveyed

An arbitrary mesh network scheme using rational splines

A C1 surface scheme is described which interpolates points defined on an arbitrary mesh network. The scheme involves the blending of strip ‘functions’ developed from a rational spline method. The rational spline provides interval and point tension weights which can be used to control the shape of the surface scheme

Filling triangular holes by convex combination of surfaces

A surface generation method is presented based on convex combination of surfaces with rational weight functions. The three constituents and the resulting surface are defined over the same triangular domain. The constructed surface matches each component along one of its boundary curves with C0 or C1 continuity depending on the weight functions in the combination. The method can be applied in surface modelling for filling triangular holes

A rational cubic spline with tension

A rational cubic spline curve is described which has tension control parameters for manipulating the shape of the curve. The spline is presented in both interpolatory and rational B-spline forms, and the behaviour of the resulting representations is analysed with respect to variation of the control parameters

Convergence of linear barycentric rational interpolation for analytic functions

Polynomial interpolation to analytic functions can be very accurate, depending on the distribution of the interpolation nodes. However, in equispaced nodes and the like, besides being badly conditioned, these interpolants fail to converge even in exact arithmetic in some cases. Linear barycentric rational interpolation with the weights presented by Floater and Hormann can be viewed as blended polynomial interpolation and often yields better approximation in such cases. This has been proven for differentiable functions and indicated in several experiments for analytic functions. So far, these rational interpolants have been used mainly with a constant parameter usually denoted by d, the degree of the blended polynomials, which leads to small condition numbers but to merely algebraic convergence. With the help of logarithmic potential theory we derive asymptotic convergence results for analytic functions when this parameter varies with the number of nodes. Moreover, we present suggestions on how to choose d in order to observe fast and stable convergence, even in equispaced nodes where stable geometric convergence is provably impossible. We demonstrate our results with several numerical examples