Tom Lyche

Discrete B-splines and subdivision techniques in computer-aided geometric design and computer graphics

Abstract The relevant theory of discrete B-splines with associated new algorithms is extended to provide a framework for understanding and implementing general subdivision schemes for nonuniform B-splines. The new derived polygon corresponding to an arbitrary refinement of the knot vector for an existing B-spline curve, including multiplicities, is shown to be formed by successive evaluations of the discrete B-spline defined by the original vertices, the original knot vector, and the new refined knot vector. Existing subdivision algorithms can be seen as proper special cases. General subdivision has widespread applications in computer-aided geometric design, computer graphics, and numerical analysis. The new algorithms resulting from the new theory lead to a unification of the display model, the analysis model, and other needed models into a single geometric model from which other necessary models are easily derived. New sample algorithms for interference calculation, contouring, surface rendering, and other important calculations are presented.

Mathematical methods for curves and surfaces

We will deal with the translation surfaces which are the shapes generated by translating one curve along another one. We focus on the geometry of translation surfaces generated by two algebraic curves in space and study their properties, especially those useful for geometric modelling purposes. It is a classical result that each minimal surface may be obtained as a translation surface generated by an isotropic curve and its complex conjugate. Thus, we can study the minimal surfaces as special instances of translation surfaces. All the results about translation surfaces will be directly applied also to minimal surfaces. Finally, we present a construction of rational isotropic curves with a prescribed tangent field which leads to the description of all rational minimal surfaces. A close relation to surfaces with Pythagorean normals will be also discussed.

T-spline simplification and local refinement

A typical NURBS surface model has a large percentage of superfluous control points that significantly interfere with the design process. This paper presents an algorithm for eliminating such superfluous control points, producing a T-spline. The algorithm can remove substantially more control points than competing methods such as B-spline wavelet decomposition. The paper also presents a new T-spline local refinement algorithm and answers two fundamental open questions on T-spline theory.

Local Spline Approximation Methods

Abstract : The construction of explicit polynomial spline approximation operators for real-valued functions defined on intervals or on reasonably behaved sets in higher dimensions is studied. The operators take the form Qf = the summation of lambda sub i f N sub i, where the N sub i are B-splines and the lambda sub i are appropriate linear functionals. Explicit operators are found which apply to wide classes of functions including continuous or integrable functions. Moreover, the operators are local and approximate smooth functions with an accuracy comparable to best spline approximation. They can be constructed without matrix inversion, local as well as global error bounds are obtained, and some of the error bounds are free of mesh restrictions. (Author)

Polynomial splines over locally refined box-partitions

We address progressive local refinement of splines defined on axes parallel box-partitions and corresponding box-meshes in any space dimension. The refinement is specified by a sequence of mesh-rectangles (axes parallel hyperrectangles) in the mesh defining the spline spaces. In the 2-variate case a mesh-rectangle is a knotline segment. When starting from a tensor-mesh this refinement process builds what we denote an LR-mesh, a special instance of a box-mesh. On the LR-mesh we obtain a collection of hierarchically scaled B-splines, denoted LR B-splines, that forms a nonnegative partition of unity and spans the complete piecewise polynomial space on the mesh when the mesh construction follows certain simple rules. The dimensionality of the spline space can be determined using some recent dimension formulas.

Knot removal for parametric B-spline curves and surfaces

This paper is the third in a sequence of papers in which a knot removal strategy for splines, based on certain discrete norms, is developed. In the first paper, approximation methods defined as best approximations in these norms were discussed, while in the second paper a knot removal strategy for spline functions was developed. In this paper the knot removal strategy is extended to parametric spline curves and tensor product surfaces. The method has been implemented and thoroughly tested on a computer. We illustrate with several examples and applications.

A stable recurrence relation for trigonometric B-splines

Abstract In this paper we give results that lead to stable algorithms for computing with trigonometric splines. In particular we show that certain trigonometric B -splines satisfy a recurrence relation similar to the one for polynomial splines. We also show how these trigonometric B -splines can be differentiated, and a trigonometric version of Marsdens identity is given. The results are obtained by studying certain trigonometric divided differences.

On a class of weak Tchebycheff systems

AbstractIn this paper we study the approximation power, the existence of a normalized B-basis and the structure of a degree-raising process for spaces of the form requiring suitable assumptions on the functions u and v. The results about degree raising are detailed for special spaces of this form which have been recently introduced in the area of CAGD.

Algorithms for degree-raising of splines

Stable and efficient algorithms for degree-raising of curves (or surfaces) represented as arbitrary B-splines are presented as a application of the solution to the theoretical problem of rewriting a curve written as a linear combination of mth order B-splines as a linear combination of (m + 1)st order B-splines with a minimal number of knot insertions. This approach can be used to introduce additional degrees of freedom to a curve (or surface), a problem which naturally arises in certain circumstances in constructing mathematical models for computer-aided geometric design.

Computation of Smoothing and Interpolating Natural Splines via Local Bases

It is shown how smoothing splines can be represented in terms of a local basis, and that the coefficients can be obtained by solution of a banded linear system. Recursion relations are developed which permit rapid and accurate calculation of the necessary basis elements.