Wayne Tiller

Rational B-Splines for Curve and Surface Representation

Nonuniform, rational B-splines, capable of representing both precise quadric primitives and free-form curves and surfaces, offer an efficient mathematical form for geometric modeling systems.

Curve and surface constructions using rational B-splines

Abstract This paper presents the non-uniform rational B-spline approximation form as a unified approach to representing free-form as well as standard analytic curves and surfaces commonly used in CADCAM. The emphasis is on geometric constructions, not on the underlying mathematical theory.

Offsets of Two-Dimensional Profiles

An offset capability for planar curves and profiles in a solid modeler leads to the solution of some practical design problems.

Parametrization for surface fitting in reverse engineering

Abstract Given four boundary curves and a set of random points lying on a surface patch, a method for assigning parameters to these points is presented. The algorithm uses various base surfaces to project the points onto these surfaces to recover the parameters based on the surfaces’ underlying parametrization. Several techniques for speeding up the time consuming projection process are also presented. A thinning method is introduced as well to select a subset of the points that may have been sampled at a much higher rate than necessary. The thinning is based on the geometry of the base surface and relies on a meaningful geometric tolerance.

Computing offsets of NURBS curves and surfaces

This paper presents algorithms for computing offsets of NURBS curves and surfaces. The basic approach consists of four steps: (1) recognition of special curves and surfaces; (2) sampling the offset curve or surface based on bounds on second derivatives; (3) interpolating these points; and (4) removing all unwanted knots using the offset tolerance. The method provides a good handle on error control and results in the fewest number of control points compared to all published work. It also allows one to control the degree and the parametrization of the offset approximation.

Algorithm for approximate nurbs skinning

Abstract An algorithm for approximate skinning through cross-sectional nurbs curves is presented. The method eliminates the problem of dealing with huge amounts of control points obtained during the curve compatability process. It also allows the designer to specify large numbers of cross-sections and approximately fit a smooth surface to these curves to any given tolerance. Depending on the tolerances used, up to 99% of the control points can be eliminated.

Geometry-based triangulation of trimmed NURBS surfaces

Abstract An algorithm for obtaining a piecewise triangular approximation of a trimmed NURBS surface is presented. The algorithm is geometry based, i.e. the surface is subdivided into triangular facets based on its geometric characteristics and not on its parametrization. No assumption is made about the surfaces parametrical representation; it does not have to be continuously differentiable, only Co continuity is assumed. The surface subdivision is performed in model space, however, the triangulation is carried out in parameter space using the parametric vertices of subdivision rectangles. Along with computing the triangulation, the method produces a compact database for browsing in the triangular irregular network, e.g. finding all neighbors of a given triangle.

Knot-removal algorithms for NURBS curves and surfaces

Abstract The paper presents pseudocode algorithms, c-language code, and error analysis for removing knots from rational B-spline curves and surfaces. Efficient and easy-to-use algorithms are presented that, with one call, remove all the removable knots from a B-spline curve or surface.

A menagerie of rational B-spline circles

The article was motivated by J. Blinns column on the many ways to draw a circle (see ibid., vol.7, no.8, p.39-44, 1987). The authors have found several other ways to represent the circle as a nonuniform rational B-spline curve, which they present. Square-based methods, infinite control points, triangle-based methods, general circular arcs and rational cubic circles are some of the methods and types of circle discussed.<<ETX>>

Surface approximation to scanned data

A method to approximate scanned data points with a B-spline surface is presented. The data are assumed to be organized in the form of Qi,j, i=0,…,n; j=0,…,mi, i.e., in a row-wise fashion. The method produces a C(p-1, q-1) continuous surface (p and q are the required degrees) that does not deviate from the data by more than a user-specified tolerance. The parametrization of the surface is not affected negatively by the distribution of the points in each row, and it can be influenced by a user-supplied knot vector.