Les A. Piegl

On NURBS: a survey

Nonuniform rational B-spline (NURBS) curves and surfaces, which are based on rational and B-splines, are defined. The important characteristics of NURBS that have contributed to their wide acceptance as standard tools for geometry representation and design are summarized. Their application to representing conic sections and commonly used surfaces, designing curves and surfaces, and modifying shapes is examined.<<ETX>>

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.

Delaunay triangulation using a uniform grid

An algorithm for triangulating 2-D data points that is based on a uniform grid structure and a triangulation strategy that builds triangles in a circular fashion is discussed. The triangulation strategy lets the algorithm eliminate points from the internal data structure and decreases the time used to find points to form triangles, given an edge. The algorithm has a tested linear time complexity that significantly improves on that of other methods. As a by-product, the algorithm produces the convex hull of the data set at no extra cost. Two ways to compute the convex hull using the algorithm are presented. The first is based on the edge list and the second is based on the grid structure.<<ETX>>

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.

Delaunay triangulation in three dimensions

A direct algorithm for computing the Delaunay triangulation in three dimensions is presented. The algorithm uses a 3D cell data structure to preprocess the data, a range searching procedure to find...Triangulation in two and higher dimensions began with Dirichlet, Voronoi, Thiessen, and Delaunay. A number of textbooks and papers have extensively covered the properties of triangulations and algorithms for their construction. Most dealt with theoretical aspects of the algorithms and gave upper bounds on their complexity. Here we present a new algorithm and its implementation. Instead of providing a theoretical analysis, we present implementation details, and tests and examples. The algorithm is a generalization of our previous method. >

Ten challenges in computer-aided design

Abstract This short paper presents 10 challenging research areas in the general field of computer-aided design. The research problems come from the authors personal experience, and as such are highly subjective. All findings and opinions are those of the author and do not represent any of the institutions the author is affiliated with.

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.

Symbolic operators for NURBS

Symbolic operators for NURBS curves and surfaces are presented in this paper. The operators are used to compute NURBS entities by performing algebraic operations using NURBS curves and surfaces as variables. Dot and cross products, sum/difference and derivative operators are presented. An application to construct ruled surfaces to rational rail curves is also included.