Computer aided design

technical reportThe report is based on the proposal submitted to the National Science Foundation in September 1981, as part of the Coordinated Experimental Computer Science Research Program. The sections covering the budget and biographical data on the senior research personnel have not been included. Also, the section describing the department facilities at the time of the proposal submission is not included, because it would be only of historical interest

An experimental system for computer aided geometric design

technical reportThe main goal of this proposed level-of-effort project is to extend present capabilities in the area of Computer Aided Geometric Design (CAGD) and to develop custom VLSI support for some special geometric functions

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

Journal ArticleThe relevant theory of discrete 5-sphnes 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

The application of total positivity to computer aided curve and surface design

technical reportOf particular importance in an interactive curve and surface design system is the interface to the user. The mathematical model employed in the system must be sufficiently flexible for interaction between designer and machine to converge to a satisfactory result. The mathematical theory of Total Positivity is combined with the interactive techniques of Bezier and Riesenfeld in developing new methods of shape representation which retain the valuable variation-diminishing and convex hull properties of Bernstein and B-spline approximation, while providing improvements in the interactive interface to the user. Specifically, extending the Bezier notion of using a polygon to describe a smooth curve, methods of assigning a weight to each vertex which will control the amount of local fit to the polygon or polygonal net are provided. Thus, the designer can cause "cusps" and "flats" easily by manipulating the "tension" at each vertex. Further, the generalization from curves to surfaces can be done with rectilinear data or triangular data. Illustrations are provided from an experimental implementation of the newly constructed models as a demonstration of their feasibility and utility in computer aided curve and surface design

Computer aided geometric design

Journal ArticleThis book contains the edited proceedings of the first International Conference on Computer Aided Geometric Design, an important new field that draws on the principles of computer science, mathematics, and geometric design. The list of contributors includes most of the leading researchers in the field in North America and Europe. The papers, containing results that are not available elsewhere, are principally concerned with Coons patches, Bezier curves, and various kinds of splines, with their applications to computer aided geometric design. The book will prove of great value to computer scientists (especially those in computer graphics), numerical analysts, applied mathematicians, mechanical, civil, aeronautical, automotive engineers, and naval architects in academic or industrial positions and government laboratories

A geometric proof for the variation diminishing property of B-spline approximation

AbstractA geometric proof for the variation diminishing property of B-spline approximation is given. The proof is based primarily upon a generalized form of the de Boor-Cox algorithm and the intuitively obvious fact that piecewise linear interpolation is variation diminishing. Previous proofs [4, 8] employed the mathematical methods of total positivity, a machinery which is available only after reading [8]

Discrete B-Splines as an Approach to Computer Aided Geometric Design

technical reportInvestigations have been made of many interesting problems deriving from the applications of discrete splines to the problems of computer aided geometric design. New theory and algorithms have been developed and support mechanisms based on the Oslo algorithm have been incorporated in order to help use the geometry model directly to calculate many geometric attributes needed for the design process including surface rendering and intersections. (Author