One Curve and Surface Basics.- 1.1 Implicit and Parametric Forms.- 1.2 Power Basis Form of a Curve.- 1.3 Bezier Curves.- 1.4 Rational Bezier Curves.- 1.5 Tensor Product Surfaces.- Exercises.- Two B-Spline Basis Functions.- 2.1 Introduction.- 2.2 Definition and Properties of B-spline Basis Functions.- 2.3 Derivatives of B-spline Basis Functions.- 2.4 Further Properties of the Basis Functions.- 2.5 Computational Algorithms.- Exercises.- Three B-spline Curves and Surfaces.- 3.1 Introduction.- 3.2 The Definition and Properties of B-spline Curves.- 3.3 The Derivatives of a B-spline Curve.- 3.4 Definition and Properties of B-spline Surfaces.- 3.5 Derivatives of a B-spline Surface.- Exercises.- Four Rational B-spline Curves and Surfaces.- 4.1 Introduction.- 4.2 Definition and Properties of NURBS Curves.- 4.3 Derivatives of a NURBS Curve.- 4.4 Definition and Properties of NURBS Surfaces.- 4.5 Derivatives of a NURBS Surface.- Exercises.- Five Fundamental Geometric Algorithms.- 5.1 Introduction.- 5.2 Knot Insertion.- 5.3 Knot Refinement.- 5.4 Knot Removal.- 5.5 Degree Elevation.- 5.6 Degree Reduction.- Exercises.- Six Advanced Geometric Algorithms.- 6.1 Point Inversion and Projection for Curves and Surfaces.- 6.2 Surface Tangent Vector Inversion.- 6.3 Transformations and Projections of Curves and Surfaces.- 6.4 Reparameterization of NURBS Curves and Surfaces.- 6.5 Curve and Surface Reversal.- 6.6 Conversion Between B-spline and Piecewise Power Basis Forms.- Exercises.- Seven Conics and Circles.- 7.1 Introduction.- 7.2 Various Forms for Representing Conics.- 7.3 The Quadratic Rational Bezier Arc.- 7.4 Infinite Control Points.- 7.5 Construction of Circles.- 7.6 Construction of Conies.- 7.7 Conic Type Classification and Form Conversion.- 7.8 Higher Order Circles.- Exercises.- Eight Construction of Common Surfaces.- 8.1 Introduction.- 8.2 Bilinear Surfaces.- 8.3 The General Cylinder.- 8.4 The Ruled Surface.- 8.5 The Surface of Revolution.- 8.6 Nonuniform Scaling of Surfaces.- 8.7 A Three-sided Spherical Surface.- Nine Curve and Surface Fitting.- 9.1 Introduction.- 9.2 Global Interpolation.- 9.2.1 Global Curve Interpolation to Point Data.- 9.2.2 Global Curve Interpolation with End Derivatives Specified.- 9.2.3 Cubic Spline Curve Interpolation.- 9.2.4 Global Curve Interpolation with First Derivatives Specified.- 9.2.5 Global Surface Interpolation.- 9.3 Local Interpolation.- 9.3.1 Local Curve Interpolation Preliminaries.- 9.3.2 Local Parabolic Curve Interpolation.- 9.3.3 Local Rational Quadratic Curve Interpolation.- 9.3.4 Local Cubic Curve Interpolation.- 9.3.5 Local Bicubic Surface Interpolation.- 9.4 Global Approximation.- 9.4.1 Least Squares Curve Approximation.- 9.4.2 Weighted and Constrained Least Squares Curve Fitting.- 9.4.3 Least Squares Surface Approximation.- 9.4.4 Approximation to Within a Specified Accuracy.- 9.5 Local Approximation.- 9.5.1 Local Rational Quadratic Curve Approximation.- 9.5.2 Local Nonrational Cubic Curve Approximation.- Exercises.- Ten Advanced Surface Construction Techniques.- 10.1 Introduction.- 10.2 Swung Surfaces.- 10.3 Skinned Surfaces.- 10.4 Swept Surfaces.- 10.5 Interpolation of a Bidirectional Curve Network.- 10.6 Coons Surfaces.- Eleven Shape Modification Tools.- 11.1 Introduction.- 11.2 Control Point Repositioning.- 11.3 Weight Modification.- 11.3.1 Modification of One Curve Weight.- 11.3.2 Modification of Two Neighboring Curve Weights.- 11.3.3 Modification of One Surface Weight.- 11.4 Shape Operators.- 11.4.1 Warping.- 11.4.2 Flattening.- 11.4.3 Bending.- 11.5 Constraint-based Curve and Surface Shaping.- 11.5.1 Constraint-based Curve Modification.- 11.5.2 Constraint-based Surface Modification.- Twelve Standards and Data Exchange.- 12.1 Introduction.- 12.2 Knot Vectors.- 12.3 Nurbs Within the Standards.- 12.3.1 IGES.- 12.3.2 STEP.- 12.3.3 PHIGS.- 12.4 Data Exchange to and from a NURBS System.- Thirteen B-spline Programming Concepts.- 13.1 Introduction.- 13.2 Data Types and Portability.- 13.3 Data Structures.- 13.4 Memory Allocation.- 13.5 Error Control.- 13.6 Utility Routines.- 13.7 Arithmetic Routines.- 13.8 Example Programs.- 13.9 Additional Structures.- 13.10 System Structure.- References.

https://www.springer.com/productFlyer_978-3-540-61545-3.pdf?SGWID=0-0-1297-1491853-0

The NURBS Book

Solid bodies roll and tumble through space. In computer animation, so do cameras. The rotations of these objects are best described using a four coordinate system, quaternions, as is shown in this paper. Of all quaternions, those on the unit sphere are most suitable for animation, but the question of how to construct curves on spheres has not been much explored. This paper gives one answer by presenting a new kind of spline curve, created on a sphere, suitable for smoothly in-betweening (i.e. interpolating) sequences of arbitrary rotations. Both theory and experiment show that the motion generated is smooth and natural, without quirks found in earlier methods.

Animating rotation with quaternion curves

It is shown that mean queue sizes, mean waiting times, and throughputs in closed multiple-chain queuing networks which have product-form solution can be computed recursively without computing product terms and normalization constants. The resulting computational procedures have improved properties (avoidance of numerical problems and, in some cases, fewer operations) compared to previous algorithms. Furthermore, the new algorithms have a physically meaningful interpretation which provides the basis for heuristic extensions that allow the approximate solution of networks with a very large number of closed chains, and which is shown to be asymptotically valid for large chain populations.

Mean-Value Analysis of Closed Multichain Queuing Networks

Basic error estimates for elliptic problems

This paper provides a tutorial and survey of methods for parameterizing surfaces with a view to applications in geometric modelling and computer graphics. We gather various concepts from differential geometry which are relevant to surface mapping and use them to understand the strengths and weaknesses of the many methods for parameterizing piecewise linear surfaces and their relationship to one another.

/pdf/surface-parameterization-a-tutorial-and-survey-3u3e0o47hm.pdf

Surface Parameterization: a Tutorial and Survey

Computer-oriented mesh generators, which serve as pre-processors to finite element programs, have recently been developed by several investigators to alleviate the frustration and to reduce the amount of time involved in the tedious manual subdividing of a complex structure into finite elements. Our purpose here is to describe how the techniques of bivariate ‘blending-function’ interpolation, which were originally developed for, and applied to, geometric problems of computer-aided design and numerically controlled machining of free-form surfaces such as automobile exterior panels, can be adapted and applied to the problems of mesh generation for finite element analyses. We concentrate attention on the problem of curvilinearly co-ordinating simply connected planar domains ℛ by constructing invertible maps of the unit square [0, 1] × [0, 1] onto ℛ. Extensions of the methods described herein to shells in 3-space is straightforward and is illustrated by a practical example taken from the automobile industry. Analogous mesh generators for three-dimensional solids can be developed on the basis of the trivariate ‘blending-function’ formulae found at the end of the second section.

Construction of curvilinear co-ordinate systems and applications to mesh generation

The results contained herein pertain to the problem of determining the equilibrium distribution of customers in closed queuing systems composed of M interconnected stages of service. The number of customers, N, in a closed queuing system is fixed since customers pass repeatedly through the M stages with neither entrances nor exits permitted. At the ith stage there are ri parallel exponential servers all of which have the same mean service rate µi. When service is completed at stage i, a customer proceeds directly to stage j with probability pij. Such closed systems are shown to be stochastically equivalent to open systems in which the number of customers cannot exceed N. The equilibrium equations for the joint probability distribution of customers are solved by a separation of variables technique. In the limit of N â ∞ it is found that the distribution of customers in the system is regulated by the stage or stages with the slowest effective service rate. Asymptotic expressions are given for the marginal distributions of customers in such systems. Then, an asymptotic analysis is carried out for systems with a large number of stages i.e., M â« 1 all of which have comparable effective service rates. Approximate expressions are obtained for the marginal probability distributions. The details of the analysis are illustrated by an example.

Closed Queuing Systems with Exponential Servers

In order to better conform to curved boundaries and material interfaces, curved finite elements have been widely applied in recent years by practicing engineering analysts. The most well known of such elements are the "isoparametric elements". As Zienkiewicz points out in [18, p. 132] there has been a certain parallel between the development of "element types" as used in finite element analyses and the independent development of methods for the mathematical description of general free-form surfaces. One of the purposes of this paper is to show that the relationship between these two areas of recent mathematical activity is indeed quite intimate. In order to establish this relationship, we introduce the notion of a "transfinite element" which, in brief, is an invertible mapping $$\vec T$$ from a square parameter domainJ onto a closed, bounded and simply connected region? in thexy-plane together with a "transfinite" blending-function type interpolant to the dependent variablef defined over?. The "subparametric", "isoparametric" and "superparametric" element types discussed by Zienkiewicz in [18, pp. 137---138] can all be shown to be special cases obtainable by various discretizations of transfinite elements Actual error bounds are derived for a wide class of semi-discretized transfinite elements (with the nature of the mapping $$\vec T$$ :J?? remaining unspecified) as applied to two types of boundary value problems. These bounds for semi-discretized elements are then specialized to obtain bounds for the familiar isoparametric elements. While we consider only two dimensional elements, extensions to higher dimensions is straightforward.

Transfinite element methods: Blending-function interpolation over arbitrary curved element domains

B-spline curves and surfaces

The mth degree Bernstein polynomial approximation to a function ƒ defined over [0, 1] is Σmμ=0 ƒ(μ/m)φμ(s), where the weights φμ(s) are binomial density functions. The Bernstein approximations inherit many of the global characteristics of ƒ, like monotonicity and convexity, and they always are at least as “smooth” as ƒ, where “smooth” refers to the number of undulations, the total variation, and the differentiability class of ƒ. Historically, their relatively slow convergence in the L∞-norm has tended to discourage their use in practical applications. However, in a large class of problems the smoothness of an approximating function is of greater importance than closeness of fit. This is especially true in connection with problems of computer-aided geometric design of curves and surfaces where aesthetic criteria and the intrinsic properties of shape are major considerations. For this latter class of problems, P. Bezier of Renault has successfully exploited the properties of parametric Bernstein polynomials. The purpose of this paper is to analyze the Bezier techniques and to explore various extensions and generalizations. In a sequel, the authors consider the extension of the results contained herein to free-form curve and surface design using polynomial splines. These B-spline methods have several advantages over the techniques described in the present paper.

William J. Gordon

Papers

Construction of curvilinear co-ordinate systems and applications to mesh generation

Closed Queuing Systems with Exponential Servers

Transfinite element methods: Blending-function interpolation over arbitrary curved element domains

B-spline curves and surfaces

Bernstein-Bézier Methods for the Computer-Aided Design of Free-Form Curves and Surfaces