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

This paper is concerned with an analysis of the stochastic behavior of a system of tandem queuing stations in which capacity restrictions are imposed on the queue lengths. The closed, cyclic systems that we consider are shown to be stochastically equivalent to open systems in which the number of customers is a random variable. A concept of duality is introduced on the basis of the simple observation that the sequential movement of customers through the stages generates a counter-sequential motion of "holes." The implications of the duality relation are discussed at some length. The differential-difference equations for the time-dependent stochastic structure of the system are derived, and the remainder of the paper is devoted to the solution of the equilibrium equations for several special systems. First, we analyze completely systems with only two stages. The well-known results for a finite capacity queue appear as a special case. Next, systems are considered for which the number of customers is so small that there is no possibility of blocking. Then, by a duality argument, an analysis is carried out for systems with a very large number of customers in which the blocking effect dominates. Finally, we compare two extreme systems. In the one system there is no blocking and customers may queue at each stage. The other system has unit capacity at each of the M stages so that the distribution of customers is determined solely by the effect of blocking.

Cyclic Queuing Systems with Restricted Length Queues

This paper considers the theoretical development of finite dimensional bivariate blending function spaces and the problem of implementing the Ritz-Galerkin method in these approximation spaces. More specifically, the approximation theoretic methods of polynomial blending function interpolation and approximation developed in [2, 11---13] are extended to the general setting of L-splines, and these methods are then contrasted with familiar tensor product techniques in application of the Ritz-Galerkin method for approximately solving elliptic boundary value problems. The key to the application of blending function spaces in the Ritz-Galerkin method is the development of criteria which enable one to judiciously select from a nondenumerably infinite dimensional linear space of functions, certain finite dimensional subspaces which do not degrade the asymptotically high order approximation precision of the entire space. With these criteria for the selection of subspaces, we are able to derive a virtually unlimited number of new Ritz spaces which offer viable alternatives to the conventional tensor product piecewise polynomial spaces often employed. In fact, we shall see that tensor product spaces themselves are subspaces of blending function spaces; but these subspaces do not preserve the high order precision of the infinite dimensional parent space.

Considerable attention is devoted to the analysis of several specific finite dimensional blending function spaces, solution of the discretized problems, choice of bases, ordering of unknowns, and concrete numerical examples. In addition, we extend these notations to boundary value problems defined on planar regions with curved boundaries.

Ritz-Galerkin approximations in blending function spaces

The draftsman's and related equations

In this paper we describe a new class of locally refined macro finite elements which are especially amenable to the use of substructuring techniques for the efficient solution of the resulting idealization. The tools and guidelines illustrated by the examples of modelling crack tips, point load singularities and singularities at re-entrant corners should enable an analyst to construct other such blended macro elements specifically tailored to his particular class of problems. The use of such substructured macro elements in finite element calculations permits substantial reduction in the manual effort of data preparation and the computational cost of numerical solution.

Substructured macro elements based on locally blended interpolation

1. Summary The two essential ingredients of any boundary value problem are the field equations which describe the physics and a set of relations which specify the geometry of the problem domain. Current finite element methodology is only superficially concerned with the global problem geometry. It focuses attention principally upon the discretized problem in the small, i.e., upon the individual elements and their local properties. As an alternative, this paper examines the possibility of a global treatment of the original problem domain. By exploiting the notion of vector-valued “blending function” interpolation, the authors propose to map the given problem domain one-to-one onto a canonical rectangular parameter domain. This is equivalent to the introduction of a generalized curvilinear coordinate system on the manifold representing the global geometry of the original problem. This method of treating the complex geometries of two and three dimensional boundary value problems has several significant advantages over the ad hoc methods currently employed, since it provides a systematic approach to topologically decomposing a manifold into finite elements and leads inevitably to well-structured sparse matrices.

William J. Gordon

Papers

Cyclic Queuing Systems with Restricted Length Queues

Ritz-Galerkin approximations in blending function spaces

The draftsman's and related equations

Substructured macro elements based on locally blended interpolation

Geometric aspects of the finite element method