Knot Insertion and Deletion Algorithms for B-Spline Curves and Surfaces (Geometric Design Publications)

Knot Insertion and Deletion Algorithms for B-Spline Curves and Surfaces

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Knot Insertion and Deletion Algorithms for B-Spline Curves and Surfaces

Edited by Ronald N. Goldman

Rice University Tom Lyche University of Oslo

Philadelphia

Society for Industrial and Applied Mathematics

Geometric Design Publications Editor Gerald E. Farin Arizona State University

Farin, Gerald E., editor, Geometric Modeling: Algorithms and New Trends (1987) Farin, Gerald E., editor, NURBS for Curve and Surface Design (1991) Barnhill, Robert E., editor, Geometry Processing for Design and Manufacturing (1992) Hagen, Hans, editor, Curve and Surface Design (1992) Hagen, Hans, editor, Topics in Surface Modeling (1992) Goldman, Ronald N., and Lyche, Tom, editors, Knot Insertion and Deletion Algorithms for B-Spline Curves and Surfaces (1993)

Cover art reprinted with permission from An Envelope Approach to a Sketching Editorfor Hierarchical Free-form Curve Design and Modification, Chapter 7 of this book.

Library of Congress Cataloging-in-Publication Data Knot insertion and deletion algorithms for B-spline curves and surfaces / edited by Ronald N. Goldman and Tom Lyche. p. cm. — (Geometric design publications) Includes bibliographical references and index. ISBN 0-89871-306-4 1. Geometrical drawing—Data processing. 2. Spline theory. 3. Knot insertion and deletion algorithms. 4. Blossoming (Mathematics) I. Goldman, Ron, 1952 - . II. Lyche, Tom. III. Series. QA497.K6 1993 511'.42—dc20

92-42080

Sponsored by SIAM Activity Group on Geometric Design. All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the Publisher. For information, write the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688. 1993 by the Society for Industrial and Applied Mathematics. is a registered trademark.

Preface

Knot insertion and deletion algorithms are the two fundamental procedures needed for understanding, analyzing, and rendering B-spline curves and surfaces. This approach to splines, however, is not traditional; originally divided differences were used to develop almost all of the theory of univariate B-splines [4]. This point of view began to change in the mid-1970s with the publication of the Cox-de Boor-Mansfield recurrence [3]. Many computer scientists and engineers not familiar with divided differences wanted to apply splines to computer graphics and computer aided design; this recurrence provided a much easier entry for them into the subject. Since this recurrence is numerically stable, it lends itself very naturally to computer programming, and therefore, as well, to computer aided design. Knot insertion algorithms were originally developed by the computer aided design community. These procedures grew out of a need for algorithms to render and intersect B-spline curves and surfaces. Divide and conquer is a standard paradigm in computer science, and subdivision is its incarnation in the study of free-form curves and surfaces. Subdivision had been applied successfully in the analysis of Bezier curves and surfaces [12]. A similar technique was clearly desirable for B-spline curves and surfaces, and this motivated the research for knot insertion procedures. Since Bezier curves and surfaces are just a special case of B-spline curves and surfaces with multiple knots, general knot insertion procedures would allow users to transform Bspline curves and surfaces into piecewise Bezier form and then to apply standard Bezier subdivision techniques. The first knot insertion procedures discovered was Chaiken's algorithm [8], [18] for uniform quadratic B-splines; this algorithm was later generalized by Lane and Riesenfeld [12] to uniform B-splines of arbitrary degree. Recently, Goldman and Warren [11] discovered an extension of this algorithm to B-splines of arbitrary degree with knots in geometric or affine progression. Sablonniere's algorithm [19] for converting from B-spline to piecewise Bezier form can also be considered a special kind of knot insertion procedure. General knot insertion algorithms for B-spline curves and surfaces of arbitrary degree with arbitrary knots were originally developed by two groups v

vi

Preface

simultaneously: in 1980 Boehm [2] discovered his knot insertion algorithm, and in the same year Cohen, Lyche, and Riesenfeld [9] collaborated on the Oslo algorithm. In Boehm's algorithm, the basic step is inserting a single knot; in the Oslo algorithm, the fundamental step is computing a single new control point. Both of these algorithms are grounded in the basic recurrence for the Bsplines, though the original proofs still made extensive use of divided difference techniques. As a bonus, Lane and Riesenfeld [13] showed how to use knot insertion to provide a simple and elegant proof of the variation diminishing property for B-spline curves. Thus, knot insertion proved to be not only a very practical tool for rendering and intersecting curves and surfaces, but also a powerful technique for theoretical analysis. A rivalry ensued concerning the efficiency of these two basic knot insertion algorithms. Soon the speed of the Oslo algorithm was improved by Morken and Lyche [14] and later another fast variant was provided by Goldman [10]. Two further developments drew attention away from divided differences and towards further reliance on the B-spline recurrence. De Boor and Hollig [6] published a paper on B-splines Without Divided Differences, whose theme was that the entire theory of univariate B-splines could be developed using only the de Boor-Fix dual functionals [5] together with the B-spline recurrence. This refocused attention on the dual functionals and presaged a development which soon became known as the blossoming approach of Rams haw [15]-[17] or the polar form approach of de Casteljau [7]. Blossoming or polar forms is an approach to dual functionals which is more directly tied to the B-spline recurrence than the de boor-Fix formula. It soon became clear that the B-spline recurrence, blossoming, and knot insertion were all intimately related. Seidel [21] rederived Boehm's knot insertion algorithm and Goldman [10] developed a variant of the Oslo algorithm using blossoming. Knot insertion adds knots in; knot deletion takes them out. Knot insertion is exact; knot deletion approximate. Knot insertion tells us how to represent exactly a given curve or surface with additional knots; knot deletion tells us how well we can approximate a given curve or surface with fewer knots. Both are important theoretical tools with many useful applications. Knot insertion introduces more data which can be used to provide piecewise linear approximations for rendering and intersection algorithms; knot deletion compresses the data, providing more compact representations, which may lead, in turn, to faster algorithms. In the fall of 1989, a SIAM conference on Computer Aided Geometric Design was held in Tempe, Arizona. At this time, Phil Barry and Ron Goldman were reexamining algorithms for B-spline curves from a blossoming point of view, and Knut Morken and Tom Lyche were deeply into their ongoing study of knot insertion and deletion algorithms for B-spline curves and surfaces. Considering these algorithms to be the basic procedures for univariate Bsplines, we collectively decided to combine our efforts to produce a single book covering as many aspects as we could on Knot Insertion and Deletion

Preface

vii

Algorithms for B-spline Curves and Surfaces. The purpose of this book is to provide new material on knot insertion and deletion as well as a fresh look at old results. To include not only theoretical material, but also a very concrete practical application, we asked Elaine Cohen to contribute a chapter on applications to curve sketching in computer aided geometric design. Goldman and Barry came to the conclusion that many well-known algorithms for B-spline curves and surfaces, and for their local polynomial generalizations called progressive curves, were simply variations of knot insertion procedures. These algorithms include recursive evaluation, differentiation, and integration. They also discovered some new variants of knot insertion procedures. These ideas are investigated in the first four chapters of this book. "Algorithms for Progressive Curves" extends B-spline and blossoming techniques to three important bases—the monomial, power, and Newton dual bases—and shows how to understand evaluation, differentiation, and integration from the knot insertion and blossoming points of view. "Factored Knot Insertion" explains how to apply the Newton dual basis, introduced in the previous chapter, to provide an alternative approach to knot insertion that can sometimes speed up the process. This factored approach bases knot insertion on recursive differentiation rather than recursive evaluation and is closely related to a similar approach recently developed independently by Sankar, Silbermann, and Ferrarri [20]. "Knot Insertion Algorithms" is our attempt to explain, compare, and contrast many of the possible alternative approaches to knot insertion and to unify these disparate techniques using the common perspective of blossoming. Thus, the unifying themes of these three chapters is the examination of the various aspects of knot insertion from the modern blossoming point of view. To keep the book self-contained, "An Introduction to Blossoming " is provided first to initiate readers into this central subject. Morken and Lyche approached some of these same problems with different motivation and different techniques. Given two spline spaces with distinct knot sequences, they wanted to know how to best represent a curve in one space by a curve in the other space. If one knot sequence is a subset of the other, then this is just the knot insertion problem; if not, then this problem has aspects of both knot insertion and knot deletion. In their first chapter in this volume, Morken and Lyche discuss the general problem of "Conversion Between BSpline Bases Using the Generalized Oslo Algorithm." Their main tools are the discrete B-Splines and the de Boor-Fix dual functionals. This chapter provides a different perspective from the chapters by Barry and Goldman, which focus on the change of basis problem from a blossoming point of view. Numerical stability issues are critical for implementation. The second chapter by Morken and Lyche focuses on these issues for knot insertion and deletion algorithms. In particular, they ask the question: "How Much Can the Size of the B-Spline Coefficients Be Reduced by Inserting a Knot?" Their answer has implications for the stability of both knot insertion and deletion algorithms for B-spline curves and surfaces.

viii.

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We close this volume with yet another perspective on knot insertion and knot deletion. In their chapter "An Envelope Approach to a Sketching Editor for Hierarchical Free-form Curve Design and Modification," Mike Banks, Elaine Cohen, and Tim Mueller discuss techniques for creating and modifying freeform curves by mimicking the natural actions used in ordinary freehand sketching. Here knot insertion and deletion are used in a different sense to modify the knot vector of a B-spline curve in order to conform more closely to the intent of the designer. Knot insertion and deletion are rich themes that we feel lie at the heart of much of the theory and application of univariate B-splines. This volume is an attempt to look at these themes from various points of view, but it is far from the last word on this subject. Recently, attempts have been made by several people to extend knot insertion algorithms to geometrically continuous splines [1], [22], and the connection between knot insertion, dual functionals, and blossoming is beginning to become clear in this setting. Extensions of knot insertion algorithms, dual functionals, and blossoming techniques to multivariate splines are now under investigation, but at this time the subject is not well understood. Perhaps this topic will be the subject of some future volume. It has taken us rather longer than we had hoped to produce this book. Although we started writing these papers in the autumn of 1989, we did not finish our work until almost the very end of 1991. Nevertheless, we think that these ideas are still timely, and we trust that these topics are of ongoing interest to a large segment of the community devoted to research and development of splines for use in computer graphics and computer aided design. We can only hope that the readers learn as much from reading these papers as the authors did from writing them. We are grateful to the following people for refereeing: Elaine Cohen Tony DeRose Tor Dokken Knut Morken Hartmut Prautzsch Kyrre Strom Phil Barry Hans-Peter Seidel Ron Goldman December 31, 1991

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ix

References [1] P. J. Barry, R. N. Goldman, and C. A. Micchelli, Knot insertion algorithms for geometrically continuous splines, 1991, in preparation. [2] W. Boehm, Inserting new knots into E-spline curves, Comput. Aided Design, 12 (1980), pp. 199-201. [3] C. de Boor, On calculating with E-splines, J. Approx. Theory, 6 (1972), pp. 50-62. [4] , A Practical Guide to Splines, Springer-Verlag, New York, 1972. [5] C. de Boor and G. Fix, Spline approximation by quasi- interpolants, J. Approx. Theory, 8 (1973), pp. 19-45. [6] C. de Boor and K. Hollig, E-splines without divided differences, in Geometric Modeling: Algorithms and New Trends, G. Farin, ed., Society for Industrial and Applied Mathematics, Philadelphia, PA, 1987, pp. 21-27. [7] P. de Casteljau, Formes a Poles, Hermes, Paris, 1985. [8] G. Chaiken, An algorithm for high speed curve generation, Comput. Graphics and Image Processing, 3 (1974), pp. 346-349. [9] E. Cohen, T. Lyche, and R. Riesenfeld, Discrete E-splines and subdivision techniques in computer-aided geometric design and computer graphics, Comput. Graphics and Image Processing, 14 (1980), pp. 87-111. [10] R. N. Goldman, Blossoming and knot insertion algorithms for B-spline curves, Comput. Aided Geometric Design, 7 (1990), pp. 69-81. [11] R. N. Goldman and J. Warren, An extension of Chaiken's algorithm to E-spline curves with knots in geometric progression, 1991, submitted for publication. [12] J. Lane and R. Riesenfeld, A theoretical development of the generation and display of piecewise polynomial surfaces, IEEE Trans. Pattern Analysis and Machine Intelligence, 2 (1980), pp. 35-46. [13] , A geometric proof for the variation diminishing proprerty of E-spline approximation, J. Approx. Theory, 37 (1983), pp. 1-4. [14] T. Lyche and K. Morken, Making the Oslo algorithm more efficient, SIAM J. Numer. Anal., 23 (1986), pp. 663-675. [15] L. Ramshaw, Blossoming: A Connect-the-Dots Approach to Splines, Tech. Report 19, Digital Equipment Systems Research Center, Palo Alto, CA, 1987. [16] , Beziers and E-splines as multiaffine maps, in Theoretical Foundations of Computer Graphics and CAD, R. A. Earnshaw, ed., NATO ASI Series F, 40, Springer-Verlag, New York, 1988, pp. 757-776. [17] , Blossoms are polar forms, Comput. Aided Geom. Design, 6 (1989), pp. 323358. [18] R. Riesenfeld, On Chaiken's algorithm, Comput. Graphics and Image Processing, 4 (1975), pp. 304-310. [19] P. Sablonniere, Spline and Bezier polygons associated with a polynomial spline curve, Comput. Aided Design, 6 (1978), pp. 257-261. [20] P. V. Sankar, M. J. Silbermann, and L. A. Ferrari, Curve and surface generation and refinejnent based on a high speed derivative algorithm, submitted for publication. [21] H.-P. Seidel, Knot insertion from a blossoming point of view, Comput. Aided Geom. Design, 5 (1988), pp. 81-86. [22] , Geometric constructions and knot insertion for /3-splines of arbitrary degree, submitted for publication.

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List of Contributors

Michael J. Banks, Hewlett-Packard Fort Collins, Mail Stop 74, 3404 E. Harmony Rd., Fort Collins, CO 80525 Phillip J. Barry, Department of Computer Science, University of Minnesota, Minneapolis, MN 55455 Elaine Cohen, Department of Computer Science, University of Utah, Salt Lake City, UT 84112 Ronald N. Goldman, Department of Computer Science, Rice University, Houston, TX 77251 Tom Lyche, Institutt for Informatikk, University of Oslo, Blindern 0316, Oslo 3, Norway Knut M0rken, Institutt for Informatikk, University of Oslo, Blindern 0316, Oslo 3, Norway Timothy I. Mueller, Department of Computer Science, University of Utah, Salt Lake City, UT84112 Kyrre Str0m, Institutt for Infonnatikk, University of Oslo, Blindern 0316, Oslo 3, Norway

XI

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Contents

1

Chapter 1 An Introduction to Blossoming Phillip J. Barry

11

Chapter 2 Algorithms for Progressive Curves: Extending B-Spline and Blossoming Techniques to the Monomial, Power, and Newton Dual Bases Phillip J. Barry and Ronald N. Goldman

65

Chapter 3 Factored Knot Insertion Phillip J. Barry and Ronald N. Goldman

89

Chapter 4 Knot Insertion Algorithms Phillip J. Barry and Ronald N. Goldman

135

Chapter 5 Conversion Between B-Spline Bases Using the Generalized Oslo Algorithm Tom Lyche, Knut M0rken, and Kyrre Str0m

155

Chapter 6 How Much Can the Size of the B-Spline Coefficients Be Reduced by Inserting One Knot? Tom Lyche and Knut M0rken

179

Chapter 7 An Envelope Approach to a Sketching Editor for Hierarchical Free-form Curve Design and Modification Michael J. Banks, Elaine Cohen, and Timothy 1. Mueller

195

Index

Xlll

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CHAPTER

J_ An Introduction to Blossoming Phillip J. Barry

1.1.

Introduction

B-spline curves and surfaces are not only practically useful, but also theoretically elegant. One reason for this is the many diverse approaches to B-splines. B-spline curves and surfaces have been defined and studied through divided differences, convolutions, recursion formulas, dual functionals, and many other approaches. One recent approach is blossoming (also known as polar forms). While all B-spline approaches have advantages, blossoming is a particularly fruitful approach to many aspects of B-splines, including knot insertion. The next three chapters in this book rely heavily on the blossoming approach to B-spline curves and surfaces. Two of these chapters use blossoming in presenting knot insertion algorithms, the other extends blossoming and knot insertion techniques to a class of curves closely related to B-spline curves. Each of these chapters is intended to be self-contained, and therefore contains some minimal background material on blossoming. However, because of the heavy reliance of these chapters on blossoming, readers unfamiliar with this approach may desire a more detailed background. This chapter, then, is a brief introduction to blossoming. More specifically, this chapter has three purposes: to define what blossoming is, to give examples of how it is used, and to motivate why it is useful. The remainder of this chapter is structured as follows: first, in §1.2 we provide a few remarks on why blossoming is a valuable approach to B-spline curves. We next define the blossom in §1.3, and in §1.4 we give some simple examples illustrating the use of blossoming in discussing B-spline algorithms. Section 1.5 covers a variant of the blossom which is particularly useful when working with derivatives, and §1.6 contains concluding remarks. This chapter is intended to be an introduction, rather than a comprehensive discussion, of blossoming. Readers interested in further details are referred to [5], [7]-[9]. In particular, this chapter relies heavily on [7]. Readers interested in still other research involving blossoming are referred to [2], [7], [10], [11], where blossoming is extended to surfaces and to geometrically continuous splines. 1

2 1.2.

Knot Insertion and Deletion Algorithms Motivation

Blossoming is closely related to other B-spline approaches—most particularly to knot insertion algorithms and to the de Boor-Fix dual functionals [1], [4]; however, in certain contexts it is more efficient or elegant. One reason for this is the relationship between a B-spline curve and its blossom. As we shall see, the blossom is a single construct which encompasses the curve itself, its derivatives, its control vertices, and any new control vertices that may arise from knot insertion. Moreover, the properties of the blossom make clear the relationship between each of these entities. These two observation enable blossoming to provide straightforward explanations of many B-spline algorithms: both the input to, and output from these algorithms is simply expressed in terms of the blossom, and explaining the algorithms only requires using the properties of the blossom to relate the input and output. Below we will give a few specific examples of using blossoming to explain B-spline algorithms, but first we define the blossom. 1.3. Definition of the Blossom

The core idea in blossoming is that there is a one-to-one correspondence between polynomials of degree at most n, and a certain class of polynomials of n variables. While going from a function of a single parameter to a function of several variables may seem like a step in the wrong direction, the properties of these multivariate functions will actually make discussion of many aspects of B-spline curves less, rather than more, complicated. Let G(i) be a polynomial curve of degree n or less. The blossom BG(UI, ..., Un) of the polynomial G(i) is the unique multivariate polynomial with the following properties: • multiaffine: for all indices i and all real numbers c

symmetry: for any permutation TT of {1,2,..., n}

dicigonal: BQ reduces to G when evaluated on its diagonal; i.e,

Before proceeding further, consider a simple example. Let G(i) = 5t3 — 6t2 + 8t — 31. The blossom is a function of three variables ^1,^2,^3. The

Introduction to Blossoming

3

multiaffine property is equivalent to no ul appearing to more than the first power [7]. We therefore get

for some values at,J}k- Because BQ is symmetric we must have «i,i,o = a i,o,i = a o,i,i aild a i,o,o = fto,i,o — ^0,0,1- Using the diagonal property and equating the coefficients of G with the remaining unknown coefficients of BQ yields

We next extend the blossom to piecewise polynomials. The blossom of a piecewise polynomial will also be piecewise, with each polynomial segment of the curve having a corresponding blossom segment. Moreover, if the polynomial segments of the curve join smoothly, as they will in the case of B-spline curves, then the blossom will have a related property. Let G(t) be a B-spline curve with knots {tz} and control vertices {V;}, and suppose tt has multiplicity // in the knot vector. Denote by #G,L the segment of the blossom corresponding to the interval which ends at £ z , and by #G,R the segment of the blossom corresponding to the interval immediately to the right of tt. Then

for all choices of t t i , . . ., w ?l _ M [7], The multiaffine, symmetry, and diagonal properties serve to characterize the blossom, and to show the close connection, given by the diagonal property, between the curve and the blossom; however, they do not explicitly link the blossom to the curve's B-spline representation. This is done by a fourth property, called the dual functional property: dual functional: any control vertex

where the blossom segment used is the segment corresponding to any of the intervals [i,,i z + 1 ),.. ., [tl+n,tl+n+] )• In other words, we can obtain a control vertex by taking any curve segment affected by that vertex, and evaluating the corresponding blossom segment at the appropriate n consecutive knots. The dual functional property applies not only to the original knot vector, but also to any refinement of that knot vector. If {u{} is a refinement of |t t ), then the control vertices {Wl} expressing the curve G(t] over the refined knot vector are

4

Knot Insertion and Deletion Algorithms

where the blossom segment used is any segment corresponding to an interval [ipij+i) such that ( t j , t j + i ) has a nonempty intersection with (w z -, w z - +n+ i). That is, the new control point W{ must affect the curve over some subsegment of (ij,£j_|_i). The dual functional property is crucial in our use of blossoming in the next few chapters, since it expresses both the new and the old control vertices in terms of the blossom. In fact, there is a natural restriction of the arguments for the blossom so that the range of the blossom is precisely the set of new control points that may arise through knot insertion. In other words, with this restriction on the arguments, every point in the range of the blossom will be a point that, if we were to insert the right knots, would be a new control point of the curve. Consider the blossom segment corresponding to the knot interval [i;,i;+i). We restrict the arguments to this blossom segment to arguments u\,..., un which (1) are consecutive values in some valid refinement of the original knot vector, and, (2a) if they contain a value less than t{ must have ti appearing in the argument list at least as often as it appears in the original knot vector, and, (2b) if they contain a value greater than tl+i must have tl+\ appearing in the argument list at least as often as it appears in the original knot vector. (See p. 86 of [7] for further details). Restriction (1) ensures that some blossom segment, when evaluated at the specific arguments, will yield a potential new control vertex, and restrictions (2a) and (2b) ensure that the blossom segment corresponding to [tf;,^+i) is one segment that does so. One added benefit of this restriction is that we do not have to keep specifying which blossom segment is under consideration. This is because a given argument will only be valid for certain segments, and each of these segments evaluated at the arguments will yield the same value, by (1.1). 1.4. Uses of the Blossom In this section we give a few examples of using the blossom to explain algorithms. While the examples given in this section are elementary, the techniques employed here are similar to the techniques used when explaining more involved algorithms in later chapters. Let us consider first using the de Casteljau algorithm (see, e.g., [6]) to subdivide cubic Bezier curves. Bezier curves are known to be a special, polynomial type of B-spline curve with the knot vector given by n knots at 0 and n knots at 1. (Since each degree n B-spline is actually defined by '2n + 2 knots, Bezier curves are sometimes presented as having a knot vector with n-\-1 knots at 0 and another n + 1 at 1; however, because Bezier curves are defined only over a single interval, only n knots at each value are needed.) By the dual functional property the control vertices Pi for the Bezier representation of the curve are given in terms of the blossom by

where 0 appears as an argument n — i times and 1 appears i times.

Introduction to Blossoming

5

FIG. 1.1. The de Casteljau algorithm for a (cubic) Bezier curve. The. values at the. base of the triangle, are original control vertices, those of the left side are control vertices for the curve segment over [0,a], and those on the right for the curve segment over [a, 1]. Each computed point is found by taking the two points immediately below it, multiplying them by the respective labels on the edges incident to the point being found, and summing the two resulting products.

For the sake of simplicity, consider the cubic case. We begin with the control vertices # G (0,0,0), £ G (0,0,1)> #c;(0,1, l),and £ G (1,1,1). From these we wish to get new control vertices representing the curve over segments [0,a] and [a, 1], respectively, as Bezier curves. The new representation can be viewed as a B-spline curve with knot vector consisting of three values, 0,a, 1, each having multiplicity n. The new control vertices are therefore # G (0,0,0), £ G (0,0,a), £ G (0,a,a), # G (a,a,a), £ G ( a , a , l ) , BG(a, 1,1), and B G (1,1,1). The question is now how to find all the new control vertices from the old control vertices. Up to this point we have not made extensive use of the multiaffine property. One power of this property is that it allows us to derive new blossom values from old ones. For example, consider finding the value 5 G (0,0,a). Note

expressing one of the new control vertices in terms of two old control vertices. In fact, if we take all consecutive pairs of the old control vertices, and combine them in the ratio 1 — a : a, then we get the values £? G (0,0,a), # G (0,a, 1), and Ba(a, 1,1). Note the first and last of these are desired new points. If we then take these three points and similarly combine them in the ratio 1 — a : a, we get the new points £? G (0,a,a) and # G (a,a, 1), and combining these yields f? G (a,a,a). This process is illustrated diagrammatically in Fig. 1.1 and graphically in Fig. 1.2. Referring to Fig. 1.1, observe that the edge labels in that figure are redundant—they follow from the labels at the nodes and the multiaffine property. Also, note that the process just described

6

Knot Insertion and Deletion Algorithms

FlG. 1.2. Applying the de Casteljau algorithm to a cubic Bezier curve with a = .6. To conserve space, the points are labelled using only the blossom arguments.

actually evaluates the curve by the standard de Casteljau algorithm. Thus, this algorithm is an example of how the blossom unites the original control vertices, the curve itself, and the new control vertices. As a second example, consider the de Boor algorithm [3] for evaluating a (cubic) B-spline curve at a value a £ [tq,tq+i]. The input to the algorithm, expressed in terms of the blossom, is the control vertices Bcwk) is called affine if w^ = 1 and vector-valued if w^ = 0. (These terms are adopted because the values (i, 1) correspond to points in a one-dimensional affine space and (£,0) correspond to vectors in the same space.) Affine parameters and affine knots are what appear in the standard version of the de Boor algorithm. In general, the homogeneous de Boor algorithm is somewhat more complicated than the standard de Boor algorithm, and little is gained by considering the most general case. However, when some of the knots are affine and the remainder are vector-valued, certain terms vanish and there can be a good deal of simplification in the algorithm. We shall return to this observation in more detail in §2.4 where we shall look at some particularly interesting examples. 2.3.2. The Multilinear Blossom. Given a univariate polynomial of degree n, we have seen that there exists a unique symmetric multiaffine polynomial in n affine variables whose diagonal reduces to the original polynomial. Similarly, given a bivariate homogeneous polynomial of degree 71, there exists a unique symmetric multilinear polynomial in n homogeneous variables whose diagonal reduces to the original homogeneous polynomial. The multilinear function corresponding to the homogeneous polynomial p(t, w) is denoted by BP[(UI,VI), ..., (w n , vn}} and is called the multilinear blossom of p(t, w\ For homogeneous polynomials expressed in monomial form, the correspondence between p(t, w) and BP[(U\,VI), ..., (w n , vn]] is given by

where the sum in brackets is taken over all subsets of order k of the integers {!,...,»}. Like ordinary and homogeneous polynomials, the multiaffine and multilinear blossoms are intimately related. Given the multiaffine blossom, we can

Knot Insertion and Deletion Algorithms

33

linearize it by multiplying each term Ui\ — -Uik by f^.+i • • -Vin. Similarly, given the multilinear blossom, we can dehomogenize it by setting Vk — 1, k — 1,. . ., ri. Moreover, by examining the monomial basis it is easy to see that blossoming and homogenization commute (see Fig. 2.7).

FlG. 2.7. Blossoming and homogemzation commute.

Just like symmetric multiaffine polynomials, symmetric multilinear polynomials can be computed from a recurrence. In general, we have the following identity:

If B is a symmetric multilinear polynomial, then it follows from this identity that

where

Thus, using the following algorithm, we can compute an arbitrary value of B[(UI, t > i ) , . . . , (un, Vn)] recursively once we know the n + 1 values B[(ti, t u i ) , . . . , (tn, wn)},..., B[(tn+i, wn+i),..., (*2n, w2n)} for any progressive sequence (i 1? w - [ ) , . . . , (t 2n , w^n)-

34

Algorithms for Progressive Curves

Recursive Evaluation Polynomials.

Algorithm

for

Symmetric

Multilinear

Let

Then

We illustrate this algorithm graphically for symmetric multilinear functions of three variables in Fig. 2.8. Notice that this algorithm can be obtained from the recursive evaluation algorithm for symmetric multiaffine polynomials by independently homogenizing the numerator and denominator (compare Fig. 2.8 to Fig. 2.3). Moreover, if we set (ur,vr) = (t,w), r = l , . . . , ? i , then this algorithm reduces to the homogeneous de Boor algorithm for the homogeneous polynomial p(t,w) = B[(t, i t > ) , . . . , (£, w)] relative to the homogeneous knots is a symmetric multilinear polynomial in the variables (tii, ^i), • • • , (UTI-, vn), then B[(u\^ 1),..., (un, 1)] is a symmetric multiaffine polynomial in the variables u\,..., un. Indeed by dehomogenization, the m u l t ili n ea r and m ulti aff ine bl oss om s are relat ed b y

Thus if we know the multilinear blossom, we know the multiaffine blossom. Conversely, if we know the multiaffine blossom, then we can use the recursive evaluation algorithm to compute the multilinear blossom. Thus it follows from the recursive evaluation algorithm that each multiaffine blossom has a unique multilinear extension. Therefore if we know one blossom, we know the other. The main tool for studying ordinary progressive curves is the blossoming form of the dual functionals. An analogous result holds for homogeneous progressive curves. Let (t\,w\),... ,(t 2n , w^n] be a progressive sequence, and

Knot Insertion and Deletion Algorithms

35

FIG. 2.8. Recursive evaluation algorithm (unnormalizecT) for symmetric jjiultilinear polynomials in three variables. To save space, the symbol "" " is used to denote homogeneous parameters; thus, for example, Uj = ( W J , D J ) .

let p(t, w) be a progressive curve with control points PQ, ..., Pn relative to these homogeneous knots. Then by much the same arguments we used in §2.2 for the multiafrme case, we can show that

That is, the control points are given by the multilinear blossom evaluated at consecutive knots. (For further details, see Ramshaw [21].) Thus once again we have a blossoming form for the dual functionals, this time for the progressive basis corresponding to the homogeneous knots (/i, w\),..., (£2™, W2n)Now all the results derived in §2.2 for ordinary progressive curves extend quite readily to homogeneous progressive curves. In particular, Ramshaw's blossoming algorithm, Boehm's knot insertion algorithm, the Oslo algorithm, Sablonniere's change of basis algorithm, Marsden's identity, and the de BoorFix form of the dual functionals all remain valid. The proofs are much the same. Simply replace affine parameters by homogeneous parameters, affine knots by homogeneous knots, and the multiafrme blossom by the multilinear blossom. Notice that in the homogeneous version of Marsden's identity and the de Boor-Fix dual functionals, the dual basis functions ^o(t),..., fy n (t] must be homogenized with respect to both the parameter t and the knots t^.. Thus the Polya basis

must be replaced by the homogeneous Polya basis

36

Algorithms for Progressive Curves

With this substitution Marsden's identity becomes

where &o(#, w x ) , . . . , b™(x, wx} are the homogeneous progressive basis functions. Similarly, for the de Boor-Fix dual functionals, we have

where the differentiation on the right-hand side is with respect to t. 2.3.3. Differentiation by Knot Insertion. The multilinear blossom can be used to compute the derivatives of a progressive curve by evaluating the blossom at vector-valued knots. This observation, originally due to Ramshaw [21], will allow us to perform differentiation by applying knot insertion procedures. Before we develop this idea further, we introduce some notation. When applying the multilinear blossom, we shall need to use simple identities of the form As is standard practice, we shall often abuse notation and simply write t to represent the affine parameter (/, 1). (We shall also use t to denote the scalar t as in the left-hand side of (2.13); the meaning we assign to t should be clear from the context.) However, we need some notation for the vector (/i, 0). We shall adopt the notation for the unit vector. Thus we have the identity

The following results are given in Ramshaw [21]; we include them here for completeness. PROPOSITION 2.3.1. Let B[(ti, w>i),.. .,(t n ,w n )] be a symmetric multilinear function. Then

where 8 appears as an argument k times in the coefficient of hk. Proof. By induction on n. Clearly this result is true for n = 1. Now suppose it is true for n — 1. Let

Knot Insertion and Deletion Algorithms

37

Then C is a symmetric multilinear function in n — 1 variables. Therefore by the inductive hypothesis

COROLLARY 2.3.1. Letp(i] be a degree n polynomial curve with multilinear blossom Bp. Then

where 8 appears as an argument k times on the right-hand side of this equation. Proof. By Taylor's theorem

But by the preceding proposition we also have

Equating the coefficients of these two polynomials in /i, we obtain

Now the result follows by multiplying both sides of this equation by k\. D Corollary 2.3.1 has an interesting interpretation in terms of the diagram for the homogeneous de Boor algorithm (Fig. 2.6). Suppose we want to compute the &th derivative of a progressive curve p(£). By Corollary 2.3.1 we need to compute Bp(8,..., , £ , . . . , £). But by Ramshaw's blossoming algorithm, we can compute B p ( f i , . . . , , £ , . . . , / ) is symmetric, we could differentiate the labels on any arbitrary k levels of the diagram and this algorithm would still be valid. We illustrate this procedure for finding the first derivative of a progressive cubic curve in Fig. 2.9.

Original Control Points FIG. 2.9. Recursive algorithm (unnormalized) for computing the first derivative of a progressive cubic curve with knots ( ^n-i(^) ^e ^le progressive basis for the knots (^2,^2), • • .?(*2n-i?^2n-i)- Then

Proof. From Corollary 2.3.1

Now by the recursive evaluation algorithm for symmetric multilinear polynomials

(see Fig. 2.9). But the recurrence for the last n — 1 levels of Bp(8,t, . . . , £ ) is exactly the same as the homogeneous de Boor algorithm for the progressive

Knot Insertion and Deletion Algorithms

39

curve with these control points relative to the knots ( £ 2 , 1 ^ 2 ) , . . . , (£271-1, ^2n-i)Therefore

More generally,

if p(t] is

— ^2j Pjtfj(t) is a progressive curve and the progressive basis for the knots then we can apply Corollary 2.3.1 to find the coefficients of p(k\i) with respect to the basis b 7 Q ~ k ( t ) , . . . , b]li~kk(t). For by Corollary 2.3.1,

Moreover, the last n — k levels of the recursive evaluation algorithm for Bp(6,..., (*2n-Jk, w-2ji-k)Thus to find the coefficients of p(k\t] with respect to the progressive basis 60~ ( t ) , . . . ,&™I f c (£), we need only compute the values on the A;th level of the recurrence for Bp(6,..., (t'2n-k,w'2n-k)We illustrate this algorithm for finding all the derivatives of a progressive cubic curve in Fig. 2.10. Note that to actually find the derivatives, we must

40

Algorithms for Progressive Curves

multiply the fcth coefficient by nl/(n—&)!. This derivative algorithm, but not its interpretation as knot insertion, is originally due to Boehm [9], who developed it for B-spline curves. Here we have extended it to arbitrary progressive schemes.

Original Control Points FlG. 2.10(a). The first for cubic progressive curves: level of n-fold knot insertion the curve with respect to the

step in Boehm's derivative algorithm (unnormalized) inserting a triple knot at 8. Notice that the kth at 6 provides the coefficients of the kth derivative of progressive basis for the homogeneous knot sequence

(tk+i>wk+i), • • • > (^2n-fc) W2n-k)homogeneous knots.

As in Fig. 2.8, the symbol ' is used to denote

2.3.4. Matching Derivatives at the Knots. The derivatives of adjacent segments of B-spline curves agree at the joints. For simplicity, let us restrict ourselves to knot sequences where each knot appears with multiplicity 1. Then if p(t) is the B-spline curve segment for the knots t\,..., tzn and control points PO, • • • i Pm and q(t] is the B-spline curve segment for the knots t-2,..., tzn+i and control points PI, ..., P n +i, then p ( t ) and q(i) meet with n — 1 matching derivatives at t — tn+\. (If tn+i has multiplicity m and we take q(t) to be the curve segment with knots £ m + i , . . . , ^2n+m and control points P m ,...,P n + m , then p(t] and q(t] meet with n — m matching derivatives at t = i n +i-) Using the results from §2.3.3, we shall now show that this result remains valid for arbitrary progressive schemes. Let us consider then two progressive curves p(i] and q(i] where p(t] is the progressive curve for the knots (t\, w\),..., (tzn? W2n) and control points P o 5 - - - : - f n 5 and q(t] is the progressive curve for the knots (£2, ^2),..., (^2?i+i? ^2n+i) and control points PI, ..., P n +i. Again let us assume for simplicity that each knot has multiplicity 1. Then by Corollary 2.3.1

Knot Insertion and Deletion Algorithms

41

FlG. 2.10(b). The second step of Boehm's derivative algorithm (unnormalized) for cubic progressive curves: inserting a triple knot at a. The values at the base of the triangle are taken from the left edge of the triangle computed in the first stage of the algorithm. The values which emerge along the right edge of the triangle computed in this second stage are, up to constant multiples, the derivatives of the original progressive curve at t = a. Again, as in Fig. 2.10(a), the symbol" is used to denote homogeneous knots.

for k — 0 , . . . , n — 1,

since differentiation and homogenization commute. (We shall always take derivatives with respect to /; symmetric results are valid for derivatives with respect to w.) Moreover, because of the overlap of knots and control points for the curves p(t] and