![Preface
The theory of univariate splines began its rapid development in the early
sixties, resulting in several thousand research papers and a number of books.
This development was largely over by 1980, and the bulk of what is known
today was treated in the classic monographs of deBoor [Boo78] and Schu-
maker [Sch81]. Univariate splines have become an essential tool in a wide
variety of application areas, and are by now a standard topic in numerical
analysis books.
If 1960–1980 was the age of univariate splines, then the next twenty
years can be regarded as the age of multivariate splines. Prior to 1980
there were some results for tensor-product splines, and engineers were us-
ing piecewise polynomials in two and three variables in the finite element
method, but multivariate splines had attracted relatively little attention.
Now we have an estimated 1500 papers on the subject.
The purpose of this book is to provide a comprehensive treatment of
the theory of bivariate and trivariate polynomial splines defined on triangu-
lations and tetrahedral partitions. We have been working on this book for
more than ten years, and initially planned to include details on some of the
most important applications, including for example CAGD, data fitting,
surface compression, and numerical solution of partitial differential equa-
tions. But to keep the size of the book manageable, we have reluctantly
decided to leave applications for another monograph.
For us, a multivariate spline is a function which is made up of pieces
of polynomials defined on some partition △ of a set Ω, and joined together
to ensure some degree of global smoothness. We will focus primarily on the
case where △ is a triangulation of a planar region, a triangulation on the
sphere, or a tetrahedral partition of a set Ω in R3
.
The term “multivariate spline” has been used in the literature for other
types of functions, see Remark 5.7 and the discussion in [Boo88] on “what
is a multivariate spline?”. Here we are following Schoenberg, who in 1966
discussed certain bivariate piecewise polynomials which he called splines.
In particular, in the paper [CurS66] he and Curry examined certain analogs
of the univariate B-spline. For some interesting correspondence involving
these early developments, see the discussion in [Mic95].
As we shall see, multivariate polynomial splines have many of the same
features which make the univariate splines such powerful tools for applica-
tions. In particular:
• splines are easy to work with computationally, and there are stable
and efficient algorithms for evaluating their derivatives and integrals,
• there is a very convenient representation which provides a strong con-
nection between the shape of a spline and its associated coefficients,
xi](https://image.slidesharecdn.com/53946-250415155734-16b7956f/85/Spline-functions-on-triangulations-1st-Edition-Ming-Jun-Lai-17-320.jpg)
![Preface xiii
Chapter 7 treats C2
macro-element spaces associated with various split-
ting schemes. These spaces also have stable local bases and full approxi-
mation power.
Chapter 8 is concerned with families of Cr
macro-element spaces based
on various triangle splits including the Clough–Tocher, Powell–Sabin, and
Powell–Sabin-12 splits, as well as certain splits based on quadrangulations.
For each macro-element space we give both a stable local minimal determin-
ing set and a stable local nodal minimal determining set. In addition, for
each element we construct a corresponding Hermite interpolation operator
and give error bounds for it. The methods of this chapter and the previous
two chapters have direct applications to scattered data fitting. Moreover,
the macro-element spaces discussed here and in Chapters 6 and 7 can be
used directly for the numerical solution of partial differential equations, and
should be of special interest to the finite-element community.
Chapter 9 presents what is currently known about the dimension of bi-
variate spline spaces. For general triangulations and arbitrary smoothness
r and degree d, we have to be satisfied with upper and lower bounds on
dimension. However, exact dimension results are available for several im-
portant spaces including Sr
d (△) for d ≥ 3r+2. We also give results for fairly
general superspline subspaces of Sr
d (△). To get dimension statements for
values of d < 3r + 2, we have to restrict ourselves to special partitions.
Here we give results for type-I and type-II partitions. In this chapter we
also compute the generic dimension of the space S1
3 (△). The problem of
finding the dimension of S1
3 (△) for arbitrary triangulations remains one of
the most challenging open questions in bivariate spline theory.
Chapter 10 is devoted to the question of how well smooth functions can
be approximated by bivariate splines on triangulations. In particular, we
show that for d ≥ 3r + 2, the spaces Sr
d (△) have full approximation order
d + 1, but have suboptimal approximation power for smaller d, and in fact
for d < (3r + 2)/2 have no approximation power at all.
Chapter 11 provides an explicit construction of stable local minimal de-
termining sets for the space Sr
d (△) for d ≥ 3r+2 and for certain superspline
subspaces of Sr
d(△). These results ensure that at least for d ≥ 3r+2, spline
spaces on arbitrary triangulations are guaranteed to have full approxima-
tion power. The connection between stable local bases and local linear
independence is also explored in this chapter.
Chapter 12 is devoted to a compact description of the theory of box splines
as examples of polynomial spline spaces defined on special triangulations.
Special emphasis is given to what we call type-I and type-II box splines.
For more on box splines and related simplex splines, we recommend the
survey articles of [DahM83] and [DaeL91], and the references therein. For
a comprehensive monograph on box splines, see [BooHR93].](https://image.slidesharecdn.com/53946-250415155734-16b7956f/85/Spline-functions-on-triangulations-1st-Edition-Ming-Jun-Lai-19-320.jpg)
Spline functions on triangulations 1st Edition Ming-Jun Lai
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- 5. Spline functions on triangulations 1st Edition Ming-Jun Lai Digital Instant Download Author(s): Ming-Jun Lai, Larry L. Schumaker ISBN(s): 9780521875929, 0521875927 Edition: 1 File Details: PDF, 3.23 MB Year: 2007 Language: english
- 7. ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS EDITORIAL BOARD P. Flajolet, M. Ismail, E. Lutwak Volume 110 Spline Functions on Triangulations Spline functions are universally recognized as highly effective tools in app- roximation theory, computer-aided geometric design, image analysis, and numerical analysis. The theory of univariate splines is well-known but this text is the first comprehensive treatment of the analogous multivariate theory. A detailed mathematical treatment of polynomial splines on triangulations is presented, providing a basis for developing practical methods for using splines in numerous application areas. The treatment of the Bernstein-Bézier representation of polynomials will provide a valuable source for researchers and students in CAGD. Chapters on smooth macro-element spaces provide new tools to engineers and scientists for solving partial differential equations numerically. Workers in the geosciences will find the results on spherical splines on triangulations especially useful for approximation and data fitting on the sphere. The book also includes a chapter on box splines, and four chapters on the latest research on trivariate splines. This comprehensive book is ideal as a primary text for graduate courses in approximation theory, and as a source book for courses in computer-aided geometric design or in finite-element methods. M I N G - J U N L A I is a Professor of Mathematics at the University of Georgia L A R R Y S C H U M A K E R is the Stevenson Professor of Mathematics at Vanderbilt University
- 8. ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS All the titles listed below can be obtained from good booksellers or from Cambridge University Press. For a complete series listing visit http://www.cambridge.org/uk/ series/sSeries.asp?code=EOM 68 R. Goodman and N. Wallach Representations and Invariants of the Classical Groups 69 T. Beth, D. Jungnickel, and H. Lenz Design Theory I, 2nd edn 70 A. Pietsch and J. Wenzel Orthonormal Systems for Banach Space Geometry 71 G. E. Andrews, R. Askey and R. Roy Special Functions 72 R. Ticciati Quantum Field Theory for Mathematicians 73 M. Stern Semimodular Lattices 74 I. Lasiecka and R. Triggiani Control Theory for Partial Differential Equations I 75 I. Lasiecka and R. Triggiani Control Theory for Partial Differential Equations II 76 A. A. Ivanov Geometry of Sporadic Groups I 77 A. Schinzel Polynomials with Special Regard to Reducibility 78 H. Lenz, T. Beth, and D. Jungnickel Design Theory II, 2nd edn 79 T. Palmer Banach Algebras and the General Theory of * Algebras II 80 O. Stormark Lie’s Structural Approach to PDE Systems 81 C. F. Dunkl and Y. Xu Orthogonal Polynomials of Several Variables 82 J. P. Mayberry The Foundations of Mathematics in the Theory of Sets 83 C. Foias, O. Manley, R. Rosa and R. Temam Navier–Stokes Equations and Turbulence 84 B. Polster and G. Steinke Geometries on Surfaces 85 R. B. Paris and D. Kaminski Asympiotics and Mellin–Barnes Integrals 86 R. McEliece The Theory of Information and Coding, 2nd edn 87 B. Magurn Algebraic Introduction to K-Theory 88 T. Mora Solving Polynomial Equation Systems I 89 K. Bichteler Stochastic Integration with Jumps 90 M. Lothaire Algebraic Combinatorics on Words 91 A. A. Ivanov and S. V. Schpectorov Geometry of Sporadic Groups II 92 P. McMullen and E. Schulte Abstract Regular Polytopes 93 G. Giertz et al. Continuous Lattices and Domains 94 S. Finch Mathematical Constants 95 Y. Jabri The Mountain Pass Theorem 96 G. Gasper and M. Rahman Basic Hypergeometric Series, 2nd edn 97 M. C. Pedicchio and W. Tholen (eds.) Categorical Foundations 98 M. E. H. Ismail Classical and Quantum Orthogonal Polynomials in One Variable 99 T. Mora Solving Polynomial Equation Systems II 100 E. Olivieri and M. Eulália Vares Large Deviations and Metastability 101 A. Kushner, V. Lychagin and V. Rubtsov Contact Geometry and Nonlinear Differential Equations 102 L. W. Beineke, R. J. Wilson, P. J. Cameron. (eds.) Topics in Algebraic Graph Theory 103 O. Staffans Well-Posed Linear Systems 104 J. M. Lewis. S. Lakshmivarahan and S. Dhall Dynamic Data Assimilation 105 M. Lothaire Applied Combinatorics on Words 106 A. Markoe Analytic Tomography 107 P. A. Martin Multiple Scattering 108 R. A. Brualdi Combinatorial Matrix Classes
- 9. Spline Functions on Triangulations MING-JUN LAI AND LARRY L. SCHUMAKER
- 10. C A M B R I D G E U N I V E R S I T Y P R E S S Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521875929 # M. J. Lai and L. L. Schumaker 2007 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2007 Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this publication is available from the British Library ISBN-13 978-0-521-87592-9 hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
- 11. Contents Preface xi Chapter 1. Bivariate Polynomials 1.1. Introduction 1 1.2. Norms of Polynomials on Triangles 1 1.3. Derivatives of Polynomials 2 1.4. Polynomial Approximation in the Maximum Norm 3 1.5. Averaged Taylor Polynomials 4 1.6. Polynomial Approximation in the q Norm 7 1.7. Approximation on Nonconvex Ω 9 1.8. Interpolation by Bivariate Polynomials 10 1.9. Remarks 15 1.10. Historical Notes 17 Chapter 2. Bernstein–Bézier Methods for Bivariate Polynomials 2.1. Barycentric Coordinates 18 2.2. Bernstein Basis Polynomials 20 2.3. The B-form 22 2.4. Stability of the B-form Representation 24 2.5. The deCasteljau Algorithm 25 2.6. Directional Derivatives 27 2.7. Derivatives at a Vertex 31 2.8. Cross Derivatives 34 2.9. Computing Coefficients by Interpolation 36 2.10. Conditions for Smooth Joins of Polynomials 38 2.11. Computing Coefficients From Smoothness 41 2.12. The Markov Inequality on Triangles 44 2.13. Integrals and Inner-products of B-polynomials 45 2.14. Subdivision 47 2.15. Degree Raising 49 2.16. Dual Bases for the Bernstein Basis Polynomials 49 2.17. A Quasi-interpolant 51 2.18. The Bernstein Approximation Operator 52 2.19. Remarks 57 2.20. Historical Notes 60 Chapter 3. B-Patches 3.1. Control Nets and Control Surfaces 62 3.2. The Convex Hull Property 65 3.3. Positivity of B-patches 65 3.4. Monotonicity of B-patches 70 v
- 12. vi Contents 3.5. Convexity of B-patches 72 3.6. Control Surfaces and Subdivision 77 3.7. Control Surfaces and Degree Raising 79 3.8. Rendering a B-patch 82 3.9. Parametric Patches 84 3.10. Remarks 84 3.11. Historical Notes 84 Chapter 4. Triangulations and Quadrangulations 4.1. Properties of Triangles 86 4.2. Triangulations 87 4.3. Regular Triangulations 89 4.4. Euler Relations 89 4.5. Storing Triangulations 91 4.6. Constructing Triangulations 94 4.7. Clusters of Triangles 96 4.8. Refinements of Triangulations 97 4.9. Optimal Triangulations 103 4.10. Maxmin-Angle Triangulations 104 4.11. Delaunay Triangulations 109 4.12. Constructing Delaunay Triangulations 110 4.13. Type-I and Type-II Triangulations 111 4.14. Quadrangulations 112 4.15. Triangulated Quadrangulations 117 4.16. Nested Sequences of Triangulations 120 4.17. Remarks 121 4.18. Historical Notes 124 Chapter 5. Bernstein–Bézier Methods for Spline Spaces 5.1. The B-form Representation of Splines 127 5.2. Storing, Evaluating and Rendering Splines 128 5.3. Control Surfaces and the Shape of Spline Surfaces 129 5.4. Dimension and a Local Basis for S0 d (△) 130 5.5. Spaces of Smooth Splines 132 5.6. Minimal Determining Sets 135 5.7. Approximation Power of Spline Spaces 137 5.8. Stable Local Bases 141 5.9. Nodal Minimal Determining Sets 143 5.10. Macro-element Spaces 146 5.11. Remarks 147 5.12. Historical Notes 149 Chapter 6. C1 Macro-element Spaces 6.1. A C1 Polynomial Macro-element Space 151 6.2. A C1 Clough–Tocher Macro-element Space 155
- 13. Contents vii 6.3. A C1 Powell–Sabin Macro-element Space 159 6.4. A C1 Powell–Sabin-12 Macro-element Space 163 6.5. A C1 Quadrilateral Macro-element Space 166 6.6. Comparison of C1 Macro-element Spaces 171 6.7. Remarks 172 6.8. Historical Notes 173 Chapter 7. C2 Macro-element Spaces 7.1. A C2 Polynomial Macro-element space 174 7.2. A C2 Clough–Tocher Macro-element Space 178 7.3. A C2 Powell–Sabin Macro-element Space 182 7.4. A C2 Wang Macro-element Space 186 7.5. A C2 Double Clough–Tocher Macro-element 189 7.6. A C2 Quadrilateral Macro-element Space 192 7.7. Comparison of C2 Macro-element Spaces 196 7.8. Remarks 197 7.9. Historical Notes 198 Chapter 8. Cr Macro-element Spaces 8.1. Polynomial Macro-element Spaces 199 8.2. Clough–Tocher Macro-element Spaces 203 8.3. CT Spaces with Natural Degrees of Freedom 209 8.4. Powell–Sabin Macro-element Spaces 214 8.5. PS Spaces with Natural Degrees of Freedom 220 8.6. Quadrilateral Macro-element Spaces 226 8.7. Remarks 231 8.8. Historical Notes 233 Chapter 9. Dimension of Spline Spaces 9.1. Dimension of Spline Spaces on Cells 234 9.2. Dimension of Superspline Spaces on Cells 238 9.3. Bounds on the Dimension of Sr d (△) 240 9.4. Dimension of Sr d (△) for d ≥ 3r + 2 244 9.5. Dimension of Superspline Spaces 249 9.6. Splines on Type-I and Type-II Triangulations 253 9.7. Bounds on the Dimension of Superspline Spaces 255 9.8. Generic Dimension 262 9.9. The Generic Dimension of S1 3 (△) 265 9.10. Remarks 272 9.11. Historical Notes 274 Chapter 10. Approximation Power of Spline Spaces 10.1. Approximation Power 276 10.2. C0 Splines and Piecewise Polynomials 277 10.3. Approximation Power of Sr d(△) for d ≥ 3r + 2 277
- 14. viii Contents 10.4. Approximation Power of Sr d(△) for d < 3r + 2 286 10.5. Remarks 304 10.6. Historical Notes 306 Chapter 11. Stable Local Minimal Determining Sets 11.1. Introduction 308 11.2. Supersplines on Four-cells 309 11.3. A Lemma on Near-degenerate Edges 317 11.4. A Stable Local MDS for Sr,µ d (△) 318 11.5. A Stable MDS for Splines on a Cell 325 11.6. A Stable Local MDS for Sr,ρ d (△) 327 11.7. Stability and Local Linear Independence 328 11.8. Remarks 331 11.9. Historical Notes 333 Chapter 12. Bivariate Box Splines 12.1. Type-I Box Splines 334 12.2. Type-II Box Splines 343 12.3. Box Spline Series 347 12.4. The Strang–Fix Conditions 351 12.5. Polynomial Reproducing Formulae 355 12.6. Box Spline Quasi-interpolants 359 12.7. Half Box Splines 363 12.8. Finite Shift-invariant Spaces 366 12.9. Remarks 375 12.10. Historical Notes 377 Chapter 13. Spherical Splines 13.1. Spherical Polynomials 378 13.2. Derivatives of Spherical Polynomials 391 13.3. Spherical Triangulations 396 13.4. Spaces of Spherical Splines 397 13.5. Spherical Macro-element Spaces 406 13.6. Remarks 407 13.7. Historical Notes 408 Chapter 14. Approximation Power of Spherical Splines 14.1. Radial Projection 409 14.2. Projections of Triangulations 409 14.3. Norms on the Sphere 414 14.4. Spherical Sobolev Spaces 416 14.5. Sobolev Seminorms 419 14.6. Clusters of Spherical Triangles 421 14.7. Local Approximation by Spherical Polynomials 423 14.8. The Markov Inequality for Spherical Polynomials 424
- 15. Contents ix 14.9. Spaces with Full Approximation Power 425 14.10. Remarks 432 14.11. Historical Notes 433 Chapter 15. Trivariate Polynomials 15.1. The Space Pd 434 15.2. Barycentric Coordinates 435 15.3. Bernstein Basis Polynomials 437 15.4. The B-form of a Trivariate Polynomial 438 15.5. Stability of the B-form 440 15.6. The de Casteljau Algorithm 441 15.7. Directional Derivatives 442 15.8. B-coefficients and Derivatives at a Vertex 443 15.9. B-coefficients and Derivatives on Edges 446 15.10. B-coefficients and Derivatives on Faces 449 15.11. B-Coefficients and Hermite Interpolation 451 15.12. The Markov Inequality on Tetrahedra 452 15.13. Integrals and Inner-products 452 15.14. Conditions for Smooth Joins 453 15.15. Approximation Power in the Maximum Norm 454 15.16. Averaged Taylor Polynomials 455 15.17. Approximation Power in the q-Norms 456 15.18. Subdivision 457 15.19. Degree Raising 458 15.20. Remarks 458 15.21. Historical Notes 460 Chapter 16. Tetrahedral Partitions 16.1. Properties of a Tetrahedron 461 16.2. General Tetrahedral Partitions 463 16.3. Regular Tetrahedral Partitions 464 16.4. Euler Relations 465 16.5. Constructing and Storing Tetrahedral Partitions 469 16.6. Clusters of Tetrahedra 470 16.7. Refinements of Tetrahedral Partitions 472 16.8. Delaunay Tetrahedral Partitions 479 16.9. Remarks 479 16.10 Historical Notes 480 Chapter 17. Trivariate Splines 17.1. C0 Trivariate Spline Spaces 481 17.2. Spaces of Smooth Splines 483 17.3. Minimal Determining Sets 484 17.4. Approximation Power of Trivariate Spline Spaces 486 17.5. Stable Local Bases 489
- 16. x Contents 17.6. Nodal Minimal Determining Sets 490 17.7. Hermite Interpolation 492 17.8. Dimension of Trivariate Spline Spaces 494 17.9. Remarks 499 17.10. Historical Notes 500 Chapter 18. Trivariate Macro-element Spaces 18.1. Introduction 502 18.2. A C1 Polynomial Macro-element 503 18.3. A C1 Macro-element on the Alfeld Split 508 18.4. A C1 Macro-element on the Worsey–Farin Split 513 18.5. A C1 Macro-element on the Worsey–Piper Split 517 18.6. A C2 Polynomial Macro-element 520 18.7. A C2 Macro-element on the Alfeld Split 524 18.8. A C2 Macro-element on the Worsey–Farin Split 530 18.9. Another C2 Worsey–Farin Macro-element 537 18.10. A C2 Macro-element on the Alfeld-16 Split 544 18.11. A Cr Polynomial Macro-element 548 18.12. Remarks 557 18.13. Historical Notes 558 References 559 Index 587
- 17. Preface The theory of univariate splines began its rapid development in the early sixties, resulting in several thousand research papers and a number of books. This development was largely over by 1980, and the bulk of what is known today was treated in the classic monographs of deBoor [Boo78] and Schu- maker [Sch81]. Univariate splines have become an essential tool in a wide variety of application areas, and are by now a standard topic in numerical analysis books. If 1960–1980 was the age of univariate splines, then the next twenty years can be regarded as the age of multivariate splines. Prior to 1980 there were some results for tensor-product splines, and engineers were us- ing piecewise polynomials in two and three variables in the finite element method, but multivariate splines had attracted relatively little attention. Now we have an estimated 1500 papers on the subject. The purpose of this book is to provide a comprehensive treatment of the theory of bivariate and trivariate polynomial splines defined on triangu- lations and tetrahedral partitions. We have been working on this book for more than ten years, and initially planned to include details on some of the most important applications, including for example CAGD, data fitting, surface compression, and numerical solution of partitial differential equa- tions. But to keep the size of the book manageable, we have reluctantly decided to leave applications for another monograph. For us, a multivariate spline is a function which is made up of pieces of polynomials defined on some partition △ of a set Ω, and joined together to ensure some degree of global smoothness. We will focus primarily on the case where △ is a triangulation of a planar region, a triangulation on the sphere, or a tetrahedral partition of a set Ω in R3 . The term “multivariate spline” has been used in the literature for other types of functions, see Remark 5.7 and the discussion in [Boo88] on “what is a multivariate spline?”. Here we are following Schoenberg, who in 1966 discussed certain bivariate piecewise polynomials which he called splines. In particular, in the paper [CurS66] he and Curry examined certain analogs of the univariate B-spline. For some interesting correspondence involving these early developments, see the discussion in [Mic95]. As we shall see, multivariate polynomial splines have many of the same features which make the univariate splines such powerful tools for applica- tions. In particular: • splines are easy to work with computationally, and there are stable and efficient algorithms for evaluating their derivatives and integrals, • there is a very convenient representation which provides a strong con- nection between the shape of a spline and its associated coefficients, xi
- 18. xii Preface • splines are capable of approximating smooth functions well, and we can establish the exact relationship between the smoothness of a function and its order of approximation. The book is organized as follows. Chapter 1 is a self-contained treatment of bivariate polynomials. Of spe- cial interest here is the discussion of local approximation properties of poly- nomials. These are a form of Whitney theorems, and are the basis for our later treatment of the approximation power of bivariate spline spaces, as well as error bounds for various macro-element schemes. Interpolation with bivariate polynomials is also discussed here. Chapter 2 deals with the Bernstein–Bézier representation of polynomials. This representation is the main tool for our theoretical developments, but is also critical for the efficient computational use of multivariate splines. In addition to introducing barycentric coordinates, Bernstein basis polyno- mials, and the B-form, we discuss derivatives, integrals, smoothness condi- tions, subdivision, degree raising, dual bases, quasi-interpolation, and the Bernstein operator. A thorough understanding of the notation and results of this chapter is an essential prerequisite to reading the rest of the book. Chapter 3 should be of special interest to the computer-aided geomet- ric design community as it contains a careful treatment of the connection between the shape of a polynomial surface patch and its associated set of B-coefficients. We discuss positivity, monotonicity, and convexity, as well as subdivision and degree raising as possible rendering schemes. Chapter 4 introduces triangulations, and deals with their construction, storage, and combinatorics. Here we also discuss optimal triangulations and various refinement algorithms which are of particular importance for our later discussion of macro-element spaces, which are important tools for data fitting and the numerical solution of partial differential equations. This chapter also includes a discussion of triangulated quadrangulations. Chapter 5 provides details on the Bernstein–Bézier approach to dealing with bivariate splines, and on methods for storing, evaluating, and render- ing such splines. It also contains a discussion of the spline space S0 d (△), and introduces the important concept of minimal determining sets which is the key tool in discussing various properties of spline spaces, including di- mensions, building local bases, and the construction of quasi-interpolation operators. The idea of nodal minimal determining sets is also introduced here. It is used heavily in Chapters 6–8 in the study of macro-element spaces. Chapter 6 collects results on C1 macro-element spaces associated with different splitting schemes. These spaces are particularly useful for appli- cations since they have stable local bases and full approximation power.
- 19. Preface xiii Chapter 7 treats C2 macro-element spaces associated with various split- ting schemes. These spaces also have stable local bases and full approxi- mation power. Chapter 8 is concerned with families of Cr macro-element spaces based on various triangle splits including the Clough–Tocher, Powell–Sabin, and Powell–Sabin-12 splits, as well as certain splits based on quadrangulations. For each macro-element space we give both a stable local minimal determin- ing set and a stable local nodal minimal determining set. In addition, for each element we construct a corresponding Hermite interpolation operator and give error bounds for it. The methods of this chapter and the previous two chapters have direct applications to scattered data fitting. Moreover, the macro-element spaces discussed here and in Chapters 6 and 7 can be used directly for the numerical solution of partial differential equations, and should be of special interest to the finite-element community. Chapter 9 presents what is currently known about the dimension of bi- variate spline spaces. For general triangulations and arbitrary smoothness r and degree d, we have to be satisfied with upper and lower bounds on dimension. However, exact dimension results are available for several im- portant spaces including Sr d (△) for d ≥ 3r+2. We also give results for fairly general superspline subspaces of Sr d (△). To get dimension statements for values of d < 3r + 2, we have to restrict ourselves to special partitions. Here we give results for type-I and type-II partitions. In this chapter we also compute the generic dimension of the space S1 3 (△). The problem of finding the dimension of S1 3 (△) for arbitrary triangulations remains one of the most challenging open questions in bivariate spline theory. Chapter 10 is devoted to the question of how well smooth functions can be approximated by bivariate splines on triangulations. In particular, we show that for d ≥ 3r + 2, the spaces Sr d (△) have full approximation order d + 1, but have suboptimal approximation power for smaller d, and in fact for d < (3r + 2)/2 have no approximation power at all. Chapter 11 provides an explicit construction of stable local minimal de- termining sets for the space Sr d (△) for d ≥ 3r+2 and for certain superspline subspaces of Sr d(△). These results ensure that at least for d ≥ 3r+2, spline spaces on arbitrary triangulations are guaranteed to have full approxima- tion power. The connection between stable local bases and local linear independence is also explored in this chapter. Chapter 12 is devoted to a compact description of the theory of box splines as examples of polynomial spline spaces defined on special triangulations. Special emphasis is given to what we call type-I and type-II box splines. For more on box splines and related simplex splines, we recommend the survey articles of [DahM83] and [DaeL91], and the references therein. For a comprehensive monograph on box splines, see [BooHR93].
- 20. xiv Preface Chapter 13 contains a complete theory of certain spaces of splines defined on spherical triangulations introduced and studied extensively by Alfeld, Neamtu, and Schumaker. These splines are made up of pieces of trivariate homogeneous polynomials restricted to the sphere, and thus are actually piecewise spherical harmonics. The beauty of these spaces is the fact that the entire algebraic theory of bivariate splines can be carried over immedi- ately. These spaces are valuable tools for fitting data and approximating functions defined on the sphere. In particular, there are spherical spline analogs of all of the bivariate macro-element spaces, which we expect will be useful numerical tools for the approximate solution of partial differential equations on the sphere. Chapter 14 provides approximation results for spherical spline spaces. The key tool here is a certain radial projection mapping a spherical cap into a plane which is tangent to the sphere. The mapping provides a means of transferring results about bivariate splines to spherical splines. Chapter 15 and the following three chapters are devoted to the theory of trivariate polynomial splines. This chapter lays the groundwork with a detailed discussion of trivariate polynomials paralleling our treatment of bivariate polynomials in Chapters 1 and 2. Of special importance is the discussion of trivariate Bernstein basis polynomials and the associated B- form of trivariate polynomials. A thorough understanding of the notation and results of this chapter is critical to the study of trivariate splines in the last three chapters of the book. Chapter 16 can also be regarded as preparation for our treatment of trivariate splines. Here we introduce tetrahedral partitions, and discuss Euler relations, refinement methods, and properties of clusters. Chapter 17 is our main chapter on trivariate splines defined over tetrahe- dral partitions. It contains all of the main features of our earlier develop- ment of the theory of bivariate splines, including minimal determining sets, stable local bases, dimension, and approximation power. Chapter 18 is devoted to an exposition of the properties of several different C1 and C2 trivariate macro-element spaces which are suitable for trivari- ate data fitting and as approximating spaces for use in the finite element method. General Cr polynomial macro-element spaces are also treated. Each chapter of the book includes a section with remarks, and a sec- tion with historical notes. We have collected most of the remarks in each chapter in a separate section at the end of the chapter with the aim of providing interesting and useful tangential information without interupting the flow of the book. We believe that historical notes are important to an understanding and appreciation of the development of this material, and we have made every effort to explain the history as accurately as possible.
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- 22. On advertisement chromos, torn with time Around a hearth where wasps and spiders build.— The house is dead: meseems that night of crime It, too, was shot and killed. UNANOINTED I Upon the Siren-haunted seas, between Fate’s mythic shores, Within a world of moon and mist, where dusk and daylight wed, I see a phantom galley and its hull is banked with oars, With ghostly oars that move to song, a song of dreams long dead:— “Oh, we are sick of rowing here! With toil our arms are numb; With smiting year on weary year Salt-furrows of the foam: Our journey’s end is never near, And will no nearer come— Beyond our reach the shores appear Of far Elysium.” II Within a land of cataracts and mountains old, and sand, Beneath whose heavens ruins rise, o’er which the stars burn red, I see a spectral cavalcade with crucifix in hand And shadowy armor march and sing, a song of dreams long dead:—
- 23. “Oh, we are weary marching on! Our limbs are travel-worn; With cross and sword from dawn to dawn We wend with raiment torn: The leagues to go, the leagues we’ve gone Are sand and rock and thorn— The way is long to Avalon Beyond the deeps of morn.” III They are the curs’d! the souls who yearn and evermore pursue The vision of a vain desire, a splendor far ahead; To whom God gives the poet’s dream without the grasp to do, The artist’s hope without the scope between the quick and dead:— I, too, am weary toiling where The winds and waters beat; When shall I ease the oar I bear And rest my tired feet? When will the white moons cease to glare, The red suns veil their heat? And from the heights blow sweet the air Of Love’s divine retreat? SUNSET AND STORM
- 24. Deep with divine tautology, The sunset’s mighty mystery Again has traced the scroll-like west With hieroglyphs of burning gold: Forever new, forever old, Its miracle is manifest. Time lays the scroll away. And now Above the hills a giant brow Night lifts of cloud; and from her arm, Barbaric black, upon the world, With thunder, wind and fire, is hurled Her awful argument of storm. What part, O man, is yours in such? Whose awe and wonder are in touch With Nature,—speaking rapture to Your soul,—yet leaving in your reach No human word of thought or speech Expressive of the thing you view. BEECH BLOOMS
- 25. Among the valleys The wild oxalis Lifts up its chalice Of pink and pearl; And, balsam-breathing From out their sheathing, The myriad wreathing Green leaves uncurl. The whole world brightens With spring, that lightens The foot that frightens The building thrush; Where water tosses On ferns and mosses The squirrel crosses The beechen hush. And vision on vision,— Like ships elysian On some white mission,— Sails cloud on cloud; With scents of clover The winds brim over, And in the cover The stream is loud. ’Twixt bloom that blanches The orchard branches Old farms and ranches Gleam in the gloam: Through fields for sowing, ’Mid blossoms blowing, The cows come lowing, The cows come home. Where ways are narrow, A
- 26. A vesper-sparrow Flits like an arrow Of living rhyme; The red sun poises, And farm-yard noises Mix with glad voices Of milking-time. When dusk disposes Of all its roses, And darkness closes, And work is done, A moon’s white feather In starry weather And two together Whose hearts are one. WORSHIP I The mornings raise Voices of gold in the Almighty’s praise; The sunsets soar In choral crimson from far shore to shore: Each is a blast, Reverberant, of color,—seen as vast Concussions,—that the vocal firmament In worship sounds o’er every continent. II
- 27. Not for our ears The cosmic music of the rolling spheres, That sweeps the skies! Music we hear, but only with our eyes. For all too weak Our mortal frames to bear the words these speak, Those detonations that we name the dawn And sunset—hues Earth’s harmony puts on. UNHEARD
- 28. All things are wrought of melody, Unheard, yet full of speaking spells; Within the rock, within the tree, A soul of music dwells. A mute symphonic sense that thrills The silent frame of mortal things; Its heart beats in the ancient hills, And in each flow’r sings. To harmony all growth is set— Each seed is but a music mote, From which each plant, each violet, Evolves its purple note. Compact of melody, the rose Woos the soft wind with strain on strain Of crimson; and the lily blows Its white bars to the rain. The trees are pæans; and the grass One long green fugue beneath the sun— Song is their life; and all shall pass, Shall end, when song is done. REINCARNATION
- 29. High in the place of outraged Liberty, He ruled the world, an emperor and god: His iron armies swept the land and sea, And conquered nations trembled at his nod. By him the love that fills man’s soul with light, And makes a heaven of earth, was crucified; Lust-crowned he lived, yea, lived in God’s despite, And old in infamies, a king he died. Justice begins now.—Many centuries In some vile body must his soul atone As slave, as beggar, loathsome with disease, Less than the dog at which we fling a stone. ON CHENOWETH’S RUN
- 30. I thought of the road through the glen, With its hawk’s nest high in the pine; With its rock, where the fox had his den, ’Mid tangles of sumach and vine, Where she swore to be mine. I thought of the creek and its banks, Now glooming, now gleaming with sun; The rustic bridge builded of planks, The bridge over Chenoweth’s Run, Where I wooed her and won. I thought of the house in the lane, With its pinks and its sweet mignonette; Its fence, and the gate with its chain, Its porch where the roses hung wet, Where I kissed her and met. Then I thought of the family graves, Walled rudely with stone, in the West, Where the sorrowful cedar-tree waves, And the wind is a spirit distressed, Where they laid her to rest. And my soul, overwhelmed with despair, Cried out on the city and mart!— How I longed, how I longed to be there, Away from the struggle and smart, By her and my heart. By her and my heart in the West,— Laid sadly together as one;— On her grave for a moment to rest, Far away from the noise and the sun, On Chenoweth’s Run.
- 31. REQUIESCAT
- 32. The roses mourn for her who sleeps Within the tomb; For her each lily-flower weeps Dew and perfume. In each neglected flower-bed Each blossom droops its lovely head,— They miss her touch, they miss her tread, Her face of bloom, Of happy bloom. The very breezes grieve for her, A lonely grief; For her each tree is sorrower, Each blade and leaf. The foliage rocks itself and sighs, And to its woe the wind replies,— They miss her girlish laugh and cries, Whose life was brief, Was all too brief. The sunlight, too, seems pale with care, Or sick with woe; The memory haunts it of her hair, Its golden glow. No more within the bramble-brake The sleepy bloom is kissed awake— The sun is sad for her dear sake, Whose head lies low, Lies dim and low. The bird, that sang so sweet, is still At dusk and dawn; No more it makes the silence thrill Of wood and lawn. In vain the buds, when it is near, Open each pink and perfumed ear,— The song it sings she will not hear
- 33. Who now is gone, Is dead and gone. Ah, well she sleeps who loved them well, The birds and bowers; The fair, the young, the lovable, Who once was ours. Alas! that loveliness must pass! Must come to lie beneath the grass! That youth and joy must fade, alas! And die like flowers, Earth’s sweetest flowers! THE QUEST I First I asked the honey-bee, Busy in the balmy bowers; Saying, “Sweetheart, tell it me: Have you seen her, honey-bee? She is cousin to the flowers— All the sweetness of the south In her wild-rose face and mouth.”— But the bee passed silently. II Then I asked the forest-bird, Warbling by the woodland waters; Saying, “Dearest, have you heard, Have you heard her, forest-bird? She is one of Music’s daughters— Never song so sweet by half As the music of her laugh.”— But the bird said not a word. III
- 34. Next I asked the evening-sky, Hanging out its lamps of fire; Saying, “Loved one, passed she by? Tell me, tell me, evening-sky! She, the star of my desire— Sister whom the Pleiads lost, And my soul’s high pentecost.”— But the sky made no reply. IV Where is she? ah, where is she? She to whom both love and duty Bind me, yea, immortally.— Where is she? ah, where is she? Symbol of the Earth-soul’s beauty. I have lost her. Help my heart Find her! her, who is a part Of the pagan soul of me! BEFORE THE RAIN
- 35. Before the rain, low in the obscure east, Weak and morose the moon hung, sickly gray; Around its disc the storm mists, cracked and creased, Wove an enormous web, wherein it lay Like some white spider hungry for its prey. Vindictive looked the scowling firmament, In which each star, that flashed a dagger ray, Seemed filled with malice of some dark intent. The marsh-frog croaked; and underneath the stone The peevish cricket raised a creaking cry. Within the world these sounds were heard alone, Save when the ruffian wind swept from the sky, Making each tree like some sad spirit sigh; Or shook the clumsy beetle from its weed, That, in the drowsy darkness, bungling by, Sharded the silence with its feverish speed. Slowly the tempest gathered. Hours passed Before was heard the thunder’s sullen drum Rumbling night’s hollow; and the Earth at last, Restless with waiting,—like a woman, dumb With doubting of the love that should have clomb Her casement hours ago,—avowed again, ’Mid protestations, joy that he had come. And all night long I heard the Heavens explain. AFTER RAIN
- 36. Behold the blossom-bosomed Day again, With all the star-white Hours in her train, Laughs out of pearl-lights through a golden ray, That, leaning on the woodland wildness, blends A sprinkled amber with the showers that lay Their oblong emeralds on the leafy ends. Behold her bend with maiden-braided brows Above the wildflower, sidewise with its strain Of dewy happiness, to kiss again Each drop to death; or, under rainy boughs, With fingers, fragrant as the woodland rain, Gather the sparkles from the sycamore, To set within the core Of crimson roses girdling her hips, Where each bud dreams and drips. Smoothing her blue-black hair,—where many a tusk Of iris flashes,—like the falchions keen Of Faery round blue banners of their Queen,— Is it a Naiad singing in the dusk, That haunts the spring, where all the moss is musk With footsteps of the flowers on the banks? Or but a wild-bird voluble with thanks? Balm for each blade of grass: the Hours prepare A festival each weed’s invited to. Each bee is drunken with the honied air: And all the heaven is eloquent with blue. The wet hay glitters, and the harvester Tinkles his scythe,—as twinkling as the dew,— That shall not spare Blossom or brier in its sweeping path; And, ere it cut one swath, Rings them they die, and tells them to prepare. What is the spice that haunts each glen and glade? A Dryad’s lips, who slumbers in the shade?
- 37. A Faun, who lets the heavy ivy-wreath Slip to his thigh as, reaching up, he pulls The chestnut blossoms in whole bosomfuls? A sylvan Spirit, whose sweet mouth doth breathe Her viewless presence near us, unafraid? Or troops of ghosts of blooms, that whitely wade The brook? whose wisdom knows no other song But that the bird sings where it builds beneath The wild-rose and sits singing all day long. Oh, let me sit with silence for a space, A little while forgetting that fierce part Of man that struggles in the toiling mart; Where God can look into my heart’s own heart From unsoiled heights made amiable with grace; And where the sermons that the old oaks keep Can steal into me.—And what better then Than, turning to the moss a quiet face, To fall asleep? a little while to sleep And dream of wiser worlds and wiser men. SUNSET CLOUDS
- 38. Low clouds, the lightning veins and cleaves, Torn from the wilderness of storm, Sweep westward like enormous leaves O’er field and farm. And in the west, on burning skies, Their wrath is quenched, their hate is hushed, And deep their drifted thunder lies With splendor flushed. The black turns gray, the gray turns gold; And sea’d in deeps of radiant rose, Summits of fire, manifold, They now repose. What dreams they bring! what thoughts reveal! That have their source in loveliness, Through which the doubts I often feel Grow less and less. Through which I see that other night, That cloud called Death, transformed of Love To flame, and pointing with its light To life above. RICHES
- 39. What mines the morning heavens unfold! What far Alaskas of the skies! That, veined with elemental gold, Sierra on Sierra rise. Heap up the gold of all the world, The ore that makes men fools and slaves: What is it to the gold, cloud-curled, That rivers through the sunset’s caves. Search Earth for riches all who will, The gold that soils, that turns to dust— Mine be the wealth no thief can steal, The gold of Beauty naught can rust. THE AGE OF GOLD
- 40. The clouds that tower in storm, that beat Arterial thunder in their veins; The wildflowers lifting, shyly sweet, Their perfect faces from the plains,— All high, all lowly things of Earth For no vague end have had their birth. Low strips of mist, that mesh the moon Above the foaming waterfall; And mountains that God’s hand hath hewn, And forests where the great winds call,— Within the grasp of such as see Are parts of a conspiracy; To seize the soul with beauty; hold The heart with love: and thus fulfill Within ourselves the Age of Gold, That never died, and never will,— As long as one true nature feels The wonders that the world reveals. A SONG FOR LABOR I Oh, the morning meads, the dewy meads, Where he ploughs and harrows and sows the seeds, Singing a song of manly deeds, In the blossoming springtime weather: The heart in his bosom as high as the word Said to the sky by the mating bird, While the beat of an answering heart is heard, His heart and hers together. II
- 41. Oh, the noonday heights, the sunlit heights, Where he stoops to the harvest his keen scythe smites, Singing a song of the work that requites, In the ripening summer weather: The soul in his body as light as the sigh Of the little cloud-breeze that cools the sky, While he hears an answering soul reply, His soul and hers together. III Oh, the evening vales, the twilight vales, Where he labors and sweats to the thud of flails, Singing a song of the toil that he hails, In the fruitful autumn weather: In heart and in soul as free from fears As the first white star in the sky that appears, While the music of life and of love he hears, Her life and his together. THE LOVE OF LOVES
- 42. I have not seen her face, and yet She is more sweet than anything Of earth—than rose or violet That winds of May and sunbeams bring. Of all we know, past or to come, That beauty holds within its net, She is the high compendium: And yet— I have not touched her robe, and still She is more dear than lyric words And music; or than strains that fill The throbbing throats of forest birds. Of all we mean by poetry, That rules the soul and charms the will, She is the deep epitome: And still— She is my world: ah, pity me! A dream that flies whom I pursue: Whom all pursue, whoe’er they be, Who toil for Art and dare and do. The shadow-love for whom they sigh, The far ideal affinity, For whom they live and gladly die— Ah me! THREE THINGS
- 43. There are three things of Earth That help us more Than those of heavenly birth That all implore— Than Love or Faith or Hope, For which we strive and grope. The first one is Desire,— Who takes our hand And fills our hearts with fire None may withstand;— Through whom we’re lifted far Above both moon and star. The second one is Dream,— Who leads our feet By an immortal gleam To visions sweet;— Through whom our forms put on Dim attributes of dawn. The last of these is Toil,— Who maketh true, Within the world’s turmoil The other two;— Through whom we may behold Ourselves with kings enrolled. IMMORTELLES I
- 44. As some warm moment of repose In one rich rose Sums all the summer’s lovely bloom And pure perfume— So did her soul epitomize All hopes that make life wise, Who lies before us now with lidded eyes, Faith’s amaranth of truth Crowning her youth. II As some melodious note or strain May so contain All of sweet music in one chord, Or lyric word— So did her loving heart suggest All dreams that make life blessed, Who lies before us now with pulseless breast, Love’s asphodel of duty Crowning her beauty. A LULLABY I
- 45. In her wimple of wind and her slippers of sleep The twilight comes like a little goose-girl, Herding her owls with many “Tu-whoos,” Her little brown owls in the forest deep, Where dimly she walks in her whispering shoes, And gown of glimmering pearl. Sleep, sleep, little one, sleep: This is the road to Rockaby Town. Rockaby, lullaby, where dreams are cheap; Here you can buy any dream for a crown. Sleep, sleep, little one, sleep; The cradle you lie in is soft and is deep, The wagon that takes you to Rockaby Town. Now you go up, sweet, now you go down, Rockaby, lullaby, now you go down. II And after the twilight comes midnight, who wears A mantle of purple so old, so old! Who stables the lily-white moon, it is said, In a wonderful chamber with violet stairs, Up which you can see her come, silent of tread, On hoofs of pale silver and gold. Dream, dream, little one, dream: This is the way to Lullaby Land. Lullaby, rockaby, where, white as cream, Sugar-plum bowers drop sweets in your hand. Dream, dream, little one, dream; The cradle you lie in is tight at each seam, The boat that goes sailing to Lullaby Land. Over the sea, sweet, over the sand, Lullaby, rockaby, over the sand.
- 46. III The twilight and midnight are lovers, you know, And each to the other is true, is true! And there on the moon through the heavens they ride, With the little brown owls all huddled a-row, Through meadows of heaven where, every side, Blossom the stars and the dew. Rest, rest, little one, rest: Rockaby Town is in Lullaby Isle. Rockaby, lullaby, set like a nest Deep in the heart of a song and a smile. Rest, rest, little one, rest; The cradle you lie in is warm as my breast, The white bird that bears you to Lullaby Isle. Out of the East, sweet, into the West, Rockaby, lullaby, into the West. PESTILENCE High on a throne of noisome ooze and heat, ’Mid rotting trees of bayou and lagoon, Ghastly she sits beneath the skeleton moon, A tawny horror coiling at her feet— Fever, whose eyes keep watching, serpent-like, Until her eyes shall bid him rise and strike. MUSINGS I Inspiration
- 47. All who have toiled for Art, who’ve won or lost, Sat equal priests at her high Pentecost; Only the chrism and sacrament of flame, Anointing all, inspired not all the same. II Apportionment How often in our search for joy below Hoping for happiness we chance on woe. III Victory They who take courage from their own defeat Are victors too, no matter how much beat. IV Preparation How often hope’s fair flower blooms richest where The soul was fertilized with black despair. V Disillusion Those unrequited in their love who die Have never drained life’s chief illusion dry. VI
- 48. Success Success allures us in the earth and skies: We seek to win her, but, too amorous, Mocking, she flees us.—Haply, were we wise, We should not strive and she would come to us. VII Science Miranda-like, above the world she waves The wand of Prospero; and, beautiful, Ariel the airy, Caliban the dull,— Lightning and Steam,—are her unwilling slaves. VIII The Universal Wind Wild son of Heav’n, with laughter and alarm, Now east, now west, now north, now south he goes, Bearing in one harsh hand dark death and storm, And in the other, sunshine and a rose. IX Compensation Yea, whom He loves the Lord God chasteneth With disappointments, so that this side death, Through suffering and failure, they know Hell To make them worthy in that Heaven to dwell Of Love’s attainment, where they come to be Parts of its beauty and divinity.
- 49. X Poppies Summer met Sleep at sunset, Dreaming within the south,— Drugged with his soul’s deep slumber, Red with her heart’s hot drouth, These are the drowsy kisses She pressed upon his mouth. XI Her Eyes and Mouth There is no Paradise like that which lies Deep in the heavens of her azure eyes: There is no Eden here on Earth that glows Like that which smiles rich in her mouth’s red rose. XII Her Soul To me not only does her soul suggest Palms and the peace of tropic shore and wood, But, oceaned far beyond the golden West, The Fortunate Islands of true Womanhood. XIII Her Face
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