Geometry and Topology for Mesh Generation by Herbert EdelsbrunnerReview by: Joseph O'RourkeSIAM Review, Vol. 49, No. 1 (Mar., 2007), pp. 154-156Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/20453931 .

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154 BOOK REVIEWS

vs. West, detailing the work of Weier strass as well as that of the Goettingen

school led by Hilbert and Klein. The final chapter emphasizes the important work of the Russian Jewish mathemati cian Sergei Bernstein, whose construc tive proof of the Weierstrass theorem and extension of Chebyshev's work serve to unify East and West in their approaches

to approximation theory.

Appendices containing biographical data on numerous eminent mathematicians, explanations of Russian nomenclature and academic degrees, and an excellent index round out the presentation.

R.I.P.

REFERENCE

[1] A. PINKUS, Weierstrass and approximation theory, J. Approx. Theory, 107 (2000), pp. 1-66.

PAUL NEVAI The Ohio State University

Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills. By Paul J. Nahin. Princeton University Press, Princeton, NJ, 2006. $29.95. xxii+380 pp., hardcover. ISBN 0-691 - 11822-1.

This author is an electrical engineering pro fessor who has published a number of ac cessible books displaying great enthusiasm for some mathematics, some mathemati cians, and some mathematical history. This book is a sequel to An Imaginary Tale: The Story of V-1 (Princeton University Press, 1998) and it demands a good knowledge of calculus and an enjoyment of related ma nipulations, especially those involving sums and integrals. Its dust jacket looks like a nineteenth-century patent medicine adver tisement and the book really pushes the ef fectiveness of complex variables, especially the formula e i + 1 = 0.

The first three chapters are well written and introductory, and mention a number of the author's quirky likes and dislikes, including J. L. Synge's novel Kandelman's

Krim and Pollock's drip painting. There's an occasional "wild ride," such as his ap proximation of e' near 0 by a ratio of nth

order polynomials to show that 7r2 is irra tional.

The heart and soul of the book are the fi nal three chapters on Fourier series, Fourier integrals, and related engineering. One can recommend them to all applied math stu dents for their historical development and sensible content. The treatment of the Gibbs phenomena is particularly interest ing, though there may be a bit too much concern for Wilbraham's neglect. Hardy's evaluation of an integral arising in optics is praised, as is the Paley-Wiener test for a causal filter. The descriptions of AM radio tuning, single sideband radio, Hilbert trans forms, and related material reflect Nahin's teaching and his earlier book, The Science of Radio.

The book ends with a 21-page biogra phy of Euler, including related information about the Bernoullis, the Russian Academy, and Voltaire, and with 27 pages of very in formative (and sometimes technical) notes.

Readers wanting more mathematical treatment of some of the topics covered

might refer to [1] or [2].

REFERENCES

[1] T. W. K6RNER, Fourier Analysis, Cam bridge University Press, Cambridge, UK, 1988.

[2] V. S. VARADARAJAN, Euler through Time:

A New Look at Old Themes, AMS, Providence, RI, 2006.

ROBERT E. O'MALLEY, JR. University of Washington

Geometry and Topology for Mesh Gen eration. By Herbert Edelsbrunner. Cambridge University Press, Cambridge, UK, 2006. $29.99. xii+177 pp., softcover. ISBN 0-521-68207-X.

This is a gem of a book concentrating on two- (2D) and three-dimensional (3D) tri angle and tetrahedral meshes, and on the mathematics necessary to obtain efficient and numerically robust algorithms for their construction. The author intentionally nar rows his focus to these topics, leaving aside sampling the manifold to be meshed or

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BOOK REVIEWS 155

constructing quadrilateral and hexahedral meshes, and only venturing beyond W3 when lifting into R4. The result is a compact pre sentation that achieves a level of coherence and elegance unattainable in a more com prehensive treatment.

The central structure studied is the De launay triangulation, which is in a sense the

most natural mesh for a finite set of points, for its edges record adjacency in the dual Voronoi diagram, and so connect proximate points. The author starts with a generic set of points S in R2, then complicates the set ting by introducing constraining segments and constructing the "constrained Delau nay triangulation," and finally considers "Delaunay refinements" that add points to S with the goal of removing skinny tri angles, which plague numerical algorithms employing the mesh. After a lucid chapter on combinatorial topology and another on surface simplification, he revisits the same sequence of Delaunay topics in 1R3: a generic set of points S, then S constrained by the faces of a polyhedron, and finally refine ments to the mesh to exclude tetrahedral "slivers." The lock-step repetition of these topics serves to highlight the surprising con trasts between 2D and 3D. Let me illustrate this difference through flipping algorithms, a central and clearly developed theme of the book.

It will simplify matters to assume the points of S are in "general position," an assumption Edelsbrunner shows is justified by a symbolic perturbation argument. A Delaunay triangulation is characterized by the empty-circle property: three points of S are vertices of a Delaunay triangle iff the circumcircle passing through those points contains no other points of S. Suppose one has a triangulation T of a set of points S. An edge ab in T is called locally Delaunay if either it is on the convex hull of S, or it belongs to two triangles abc and abd, and c is outside the circumcircle of abd and, sym

metrically, d is outside the circumcircle of abc. Such an edge looks like it might be part of a Delaunay triangulation.

If edge ab belongs to two triangles which form a convex quadrilateral acbd, an edge flip is possible: delete ab and add cd to T. If one lifts the quadrilateral to a paraboloid in W3, the edge flip can be viewed as flipping

between the bottom and top sides of the re sulting tetrahedron. It is a natural, albeit mindless algorithm to hope that repeatedly flipping edges that are not locally Delaunay eventually leads to the Delaunay triangula tion (Algorithm 1). Remarkably, this does work, but it might need e(n2) flips to reach the Delaunay triangulation for ISI = n.

A simple variation leads to a more ef ficient algorithm (Algorithm 2): add the points of S one by one to a growing tri angulated set, and flip to quiescence after each addition. Each new point p falls in side some triangle abc (starting from a large surrounding triangle that is eventually dis carded ensures this), and p is connected to a, b, and c before flipping. If the points are added in random order, Edelsbrunner shows that this algorithm runs in expected time O(n log n).

It is natural to hope that these beautiful flipping algorithms generalize to construct the 3D Delaunay tetrahedralization. Just as the edge flip can be viewed usefully as a pro jection from R3, so one can define "bistellar flips" between the bottom and top sides of a lifted 5-vertex simplex in R4. Unfortu nately, it is no longer true that an arbitrary tetrahedralization of a set of points S can be bistellar flipped to the Delaunay tetra hedralization: a deadlock is possible. This rules out the analogue of Algorithm 1. For tunately, the incremental Algorithm 2 does generalize, as was established by Edelsbrun ner and Shah. Points of S are added one at a time to a growing tetrahedral complex, and tetrahedra that are not "locally Delaunay" are flipped. This remains the simplest and fastest algorithm for computing Delaunay tetrahedralizations.

Another 2D/3D distinction is provided by "shellings." A triangulation K of a poly gon (a topological disk) is shellable if there is an ordering of the triangles such that each prefix of the ordering is also a topological disk. Thus K can be "shelled" one triangle at a time. He proves that any such K is shellable. However, the generalization to 3D fails: Bing's "house with two rooms" shows that not every tetrahedralization of a topo logical 3D ball is shellable. Interestingly, the Delaunay tetrahedralization is shellable because it is the projection of a polytope in R4 and all polytopes are shellable.

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156 BOOK REVIEWS

Which brings us to open problems. Edels brunners ends the book with 23 unsolved problems, one of which (P.12) asks for a polynomial-time algorithm to decide if a tetrahedral complex is shellable. Each prob lem is explained in Edelsbrunner's charac teristic lean style, and each is provided with its own bibliography.

Inevitably, such a list dates quickly. Here are a few updates. Problem P.1, Empty con vex hexagons, was recently settled by Tobias Gerken, who showed that any sufficiently large finite set of points in the plane con tains six that determine an empty convex hexagon. Problems P.8 and P.9, Union and intersection of disks, were solved by Bezdek and Connelly [BC02]: any rearrangement of a finite set of disks in the plane that does not decrease the distance between any pair of centers neither decreases the area of the union nor increases the area of the intersection.

I am not aware that any of the other 20 problems have been solved outright, but I will mention that although Problem P.7, Sorting X + Y, remains open, now the min imum entry in a matrix convolution X + Y can be found in subquadratic time [B+06]; that Problem P.11, Connecting contours, needs slight emendation because contours do not always exist [GOS96]; and that al though Problem P.18, Flip graph connectiv ity, remains open in R 3 (using here a more powerful flip than the one that can be dead locked), Santos has shown it to be false in

R5 [SanOO] by constructing polytopes with a space of triangulations not connected via bistellar flips. Hopefully the combination of this accessible and stimulating list of prob lems, and Edelsbrunner's lapidary presen tation of the mathematics behind them, will encourage engagement with this fascinating field.

REFERENCES

[BCO2] K. BEZDEK AND R. CONNELLY, Push

ing disks apart-the Kneser

Poulsen conjecture in the plane, J. Reine Angew. Math., 553

(2002), pp. 221-236.

[B+06] D. BREMNER, T. M. CHAN, E. D.

DEMAINE, J. ERICKSON, F. HUR

TADO, J. IACONO, S. LANGER

MAN, AND P. TASLAKIAN, Neck

laces, convolutions, and X + Y, in Proceedings of the 14th

Annual European Symposium

on Algorithms, Zurich, Switzer land, Springer-Verlag, Heidel berg, 2006.

[GOS96] C. GITLIN, J. O'RoURKE, AND V.

SUBRAMANIAN, On reconstruct ing polyhedra from parallel slices, Internat. J. Comput. Geom.

Appl., 6 (1996), pp. 103-122.

[SanO0] F. SANTOS, A point configuration whose space of triangulations is disconnected, J. Amer. Math. Soc., 13 (2000), pp. 611-637.

JOSEPH O'ROURKE Smith College

Convex Functions and Their Applications: A Contemporary Approach. By Constantin Niculescu and Lars-Erik Persson. Springer-Verlag, New York, 2006. $69.95. xvi+255 pp., hard cover. ISBN 0-387-24300-3.

The title of this book might be a bit mis leading to readers of SIAM Review, since the word "applications" in the title refers primarily to applications of convexity to a wide variety of inequalities, not to ap plications to other parts of analysis. (An exception is contained in the ten pages of Appendix C that describe applications of global minima of special convex function als to certain partial differential equations.) The emphasis on inequalities clearly reflects the authors' research interests; if you look under "inequality" in the index (revised see below) you'll find 58 named inequalities. They treat not just the usual convex func tions, but also (to cite a few examples) functions which are log-convex, multiplica tively convex, quasi-convex, Schur convex, or semi-convex. Thus, the book could be of interest to those applied mathematicians who feel that such material could be useful to them.

The first chapter, titled "Convexity and Majorization," comprises about one-fourth of the book and deals with these various notions of convexity for functions defined on intervals in the real line. We are given some two dozen inequalities involving them and/or their integrals, as well as some clas sical inequalities involving positive matri

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