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Reviewer: Joseph J. O'Rourke

This gem of a book concentrates on 2D and 3D triangle and tetrahedral meshes, and on the mathematics necessary to obtain efficient and numerically robust algorithms for their construction. The author intentionally narrows his focus to these topics, eschewing arbitrary dimensions, and leaving aside quadrilateral or hexahedral meshes. The result is a compact (177-page) presentation that achieves a level of coherence and elegance unattainable in a more comprehensive treatment. The theme is Delaunay triangulations, which connect proximate points, and consequently have a number of desirable properties. The author explores these for 2D points, Delaunay triangulations constrained by given segments, and Delaunay refinements for meshing. The same three topics are explored in 3D, where each is considerably more complicated. An unhackneyed emphasis I appreciated was that on “flipping” algorithms: improving a triangulation/tetrahedralization by local flipping operations. Between the chapters on 2D and 3D, a lucid chapter on combinatorial topology and another on surface simplification are included. Each topic in the book is partitioned into digestible pieces, reflecting the author’s lecturing experience. Each lemma and theorem is memorably named, and such is the logical economy that every proof fits on one page. Notation is carefully chosen, and the figures are plentiful and apropos (although the thin lines nearly disappear in my copy). Every few pages of sustained argument is relieved by useful bibliographic notes and a short bibliography. These reveal how much of the fundamentals of this topic were developed by Edelsbrunner himself. The language is not always elegant, but the style is lapidary, with not a wasted word, and with every concept explained in the most concise manner conceivable. This does not mean the material is easy. The author taught this topic as a graduate course, and the material assumes knowledge of calculus, linear algebra, data structures, algorithms, and considerable quantities of that elusive “mathematical sophistication.” But it is difficult to imagine improving the presentation: it provides us an optimal access route to this material. The only flaws I noticed—a few typographical errors, a few concepts used prior to their definition, a poor index (missing “dunce cap,” “Gabriel graph,” “Klein bottle,” “Schlegel diagram,” and “sliver,” among other concepts discussed in the book)—could all easily be corrected in a second edition. The book closes with a long (37-page) chapter describing 23 unsolved problems extending the topics in the book. This alone is worth the price of admission. Each is explained in Edelsbrunner’s characteristic clean, lean style, and each is provided with its own bibliography. Problems 8 and 9 (on the area of the union and intersection of disks) have been solved since publication [1], which (I hope) makes a second edition inevitable. Online Computing Reviews Service