discrete and computational geometry

[...]... vast area beyond our coverage And what lies beyond is indeed vast The Handbook of Discrete and Computational Geometry runs to 1,500 pages and even so is highly compressed Our coverage represents a sparse sampling of the field We have chosen to cover polygons, convex hulls, triangulations, and Voronoi diagrams, which we believe constitute the core of discrete and computational geometry Beyond this core,... less constrained partitions, and engender the fascinating question of which pairs of polygons can be dissected and reassembled into each other This so-called “scissors congruence” (Section 1.4) again highlights the fundamental difference between 2D and 3D (Section 1.5), a theme throughout the book 1.1 DIAGONALS AND TRIANGULATIONS Computational geometry is fundamentally discrete as opposed to continuous... vertices has n − 2 triangles and n − 3 diagonals Proof We prove this by induction on n When n = 3, the statement is trivially true Let n > 3 and assume the statement is true for all polygons with fewer than n vertices Choose a diagonal d joining vertices a and b, cutting P into polygons P1 and P2 having n1 and n2 vertices, respectively Because a and b appear in both P1 and P2 , we know n1 + n2 = n... as Figure 1.1(d), admits a triangulation of its interior 1.1 DIAGONALS AND TRIANGULATIONS (a) (b) (c) (d) Figure 1.5 Polyhedra: (a) tetrahedron, (b) pyramid with square base, (c) cube, and (d) triangular prism That every polygon has a triangulation is a fundamental property that pervades discrete geometry and will be used over and over again in this book It is remarkable that this notion does not generalize... some have resisted the assaults of many talented researchers and may be awaiting a theoretical breakthrough, others may be accessible with current techniques and only await significant attention by an enterprising reader The field has expanded greatly since its origins, and the new connections to areas of mathematics (such as algebraic topology) and new PREFACE application areas (such as data mining) seems... colleagues and coauthors Lauren Cowles, Erik Demaine, Jin-ichi Ito, Joseph Mitchell, Don Shimamoto, and Costin Vîlcu, who each taught me so much through our collaborations The early stages of my work on this book were funded by a NSF Distinguished Teaching Scholars award DUE-0123154 Satyan L Devadoss Williams College Joseph O’Rourke Smith College xi This page intentionally left blank Discrete and Computational. .. factorization This chapter introduces triangulations (Section 1.1) and their combinatorics (Section 1.2), and then applies these concepts to the alluring art gallery theorem (Section 1.3), a topic at the roots of computational geometry which remains an active area of research today Here we encounter a surprising difference between 2D triangulations and 3D tetrahedralizations Triangulations are highly constrained... back face The result is a deep dent at each square of the front face Repeat this procedure for the top face and the right face, staggering the squares so their respective dents do not intersect Now imagine standing deep in the interior, surrounded by dent faces above and below, left and right, fore and aft From a sufficiently central point, no vertex can be seen! 19 20 CHAPTER 1 POLYGONS (a) (b) Figure... all polygons with fewer than n vertices Using Lemma 1.3, find a diagonal cutting P into polygons P1 and P2 Because both P1 and P2 have fewer vertices than n, P1 and P2 can be triangulated by the induction hypothesis By the Jordan curve theorem (Theorem 1.1), the interior of P1 is in the exterior of P2 , and so no triangles of P1 will overlap with those of P2 A similar statement holds for the triangles... O’Rourke Smith College xi This page intentionally left blank Discrete and Computational GEOMETRY This page intentionally left blank POLYGONS Polygons are to planar geometry as integers are to numerical mathematics: a discrete subset of the full universe of possibilities that lends itself to efficient computations And triangulations are the prime factorizations of polygons, alas without the benefit of . 06:48pm fm.tex Discrete and Computational GEOMETRY This page intentionally left blank December 13, 2010 Time: 06:48pm fm.tex Discrete and Computational GEOMETRY SATYAN L. DEVADOSS and JOSEPH O’ROURKE PRINCETON. fm.tex Preface Although geometry is as old as mathematics itself, discrete geometry only fully emerged in the twentieth century, and computational geometry was only christened in the late 1970s. The terms discrete . Data Devadoss, Satyan L., 1973– Discrete and computational geometry / Satyan L. Devadoss and Joseph O’Rourke. p. cm. Includes index. ISBN 978-0-691-14553-2 (hardcover : alk. paper) 1. Geometry Data processing.