Triangulations

1. Front Matter

   Pages i-xiii

   PDF

2. Triangulations in Mathematics

   • Jesús A. De Loera, Jörg Rambau, Francisco Santos

   Pages 1-41

3. Configurations, Triangulations, Subdivisions, and Flips

   • Jesús A. De Loera, Jörg Rambau, Francisco Santos

   Pages 43-92

4. Life in Two Dimensions

   • Jesús A. De Loera, Jörg Rambau, Francisco Santos

   Pages 93-148

5. A Tool Box

   • Jesús A. De Loera, Jörg Rambau, Francisco Santos

   Pages 149-208

6. Regular Triangulations and Secondary Polytopes

   • Jesús A. De Loera, Jörg Rambau, Francisco Santos

   Pages 209-274

7. Some Interesting Configurations

   • Jesús A. De Loera, Jörg Rambau, Francisco Santos

   Pages 275-336

8. Some Interesting Triangulations

   • Jesús A. De Loera, Jörg Rambau, Francisco Santos

   Pages 337-376

9. Algorithmic Issues

   • Jesús A. De Loera, Jörg Rambau, Francisco Santos

   Pages 377-432

10. Further Topics

    • Jesús A. De Loera, Jörg Rambau, Francisco Santos

    Pages 433-511

11. Back Matter

    Pages 513-535

    PDF

Triangulations appear everywhere, from volume computations and meshing to algebra and topology. This book studies the subdivisions and triangulations of polyhedral regions and point sets and presents the first comprehensive treatment of the theory of secondary polytopes and related topics. A central theme of the book is the use of the rich structure of the space of triangulations to solve computational problems (e.g., counting the number of triangulations or finding optimal triangulations with respect to various criteria), and to establish connections to applications in algebra, computer science, combinatorics, and optimization. With many examples and exercises, and with nearly five hundred illustrations, the book gently guides readers through the properties of the spaces of triangulations of "structured" (e.g., cubes, cyclic polytopes, lattice polytopes) and "pathological" (e.g., disconnected spaces of triangulations) situations using only elementary principles.

• Graph
• algorithms
• computational geometry
• convex geometry
• polyhedral subdivisions
• polytopes
• triangulations

From the reviews:

“Focusing on the structure of the set of all possible triangulations … the current study sits at the threshold of geometry and combinatorics … . offering terra firma to students still struggling with abstraction, the central theorem … only dates to 1989, so the present elaboration carries readers to the frontiers of research. … It is unusual to find such a leisurely, generous exposition of a new subject, as replete with illustrations as contemporary calculus textbooks. … Summing Up: Recommended. Upper-division undergraduates through professionals.” (D. V. Feldman, Choice, Vol. 49 (1), September, 2011)

“This book masterfully presents the theory of triangulations of (the convex hull of) a point set alongside many appealing applications in algebra, computer science, combinatorics, and optimization. … The writing is thorough and engaging, assisted by clear (and numerous) illustrations, and many exercises for the reader. Graduate students and researchers in any area in which triangulations of points set configurations play a role will find this book a comprehensive and most useful reference.” (Matthias Beck, Zentralblatt MATH, Vol. 1207, 2011)

• Department of Mathematics, University of California, Davis, Davis, USA

  Jesús A. Loera

• Lehrstuhl für Wirtschaftsmathematik, Universität Bayreuth, Bayreuth, Germany

  Jörg Rambau

• Depto. Matemáticas Estadística y Computación, Universidad de Cantabria, Santander, Spain

  Francisco Santos

J.A. De Loera is a professor of mathematics at the University of California, Davis. His work approaches difficult computational problems in discrete mathematics and optimization using tools from algebra and convex geometry. His research has been recognized by an Alexander von Humboldt Fellowship and several national and international grants. He is an associate editor of the journal "Discrete Optimization". Jörg Rambau is the chair professor of Wirtschaftsmathematik (Business Mathematics) at the Universität of Bayreuth since 2004. Before that he was associate head of the optimization department at the Zuse Institute Berlin (ZIB). His research encompasses problems in applied optimization, algorithmic discrete mathematics and combinatorial geometry. He is the creator of the state of the art program for triangulation computations TOPCOM. He is associate editor of the "Jahresberichte der Deutschen Mathematiker-Vereinigung". Francisco Santos, a professor at the Universidad de Cantabria Spain, received the Young Researcher award from the Universidad Complutense de Madrid in 2003 and was an invited speaker in the Combinatorics Section of the International Congress of Mathematicians in 2006. He is well-known for his explicit constructions of polytopes with disconnected spaces of triangulations, some of which are featured in this book. He is an editor of Springer Verlag's journal "Discrete and Computational Geometry".

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