Triangulations Of Point Sets Applications, Structures, Algorithms

1 When solving a difficult problem it is a natural idea to decompose complicated objects into smaller, easy-to-handle pieces. In this book we study such decomposi-tions: triangulations of point sets. We will look at trian-gulations from many different points of view. We explore their combinatorial and geometric properties as well as some algorithmic issues arising along the way. This first chapter is designed to informally introduce the fundamental notions to come in later chapters. We provide motivating examples to convince the reader that tri-angulations are rather useful and that they appear in many areas of mathematics. The reader can skip most of this chapter safely: essentially only the first two pages of the chapter are needed later on. The sections in this chapter present examples which are not meant to be read in any particular order. The examples also provide non-discrete-geometers (e.g., algebraic geometers, computer scientists, linear programming enthusiasts, etc.) that wish to learn about triangulations for their research a door connecting our book to their topic. Without more delay we begin. A point configuration 1 is a finite collection of points A = {a 1 ,. .. , a n } in Euclidean space R d. The convex hull of A is by definition the intersection of all convex sets containing the points in A. We denote it by conv(A). A k-simplex is the convex hull of k + 1 affinely independent points in R d (clearly d ≥ k). Simplices are the simplest of polyhedra: points, segments, triangles, tetrahe-1 The word configuration is used to distinguish this from a set of points: in a subset of R d there can be no multiple points, whereas in a configuration we in principle allow more than one point with the same coordinate. 2 Motivation: Triangulations in Mathematics dra, etc. A j-face of a k-simplex is the convex hull of j + 1 of its vertices and thus in particular a j-simplex itself. We say that the empty set is a (−1)-dimensional face common to all simplices, so that every k-simplex has exactly k+1 j+1 j-faces for j = −1, 0, 1,. .. , k. A simplex of A is a simplex whose vertices are taken from A. Here is the main actor of this play: Definition 1.0.1. Given a point configuration A in R d , we say that a triangulation of A is a finite collection T of d-simplices …