Configurations, Triangulations, Subdivisions, and Flips

The first goal of this chapter is to introduce the necessary mathematical language to work with triangulations. This language includes the geometry of polyhedra and cones [339] and the combinatorics of point and vector configurations, as described by their oriented matroids [55]. These two books are recommended for more details. The second goal is to provide the reader with formal definitions and notation that are strong but flexible enough to cover all kinds of point configurations, and even the more general case of vector configurations. These definitions should include degeneracies, such as collinearities and repetition of points. We will do this slowly, intending to help the reader to see why a naive definition may lead to problems.