The title of this book promises a discussion of topics in geometry and topology applied to grid or mesh generation. To generate meshes we need algorithms, the subject that provides the glue for our various investigations. However, I make no attempt to cover the breadth of computational geometry. Quite to the contrary, I seek out the subarea relevant to mesh generation, and I enrich that material with concepts from combinatorial topology and a modest amount of numerical analysis. To preserve the focus, I limit attention to meshes composed of triangles and tetrahedra. The economy in breadth permits a coherent and locally self-contained treatment of all topics. My choices are guided by stylistic concerns aimed at exposing ideas and limiting the amount of technical detail.

This book is based on notes I developed while teaching graduate courses at the University of Illinois at Urbana-Champaign and Duke University. The organization into chapters, sections, exercises, and open problems reflects the teaching style I practiced in these courses. Each chapter but the last develops a major topic and is worth about 2 weeks of teaching. Some of the topics are closely related and others are independent. The chapters are divided into sections; each section corresponds to a lecture of about 75 minutes. I believe in an approach to research that complements knowing what is known with knowing what is not known. I therefore recommend spending time in each lecture to discuss one of the open problems collected in the last chapter.