The subject of this book is the approximation of curves in two dimensions and surfaces in three dimensions from a set of sample points. This problem, called reconstruction, appears in various engineering applications and scientific studies. What is special about the problem is that it offers an application where mathematical disciplines such as differential geometry and topology interact with computational disciplines such as discrete and computational geometry. One of my goals in writing this book has been to collect and disseminate the results obtained by this confluence. The research on geometry and topology of shapes in the discrete setting has gained a momentum through the study of the reconstruction problem. This book, I hope, will serve as a prelude to this exciting new line of research.

To maintain the focus and brevity I chose a few algorithms that have provable guarantees. It happens to be, though quite naturally, they all use the well-known data structures of the Voronoi diagram and the Delaunay triangulation. Actually, these discrete geometric data structures offer discrete counterparts to many of the geometric and topological properties of shapes. Naturally, the Voronoi and Delaunay diagrams have been a common thread for the materials in the book.

This book originated from the class notes of a seminar course “Sample-Based Geometric Modeling” that I taught for four years at the graduate level in the computer science department of The Ohio State University.