He has also published four books on computational geometry: Algorithms in Combinatorial Geometry (Springer-Verlag, 1987, ISBN978-3-540-13722-1), Geometry and Topology for Mesh Generation (Cambridge University Press, 2001, ISBN978-0-521-79309-4), Computational Topology (American Mathematical Society, 2009, 978-0821849255) and A Short Course in Computational Geometry and Topology (Springer-Verlag, 2014, ISBN978-3-319-05956-3).
As Edelsbrunner's Waterman Award citation states,[9]
Dr. Edelsbrunner is a pioneer in the field of computational geometry. ... Dr. Edelsbrunner has had a tremendous impact on computational geometry by his own research as well as by his 1987 book Algorithms in Combinatorial Geometry which systematized the field in its early days. This book is considered by many people to be still the best textbook and reference source on computational geometry.
Research contributions
Edelsbrunner's most heavily cited research contribution[10] is his work with Ernst Mücke on alpha shapes, a technique for defining a sequence of multiscale approximations to the shape of a three-dimensional point cloud. In this technique, one varies a parameter alpha ranging from 0 to the diameter of the point cloud; for each value of the parameter, the shape is approximated as the union of line segments, triangles, and tetrahedra defined by 2, 3, or 4 of the points respectively such that there exists a sphere of radius at most alpha containing only the defining points.[citation needed]
Another heavily cited paper, also with Mücke, concerns “simulation of simplicity.” This is a technique for automatically converting algorithms that work only when their inputs are in general position (for instance, algorithms that may misbehave when some three input points are collinear) into algorithms that work robustly, correctly, and efficiently in the face of special-position inputs.[citation needed]