His Ph.D. thesis was entitled Intersection Problems in Computational Geometry obtained under the supervision of Hermann Maurer.

[3] In 1996, with Ping Fu (then director of visualization at the National Center for Supercomputing Applications and his wife), he co-founded Geomagic, a company that develops shape modeling software.

He was elected to the American Academy of Arts and Sciences in 2005, and received an honorary doctorate from Graz University of Technology in 2006.

This book is considered by many people to be still the best textbook and reference source on computational geometry.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.

[citation needed] Edelsbrunner has also made important contributions to algorithms for intersections of line segments, construction of K-sets, the ham sandwich theorem, Delaunay triangulation, point location, interval trees, fractional cascading, and protein docking.