Computational geometry: a retrospective

BERNARD CHAZELLE Department of Computer Science Princeton University Princetoq NJ08544, USA Computational geometry is at a crossroads. New challenges and opportunities are likely to reshape the field rather drastically in the years ahead. I will survey some of its principal accomplishments, and in light of recent developments, I will discuss the profound transformations the field has begun to undergo. There are reasons to believe that computational geometry will emerge from this transition far richer and stronger but barely recognizable from what it was ten years ago. Over the last two decades the field has enjoyed tremendous successes. Some of them might be dismissed as the cheap payoffs to be expected from any field lacking maturity. But others are the products of indisputable creativity and should be held as genuirle scientific achievements. More important, the field is now able to claim a broad, solid foundation upon which its future can be securely built. To mature fully as an original subfield of computer science, however, computational geometry must broaden its connections to applied mathematics while at the same time pay more than lip service to the applications areas that it purports to serve. Happily, active efforts to meet these challenges are underway. Three recent developments are particular encouraging: one is the building of a theory of geometric sampiing and its revolutionary impact on the design of geometric algorithms. Another is the maturing of computational real-algebraic geometry and computational topology both subjects are being revitalized by the introduction of geometric (as opposed to purely algebraic) methods. On the practical end of the spectrum, the emergence of a sub-area concerned specifically with issues of This work was supported in part by NSF Grant CCR-9301254 and The Geometry Center, University of Minnesota, an STC funded by NSF, DOE, and Mimesota Technology, Inc. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Asmciation of Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. finite precision and degeneracy in geometric computing is a most welcome development.