http://www.dagstuhl.de/07111

### March 11 – 16 , 2007, Dagstuhl Seminar 07111

# Computational Geometry

## Organizers

Pankaj Kumar Agarwal (Duke University – Durham, US)

Helmut Alt (FU Berlin, DE)

Franz Aurenhammer (TU Graz, AT)

## For support, please contact

## Documents

## Motivation

The field of computational geometry is concerned with the design, analysis, and implementation of algorithms for geometric problems, which arise in a wide range of areas, including computer graphics, CAD, robotics computer vision, image processing, spatial databases, GIS, molecular biology, and sensor networks. Since the mid 1980s, computational geometry has arisen as an independent field with its own international conferences and journals.

In the early years mostly theoretical foundations of geometric algorithms were laid, and fundamental theoretical questions remain an important topic in the field. Meanwhile, as the field matured, researchers have started paying close attention to applications and implementations of geometric algorithms. Several software libraries for geometric computation have been developed. Remarkably, these implementations emerged from the originally theoretically oriented computational geometry community itself. Consequently, many researchers are now concerned with theoretical foundations as well as implementation issues.

The seminar will focus on theoretical as well as practical issues in computational geometry. Some of the currently most important topics in computational geometry will be addressed in the seminar:

- Theoretical foundations of computational geometry lie in combinatorial geometry and its algorithmic aspects. They are of an enduring relevance for the field, particularly the design and the analysis of efficient algorithms require deep theoretical insights.
- Various applications such as robotics, GIS, or CAD lead to interesting variants of the classical topics originally investigated, including convex hulls, Voronoi diagrams and Delaunay triangulations, and geometric data structures. For example, pseudo triangulations, generalization of triangulations and developed in connection with visibility and shortest-path problems, have turned out to be useful for many other applications and are being investigated intensively.
- Because of applications in molecular biology, computer vision, geometric databases, shape analysis has become an important topic.
- Another increasingly important application of computational geometry is modeling and reconstruction of surfaces. It brings about many interesting questions concerning fundamental structures like triangulations as well as new issues in computational topology.
- Implementation issues have become an integral part of the research in computational geometry. Besides general software design questions especially robustness of geometric algorithms is important. Several methods have been suggested and investigated to make geometric algorithms numerically robust while keeping them efficient, which lead to interaction with the field of computer algebra, numerical analysis, and topology.

Dagstuhl seminars on computational geometry have been organized since 1990, in a two year rhythm in the recent years, have always been very successful in both disseminating the recent research and conceiving the new ideas.

Parallel to our seminar a second one on Cutting, Packing, Layout and Space Allocation will be organized in Dagstuhl. Because of the overlap of both topics there certainly will be talks in the one seminar which are of interest to the other group, as well. In addition, there should be a beneficial exchange of problems and ideas between both groups. Therefore, the groups of organizers of both seminars agreed to give participants the opportunity to move freely between the two seminars and to have at least one common session.

## Dagstuhl Seminar Series

- 17171: "Computational Geometry" (2017)
- 15111: "Computational Geometry" (2015)
- 13101: "Computational Geometry" (2013)
- 11111: "Computational Geometry" (2011)
- 09111: "Computational Geometry" (2009)
- 05111: "Computational Geometry" (2005)
- 03121: "Computational Geometry" (2003)
- 01121: "Computational Geometry" (2001)
- 99102: "Computational Geometry" (1999)
- 9707: "Computational Geometry" (1997)
- 9511: "Computational Geometry" (1995)
- 9312: "Computational Geometry" (1993)
- 9141: "Computational Geometry" (1991)
- 9041: "Algorithmic Geometry" (1990)

## Classification

- Data Structures / Algorithms / Complexity
- Computer Graphics / Computer Vision

## Keywords

- Computational geometry
- Triangulations
- Voronoi diagrams
- Combinatorial geometry
- Robustness
- Shape