**Computational Geometry Seminar**

**Wednesday, March 7th, 2007, 16:10-18:00**

**Room 309**

**Schreiber Building**

### The Integration of Exact Arrangements with Effective Motion Planning

### Ron Wein, Tel Aviv University

**Abstract:**
The field of Computational Geometry has greatly evolved over the
last three decades, yielding a multitude of algorithms to solve
real-life problems emerging from diverse application fields, such
as Solid Modeling, Robotics and Geographical Information Systems.
Yet implementing a geometric algorithm from a textbook is far from
being a trivial task. Common assumptions in the design of algorithms
are that all calculations can be carried out using exact arithmetic,
and that the input objects are in general position. Unfortunately,
these assumptions usually do not hold in practice.

The arrangement package of CGAL, the Computational Geometry
Algorithms Library, is capable of constructing and maintaining
planar subdivisions induced by a set of arbitrary planar curves,
requiring only a small set of geometric predicates and constructions
involving the curves. This package has gone through a major redesign
process, which made it more flexible and extendible, while at the
same time it led to major improvements in the package efficiency.
The modular design of the arrangement package made it a convenient
basis for the development of several peripheral CGAL packages, for
computing Boolean operations on polygons, for constructing lower
envelopes of 3D surfaces, etc. This robust software infrastructure
has paved the way for providing more accurate solutions to
motion-planning problems that arise in fields like Robotics and
Computer-Aided Design. I will start the talk with an overview of the
thesis results related to robust computation with arrangements and
its applications.

I will then focus on a recent result regarding a fast approximation
algorithm for offsetting a polygon, namely computing the Minkowski
sum of a polygon and a disc. The offset-approximation algorithm
employs only rational arithmetic, which significantly expedites its
running times. At the same time, it provides guaranteed precision
bounds. This algorithm is based on the infrastructure provided by
the arrangement package, and it will be included in the forthcoming
release of CGAL (Version 3.3). Using the offset infrastructure it is
now possible to provide complete solutions to many practical problems
in real time, as our experiments show.