This project focuses on Effective Computational Geometry for Curves and Surfaces. This is a challenging and almost untouched research area with a huge number of potential applications in almost all application domains involving geometric computing, e.g. computer aided design and manufacturing, computer graphics and virtual reality, scientific visualization, geographic information systems, molecular biology, fluid mechanics, and robotics. We intend to develop multidisciplinary cooperative research in three main directions: computational geometry, computer algebra and numerical analysis, to develop solid theoretical foundations, to validate our theoretical advances through extensive experimental research, and to develop software packages. Special attention will be paid to the impact of our research.

OBJECTIVES

The objectives of the project are:

- To take into consideration the multidisciplinary nature of the problem and to develop cooperative research in three main directions: computational geometry, computer algebra and numerical analysis;

- To give Effective Computational Geometry for Curves and Surfaces solid mathematical and algorithmic foundations, to provide solutions to key problems and to validate our theoretical advances through extensive experimental research and the development of software packages that could serve as steps towards a standard for safe and effective geometric computing;

- To promote collaborative research, the interchange between the partners (workshops), exchanges of Ph.D. students and research staff;

- To disseminate our results through research reports, open source softwares, software packages and through a program of open activities including summer-schools and advanced courses intended to academia and industry.

DESCRIPTION OF WORK

This project is focused on effectively handling curved objects. Many application domains ranging from engineering to medicine have a demand for computer models of physical objects that are curved, moving and deformable. Our research will be guided by four different main aspects:

- Geometric algorithms for curves and surfaces. We intend to revisit the field of Computational Geometry in order to understand how structures that are well known for linear objects behave when defined on curves and surfaces;

- Algebraic issues. Several operations on non-linear geometric objects, often lying at the algorithm's bottleneck, are equivalent to manipulating polynomials. A fundamental question is the solution of algebraic systems, ubiquitous in the construction of new objects, such as intersections. Another crucial goal is the implementation of primitives with Boolean or discrete output, such as an object is contained in some bounding object;

- Robustness issues. Geometric programs are notorious for their non-robustness: algorithms are designed for a model of computation where real numbers are dealt with exactly geometric algorithms are frequently only formulated for inputs in general position. This is not simply and academic problem. It is easy to crash any commercial CAD-system. Progress has been made only in recent years. A significant part of the progress was made by the proposers and centres around the so-called exact computation paradigm. We will extend this paradigm to curved objects;

- Approximating curves and surfaces. Since algorithms for curves and surfaces are more involved, more difficult to make robust and typically several orders of magnitude slower than their linear counterparts, there is a need for approximate representations. We will provide robust and quality guaranteed approximations of curves and surfaces. Such a research program requires a multidisciplinary approach and our consortium will gather expertise from Computational Geometry and other areas in mathematics and computer science such a computer arithmetic, computer algebra and numerical analysis.