Geometric Modelling

Many scientific and industrial problems require the description of geometry. The basic techniques used today are based on numerical methods, using tools such as spline functions and numerical linear algebra, while more classical mathematical disciplines, like algebraic and differential geometry, have so far had more limited use.

• Splines and mesh-based modelling
  Spline curves and surfaces are a classical tool for geometric modelling. During the last few years, a major research focus has been the use of splines in multiresolution models. Another trend is the use of polygonal meshes to model surfaces, which has brought with it a need for supporting mathematical theory. Both of these areas have been central to the ongoing EU project MINGLE (Jan 2000 to Jan 2004). The main goal is to further extend spline theory and develop mathematical theory for mesh-based representations to improve modelling, interpretation, and visualization of large structured and unstructured 3D data sets, and large time-varying 4D data sets, such as in medical imaging and large simulation grids.
  
  
• Algebraic Methods
  A relatively new activity, led by Tor Dokken and Ragni Piene, is the combination of algebraic geometry and approximation theory using piecewise representations of algebraic curves, algebraic surfaces, and higher dimensional implicitly represented objects. The aim is a better understanding of the approximation of algebraic representations by parametric representations and vice versa, especially with regard to improved algorithms in computer-aided geometry systems. Central in this activity is the EU-supported project GAIA II addressing: intersection algorithms for geometry-based IT-applications using approximate algebraic methods (July 2002 - June 2005). A number of difficult problems in geometric modelling, such as surfacesurface intersection, closest point location, and ray tracing, are still only partially solved with today’s technology. The ultimate goal of this activity is to provide correct numerical solutions to these problems.
  
  Another problem concerns discrete optimization. The focus will be to investigate problems in linear algebra and optimization motivated by applications in image analysis, discrete tomography, and telecommunications.
  
  
• Differential Equation methods
  There is growing industrial interest in more interaction between geometric design and modelling of physical systems. This leads to various optimal design problems, where a combination of differential equations and geometry is used to describe the optimality condition. One activity will be in the EU-project Leading Edge. - Prediction of Leading Edge and Tip Flow for the Design of Quiet and Efficient Screw Propellers (July 2002 - December 2004). Here the problems where flow and boundary geometry are interrelated will be addressed. Another example is solving scattered data modelling using differential equations. The ultimate goal is to combine tools from numerical geometry, differential geometry, differential equations, and optimization, in order to develop new algorithms which will realise the potential of optimal design for industrial use.