The main research areas of the geometry group are geometrical modeling, algebraic geometry, combinatorics and optimization. Although these areas are well-established, mathematical fields individually, there has been major focus in the geometry group on bringing these areas together through joint projects.
Geometric modeling is a branch of applied mathematics and computational geometry that studies methods and algorithms for the mathematical description of shapes. These shapes are mostly two- or three-dimensional, although many of the mathematical tools and principles involved can be applied to higher dimensions. Two-dimensional models are important in computer typography and technical drawing. Three-dimensional models are central to computer-aided design and manufacturing (CAD/CAM) and widely used in many applied technical fields such as civil and mechanical engineering, architecture, geology and medical and image processing. The geometry group at CMA is involved in several applications. A common important application area for the group is medical technology, in particularly various aspects of medical imaging.
A strong research area of the geometry group has been parametric curve and surface fitting, and especially the development of new parameterization methods and other algorithms for processing triangular and polygonal meshes for modeling surfaces. This work has led to the new topic of 'generalized barycentric coordinates’ for polygons and polyhedra, which have been applied not only to surface fitting, but also to surface deformation and computer animation. This is a potentially interesting area for collaboration with the PDE group. Another focus of the geometry group with a strong links to PDE’s is isogeometric representation and analysis. Here, the spaces of splines
(piecewise polynomials), typically used to represent the geometric model, are also used to design the finite element method. A lot more research remains to be done, however, to make this approach fully applicable.
In recent years, the geometry group has been increasingly active in the topic of subdivision, which is a way of generating curves and surfaces recursively as the limits of either polygons or polygonal meshes. The group has made several important contributions to the existing body of theory, but the picture is by no means complete. Algebraic spline geometry represents research at the interface between computational geometry and algebraic geometry. Here one uses algebraic spline functions on
triangulated surfaces to set up a geometric theory. The more practical applications of this theory are expected to grow in years to come, and we plan an increased focus on this topic.