Georg Umlauf's Research Projects
               


        
Subdivision algorithms G-Spline-Surfaces
 Meshes
 Voronoi-Splines
 Geometry of Splines


Subdivision algorithms


Subdivision algorithms are a popular tool in CAD to design, create and model surfaces of arbitrary topological type. The initial input to a subdivision algorithm is an arbitrary net of control points. This net is refined to generate a finer control net. Iterating this process results in a sequence of ever finer nets that converges towards a limit surface of the desired topology and smoothness. Thereby the smoothness depends solely on the rules (subdivision masks) that have been used to compute the finer subdivided control nets.

An example sequence of control nets for Loop's algorithm is shown below. To the left is the visualization of the Gaussian curvature of the limit surface.
    





It is possible to conclude from the subdivision mask and the actual geometry of the control net the curvature behavior of the limit surface at the so-called extraordinary points. An example for Loop's algorithm is shown below. Here some free parameters in the construction of the subdivision algorithm have be used to optimize the behavior of the limit surface.

This project aims at tailoring the subdivison rules in such a way that a desired geometrical behavior of the limit surface is guaranteed.
Visualization of Gaussian curvature of surfaces from Loop's algorithm, from the modified algorithm of Loop, from datadependent modified algorithm of Loop using minimization of an energy functional and from hybrid algorithm of Loop (f.l.t.r.).

G-Spline-Surface


G-Spline-Surfaces provide a means to model freeform surfaces. These surfaces consist of polynomial pieces with geometrically smooth transitions, i.e.  the normals or curvatures are continuous.
One way to construct such G2-surfaces for triangular nets (i.e. quartic box splines) is described by H.Prautzsch and G.Umlauf. An example is shown below: to the left is the triangular control net of a generalized quartic box spline surface, in the middle the corresponding generalized quartic box spline surface and to the right the resulting G2-surfaces.
The goal of this project is to generalize this approach to arbitrary smoothness orders while keeping at the same time the number of the required patches to a minimum.





Meshes


Manipulation and transmission of large high dimensional scalar-, vector- or tensorfield data-set bears the problem of efficiently processing this data. For this efficient data structures supporting the necessary access and transmission operations must be constructed. Based on well-known representations for two-dimensional, meshed data generalizations for different application areas are designed and implemented. These include streaming data, video data but also meshes originating from FE simulation.
Voronoi-Splines


Voronoi-Diagrams induce in a natural way piecewise linear splines, that can be used for scattered-data interpolation over the Voronoi diagam. Applying this construction recursively to the Voronoi sites yields piecewise polynomial or rational splines, the so-called Voronoi splines. Furthermore, this construction can be generalized to globally smooth Voronoi splines of arbitrary smoothness order.

In this project we analyze different generalization approaches, their implementation and performance applied to different data-set. Additionally, we generalize these approaches to scattered data interpolation in arbitrary dimensions and applications of these techniques.
Sibson interpolant     Sibson interpolant     Sibson interpolant  
Sibson's C0 interpolant                     Farin's C1 interpolant                         Hiyoshi's C2 interpolant
Geometry of Splines

The geometry of a spline/subdivision curve or surface can be deduced from the geometry of its control polygon or net, respectively. For example, the curve normal of a planar spline curve lies in the cone spanned by the normals of the polygon edges. Such a simple dependency does not exist for higher order manifolds. For surfaces much larger cones must be used to bound the surface normal. This is of particular interest e.g. for back-face culling. Here tighter bounds for the normals are of great interest.
bezier curve
A Bézier curve b(t) and its hodograph b'(t).


© Georg Umlauf Last modified: 12th February 2008