When implementing algorithms for CAD-systems it is desirable to both use parametric and implicit representations of curves and surfaces to make good algorithms. E.g., when intersecting two parametric curves of degree two or higher, p(s) and r(t), only using the parametric form will resulting in finding the zero set of two nonlinear equations in two variables: However, if one of the curves has an implicit representation q_{r}(x,y)=0, we can directly express the problem as finding the zero set of one nonlinear equation in one variable q_{r}(p(s))=0.
Curves and surfaces in CAD-systems are represented as
The work on approximate implicitization in SINTEF started with the dissertation of Tor Dokken for the doctor philosophiae degree, Aspects of Intersection Algorithms and Approximation defended in 1997.
GPGPU (2004-2007), Parallel3D (2007-2011) and Heterogeneous Computing (2008-2011) are all projects addressing the use of modern heterogeneous many core processors (e.g., GPUs) for speeding up and developing new approaches to algorithms. Within algebraic geometry computation time is still an issue as many current algorithms require extensive computational power. In these projects one of the tasks is to look into real-time visualization of algebraic surface (see more info to the right). Another task is to speed up approximate implicitization by GPU-implemented algorithms.
Jan B. Thomassen, Pål H. Johansen, and Tor Dokken, Closest Points, Moving Surfaces, and Algebraic Geometry, in Mathematical Methods for Curves and Surfaces : Tromsø 2004, Nashboro Press, Brentwood, Tenn, 2005, pp. 351-382.
T. Dokken and Thomassen J. B., Overview of Approximate Implicitization, 2003,. Overview of Approximate Implicitization, in Topics in Algebraic Geometry and Geometric Modeling Ron Goldman and Rimvydas Krasauskas (eds), AMS Series on Contemporary Mathematics. 2003. Pages: 169-184.
T. Dokken, Approximate Implicitization, in Mathematical Methods for Curves and Surfaces: Oslo 2000, Vanderbilt Univ. Press, 2001, Pages 81-102 , [Paper]
Tor Dokken, Aspects of Intersection Algorithms and Approximation, Dissertation for the for the doctor philosophiae degree, University of Oslo, Norway, 1997. [Thesis]
Published November 20, 2009
Our work on Real-Time ray-casting of algebraic surfaces was inspired by a talk by Charles Loop on the use of algebraic curves for high quality visualization of fonts at the SIAM GD Conference in 2005. Returning from the conference Tor Dokken presented an approach to Johan S. Seland, resulting in the paper by Seland and Dokken in 2007. The cooperation was then extended to include Martin Reimers from the University of Oslo resulting in an improved approach presented in the paper by Reimer and Seland in 2008.
After working with the development of intersection algorithms since 1979, Tor Dokken started to work on his PhD in 1992 addressing open question related to surface intersection algorithms. One idea was to find some way to find algebraic approximations to rational parametric surfaces. After a lot of experiments in Mathematica a new numerical stable approach with good approximation properties resulted. The results were made public available though the PhD dissertation in 1997, and the published in 2001. Approximate was the basis for the EU FET Open project GAIA (2000-2005) and the current Marie Curie Initial Training Network SAGA (2008-2012).