Toric surfaces

Toric varieties were introduced in the early 1970's in algebraic geometry. Since that time their abstract theory has been rapidly developed and many new important results are achieved.

This remarkable theory appeared to be quite close to combinatorics of convex polytopes and therefore much more elementary than other parts of the sophisticated building of algebraic geometry. This makes the theory of toric varieties very attractive for different kind of applications.

There are several applications of toric geometry to geometric modeling:

• Parametrization. Data conversion from traditional solid modeling systems that deal with simplest surfaces (plane, circular cone and cylinder, sphere, torus) to NURBS based systems.
• Bezier like constructions. Bezier surfaces and volumes are generalized to toric surface patches and Bezier polytopes. Introducing non-standard real structures on toric surfaces leads to to a wide shape possibilities. Many well-known surfaces of low implicit degree are found to be toric.
• Implicitization. This is an inverse problem to the parametrization one. Dealing with toric surfaces their implicit equations are important for various purposes: for instance, they are useful for finding intersection curves between surfaces.

The paper [2] concentrates on the first application, which can be formalized to the following task: find a rational Bezier surface of minimal degree on a given algebraic surface with fixed rational boundary curves. For several rational surfaces the solution was found using universal parametrizations. Soon universal parametrizations for arbitrary toric surfaces will be presented.

The second application (standard real case) is considered in [3]. Probably J. Warren [1] was the first who used real toric surfaces for constructing multisided surface patches. Shape of toric surfaces with non-standard real structures are discussed in [5]

For implicitization problems we refer to S. Zube [4], where implicit equations of several low-degree toric surfaces in 3-space are calculated using A-resultants.

References

1. J. Warren, Creating multisided rational Bezier surfaces using base points, ACM Transactions on Graphics 11 (1992), 127-139.
2. R. Krasauskas, Universal parametrizations of some rational surfaces, in: Curves and Surfaces with Applications in CAGD, A. Le Mehaute, C. Rabut and L.L. Schumaker (eds.), Vanderbilt University Press, Nashville, 1997, 231-238.
3. R. Krasauskas, Toric surface patches I, Preprint 2000-21, Faculty of Mathematics and Informatics, Vilnius University, 2000.(pdf 434 KB)
4. S. Zube, The n-sided toric patches and A-resultants, Computer Aided Geometric Design, 17 (2000), 695-714.
5. R. Krasauskas, Shape of toric surfaces, Preprint 2001-06, Faculty of Mathematics and Informatics, Vilnius University, 2001.(pdf 355 KB)