Prof. Dr. Kestutis Karciauskas

"Rational Multisided Patches"

Probably best known rational multisided patch is Sabin (1983) pentagonal patch. The title of the corresponding paper "Non-Rectangular Surface Patches suitable for inclusion in a B-spline Surface" justifies clearly that these patches are worth to be considered seriously.

Next two types of rational multisided patches that motivated creation of M-patches are: S-patches of Loop&DeRose (1989) and Warren hexagonal patch (1992) constructed blowing-up the base points at the corners of domain triangle.

It appeared that five-sided M-patch plays very important role in this stuff. The reason -- geometrically the same patch has two different interpretations:

1. blown-up patch -- connection to Warren patches;
2. rational patch over regular pentagon -- connection to S-patches and Sabin patches; moreover, regular interpretation points out the way how M-patches can be defined over any regular m-gon.

Next it is shown how M-patches are used to construct so called tensor-border patches -- tensor-border structure simplifies an inclusion of M-patch in a B-spline surface.

Some important remarks will be mentioned briefly at the end:

1. problem of setting the coefficients of inner basis functions;
2. problem of defining inner control points in constructions of fair surfaces;
3. efficient representation of five- and six-sided M-patches;
4. efficient representation of m-sided M-patches for m>6 -- guided subdivision;
5. though six-sided Sabin, Hosaka&Kimura patches are defined over non-rational domain they are special case of rational six-sided M-patches;
6. what is true and what is not if we want to extend to multisided M-patches such standard in CAGD procedures like degree elevation and linear precision.