Peter Alfeld, --- Department of Mathematics, --- College of Science --- University of Utah

Multivariate Splines and the 4 Color Map Problem

Notes

The dimension statement for S13 is obtained as follows: Specify the value and gradient of the spline at every vertex, and also the value of the Bézier ordinate correspond to the centroid of each triangle. This gives 3V+N degrees of freedom. The only remaining smoothness conditions are described by the central quadrilaterals across each interior edge. The crucial assumption now is that these are linearly independent. In that case we can subtract their number from 3V+N.

The only case known where the central smoothness conditions are in fact linearly dependent is that of a singular vertex. It is an open, and apparently very difficult, question whether there are any other triangulations where the central smoothness conditions are in fact linearly dependent.

Click here to see a more detailed discussion of reducibility.

[01-Nov-2023]