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

Spherical Bernstein-Bézier Interpolants

To appreciate these pictures fully you should be familiar with our paper (with Mike Neamtu and Larry Schumaker ) " Fitting Scattered Data on Sphere-Like Surfaces using Spherical Splines " ( dvi file or postscript file )

The pictures here are generated by the Explorer package. However, for greater portability the Explorer interface also generates a data file that can be used to view the object with the Geomview package. To obtain the corresponding files load down the files /u/ma/alfeld/graphics/SBB-int/*.gv via anonymous ftp (see the instructions in recent papers ). For example, the triangulation picture below looks something like this on the screen when viewed by GeomView.

To see a full-size version of any picture just click on it!

Most of our testing was done with a function that Mike Neamtu came up with. One view of its graph is this. For a definition of the term graph see the section on Spherical Bernstein-Bézier Basis Functions , or better yet our relevant Papers. The color in this picture indicates the function value as described in the legend in the upper left corner. So for Neamtu's function the value of the function ranges approximately from 1.2 to 9.4.

To interpolate or approximate this surface we work with a triangulation on the sphere. In this particular example we start with a regular octahedron and subdivide each of the eight spherical triangles into 16 subtriangles to obtain a total of 128 triangles, 192 edges, and 66 points on the sphere. To render the surfaces we evaluate the function at 66 regularly spaced points in each triangle. (It's a coincidence that the number 66 occurs twice here.) You can see those small triangle in most of these pictures. We could render the surface smoothly (and they can be smoothed with GeomView) but the views given here actually contain more information. The triangulation in this particular case looks like this: The color here indicates the indices of the points used for rendering. Thus there are a total of 6,402 points. Since vertices of the big triangles are numbered before points on the edges, and those are numbered before points in the interior of the triangles, an outline of the triangulation appears. On the original sphere the triangulation looks a little more regular:

The remaining four pictures show the graphs of interpolants of Neamtu's function based on finite elements and approximating spaces that are well known from the planar case. Color now indicates the error, i.e., the difference between the true function value and it approximation.

• Piecewise quintic interpolant, using function values, gradients, and Hessians:
• Piecewise cubic Clough-Tocher, using function values and gradients:
• Piecewise quadratic Powell-Sabin, using function values and gradients:
• Minimum Energy interpolant by C1 piecewise cubic polynomials, using function values only:

All the interpolants do a reasonable job, but, as one might expect, the quintic scheme - which uses more information than all the other schemes - performs best.

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