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

Spherical Bernstein-Bézier Basis Functions

To appreciate these pictures fully you should be familiar with our paper " Bernstein-Bézier Polynomials on Spheres and Sphere-Like Surfaces" ( dvi file or postscript file ) written with Marian Neamtu and Larry Schumaker and to appear in the CAGD journal. In the context of that paper b1, b2, and b3 that appear below in the descriptions of the individual pictures are the spherical barycentric coordinates of a point on the unit sphere with respect to a particular spherical triangle. However, for the pictures that spherical triangle has as its vertices the three standard unit vectors in 3-space, and b1, b2, b3 happen to coincide with the usual cartesian coordinates. So let's explain everything in terms of cartesian coordinates. They are defined every on the sphere. We also consider functions (of those cartesian coordinates) on the sphere and define the graph of a function f(b1,b2,b3) as follows: With every point P=(b1,b2,b3) associated the point f(P) times P in 3-space. Thus we place the point corresponding to P along the line from the origin through P at a distance f(P) from the origin. (This idea goes back to Tom Foley and others.) Naturally, if f(P) is negative then we go off in the direction opposite P. In our paper mentioned above we use those barycentric coordinates to approximate functions on the sphere and in particular express the approximating polynomials in terms of Bernstein-B\'ezier polynomials whose graphs are given below. (Notice the integer factors multiplying some of the monomial, those factors turn out to be convenient in our applications, but they do cause some of the graphs to stick out of the unit sphere.)

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

The key picture is the very first one. This shows the graph of the function f(P)=1: [image] . Thus the graph of f is just the sphere itself. Color indicates which octant contains the colored graph of the sphere. For example the red triangle corresponds to the octant where all cartesian coordinates are positive. The correspondence of color and octant is the same in all of the figures. Also, all figures are drawn to the same scale and viewed from the same point. So keep a mental image of the unit sphere! (In the context of our paper, the red triangle could be any spherical triangle, and the other seven triangle would be the remaining spherical triangles defined by the great circles that make up the spherical triangle of interest.)

Graphs of linear functions

The next picture f(P) = b1: [image] is probably the hardest to understand. The point on the graph corresponding to a point on the sphere is the same as the the point corresponding to -P! The equator described by the equation b1=0 is mapped entirely to the origin. Thus the graph is a "double sphere", which in a manner of speaking has infinitely many points at the equator and two points everywhere else. Why is the graph striped? For example, the red and the purple quadrants are mapped to the same part of the sphere, and simply because of rendering artifacts, i.e., the peculiar way in which this particular version of Explorer acts, the graph appears to be striped!

The other coordinates have similar graphs: b2: [image] b3: [image]

Graphs of quadratic functions

Following are the graphs of all six spherical quadratic Bernstein Bézier polynomials.

b1^2: [image] b2^2: [image] b3^2: [image] 2b1b2: [image] 2b1b3: [image] 2b2b3: [image]

Graphs of Cubic Polynomials

It's getting a little tedious to list all the graphs. (There are a total of 10 cubic basis functions.) So here are just three cubic functions that exhibit all the types that occur:

b1^3: [image] 3b1^2b2: [image] 6b1b2b3: [image]

Some more Graphs:

12b1^2b2b3: [image] 20b1^3b2b3: [image] 30b1^2b2^2b3: [image]


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