- is used to place the spheres in this image.

A NURBS surface is a made with a bivariate function.
I.e. the value of the functions varies according
to 2 variables; e.g.  u and v

(A curve is a univariate function. It's value
varies along with a single variable; e.g.  t)

The value of a trivariate function varies along with
3 variables; e.g.   u, v and w   or   x, y and z

So a trivariate function can be used to build solid
shapes in 3D (that has a volume; like we do with
isosurfaces).

The image enclosed shows a quick try at doing this
with a trivariate NURBS function that controls
the positions of spheres in a 3D grid. (The tone
range in the image is adjusted after rendering.)

But trivariates can also be used to control other
things, like e.g. density, pigments, light,
directions of movement/acceleration, refraction
index, temperature, humidity, vector fields
(gradients) and so on...

Later I'll try to find time to show how trivariate
B-splines can be used to do a kind of 3D morphing
of shapes made from meshes.

I also hope that POV-Ray will render fast enough
to calculate some interesting isosurfaces made
with (simple) trivariate B-splines.

Higher dimensional multivariate B-spline functions
are interesting too, but I fear that the slow
speed of POV's parsing will prevent us from
exploring the possibilities that they provide.


Tor Olav

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