nurbs and bezier rendering

[Richard Bartels]

In article <5966 at eos.UUCP> prevost at eos.UUCP (Michael Prevost) writes:

>2) Is it possible to restrict the end points of nurbs.

If SGI really offers nurbs, then it allows you to specify the knot
structure of the nurb.  When you specify enough knots of equal value,
you get interpolation.  For example, a cubic Bezier segment is the same thing
as a cubic nurb segment with the nurb knots given as the sequence

	0.0, 0.0, 0.0, 0.0, 1.0, 1.0, 1.0, 1.0

If a cubic nurb curve had the knots

	0.0, 0.0, 0.0, 0.0, 1.0, 2.0, 3.5, 4.2, 6.8, 7.7, 7.7, 7.7, 7.7

for example, it would interpolate its endpoints (i.e. interpolate its
first and last control vertex) while floating near its intermediate
points (control vertices).

The same thing works in the interior of the curve.  If the knot 3.5
were replicated to 3.5, 3.5, 3.5, 3.5, then the curve would interpolate
one of the interior control vertices.  (Actually, only three 3.5's are
needed to interpolate.  The extra one would allow the curve to be
"torn apart" at the corresponding parametric point).

This means that a composite Bezier curve can be regarded as a
nurb curve with knots equal to, e.g.

	0.0, 0.0, 0.0, 0.0, 1.0, 1.0, 1.0, 2.0, 2.0, 2.0, 3.0, 3.0, 3.0, etc.
        etc. n-1, n-1, n-1, n, n, n, n

-Richard