Path: utzoo!utgpu!watserv1!watcgl!rhbartels
From: rhbartels@watcgl.waterloo.edu (Richard Bartels)
Newsgroups: comp.graphics
Subject: Re: Approx. Bezier Surfaces w/Bi-cubic B-surfaces?
Keywords: nth degree Bezier Surfaces, Bi-cubic B-surfaces, approximation
Message-ID: <13734@watcgl.waterloo.edu>
Date: 10 Mar 90 23:00:16 GMT
References: <1120@etnibsd.UUCP>
Reply-To: rhbartels@watcgl.waterloo.edu (Richard Bartels)
Distribution: na
Organization: U. of Waterloo, Ontario
Lines: 31

In article <1120@etnibsd.UUCP>
dandelion!sean@apollo.com posts the question:
>
>[cubic Bezier surfaces are]
>...special cases of the bi-cubic B-spline surfaces.  Unless I'm
>mistaken, which is entirely possible, the others, the degree >= 4
>cases, are not so obvious.
>

There is nothing special about cubics.  If a B-spline has as many
knots as its order at every joint, it reduces to the Bezier basis
polynomial.  All Bezier curves and tensor-product surfaces of whatever
order are special cases of B-spline curves and tensor-product surfaces
of the same order (in the same sense that this is true of the cubic case).
Bezier's are simply B's with the highest possible knot multiplicities.

Perhaps the person originally posing the question had something else in
mind.  If the Bezier surface had higher continuity than a Bezier surface
has a right to enjoy, for example a cubic Bezier surface with continuity
in second derivatives between the patches, then the surface can be
re-represented in terms of B-splines with lower knot multiplicity,
for example cubic B-splines with single knots (the B-splines most
people think of as the "normal and reasonable" ones).  The way to
get from the Bezier representation to the B-spline representation
(or from the "high knot multiplicity B-splines" to the "low multiplicity
knot B-splines") is by, you guessed it, "removing knots".  We didn't
cover that one in the book, but there is a brief mention of the concept
in a recent article by Piegl and Tiller in IEEE CG&A on drawing circles
with rational splines.

-Richard