Path: utzoo!utgpu!jarvis.csri.toronto.edu!mailrus!wuarchive!zaphod.mps.ohio-state.edu!uakari.primate.wisc.edu!aplcen!uunet!etnibsd!vsh
From: vsh@etnibsd.UUCP (Steve Harris)
Newsgroups: comp.graphics
Subject: Approx. Bezier Surfaces w/Bi-cubic B-surfaces?
Summary: Sources, algorithms requested
Keywords: nth degree Bezier Surfaces, Bi-cubic B-surfaces, approximation
Message-ID: <1120@etnibsd.UUCP>
Date: 8 Mar 90 20:56:56 GMT
Reply-To: dandelion!sean@apollo.com
Followup-To: comp.graphics
Distribution: na
Organization: Cognition, Inc., Billerica, MA
Lines: 39


	[This article is being posted (belatedly) for a friend.]
	[Please see the end of the article for his email address.]

Is anyone out there familiar with techniques for approximating Bezier
surfaces of degree 4-7 with B-spline surfaces of order 4 (degree == 3,
bi-cubic)?  Degree 2 and 3 Bezier surfaces can be easily approximated since
they 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.

One approach I'm considering is:

1) Generate the points on the Bezier surface (using the conventional
   approach).
2) Divide these points into sub-regions.
   a) What criteria should I use to pick the subregions?  Number of
      Control points, Tangency, curvature, other?
3) Create bicubic B-surfaces which approximate these regions.
   a) Anyone got any good ideas on how to accomplish this with finite
      workstation-type resources?

The ultimate goal here is to produce a B-surface which looks like the
Bezier used as input.  Other requirements, such as nth order continuity
at the joints, although desirable, are not necessities.

I've been looking through Faux & Pratt (Computational Geometry ...), but there
is not much mention of approximation techniques which produce b-splines.
Also I'm trying to get through Bartels, Beatty & Barsky (Intro. to
B-splines ...) but I figure it will be sometime in the middle of summer
before I've mastered the concepts presented there enough to derive the above
desiderata.  Can anyone out there suggest a source which might have a direct
bearing on this problem?  Can anyone offer a suggestion for where I may
have gone wrong in my analysis or my approach?  Any help appreciated.

  ____,.,_..,__.,_.,__     Sean Philip McElroy
   __'..__._,_.__.__.__    Cognition Corp.
   _,___`_.'__.__.__.__    755 Middlesex Turnpike, Billerica, MA 01821
  ___`..'_,___.__.__,_     dandelion!sean@apollo.com