Path: utzoo!attcan!utgpu!jarvis.csri.toronto.edu!rutgers!uwvax!rennet.cs.wisc.edu!stuart
From: stuart@rennet.cs.wisc.edu (Stuart Friedberg)
Newsgroups: comp.graphics
Subject: Re: Bezier curves
Summary: Pointers to Bezier and B-spline literature
Message-ID: <8358@spool.cs.wisc.edu>
Date: 12 Sep 89 16:55:24 GMT
References: <273@marvin.moncam.co.uk>
Sender: news@spool.cs.wisc.edu
Reply-To: stuart@rennet.cs.wisc.edu (Stuart Friedberg)
Organization: U of Wisconsin CS Dept
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kaush@moncam.co.uk (Kaush Kotak) writes:
>Help! I need info on how to split a single (cubic) bezier curve into two. 
>Given a t-value & 4 control points, how do I find values for the new control 
>points?

There is probably a more direct method, but you can convert your Bezier
curve to a B-spline, find the point at the t-value, and add a knot
there with multiplicity 4 (for a cubic curve).  The result is two
concatenated curves joined at the knot.

I used the articles listed below as sources for most of my spline and
Bezier curve manipulation routines.  The first one provides an
algorithm to introduce a new knot to a spline.  (Let me distinguish
knot vertex <x,y> from knot parameters <t>.)

You can convert a B-spline to a (sequence of) Bezier curve(s) by
expanding each knot parameter to multiplicity 4 (for cubic curves).
The control points for the (concatenated) Bezier segment(s) are the new
knot vertices, 4 for each Bezier curve.  The t parameter ranges
between the two knot parameter values at either end.

If I remember correctly (and I may be mistaken here), you can convert a
Bezier curve to a B-spline trivially by treating every control point as
a knot vertex, and use one knot parameter of multiplicity 4 (for cubic
curves) with the value for the beginning of the curve segment (typically 0).

  Discrete B-Splines and Subdivision Techniques in Computer-Aided
    Geometric Design and Computer Graphics
  Elaine Cohen, Tom Lyche, and Richard Riesenfeld
  Computer Graphics and Image Processing, Vol. 14, No. 2, pp.  87-111, 1980

  On Calculating with B-Splines
  Carl deBoor
  Journal of Approximation Theory, Vol. 6, No. 1(?), pp. 50-62, 1972

  Rational B-Splines for Curve and Surface Representation
  Wayne Tiller
  IEEE Computer Graphics and Applications, Vol. 3, No. 6, pp. 61-69, 1983

  Automatic Construction of a Cubic B-Spline Representation
    for a General Curve
  O. Lozover & K. Preiss
  Computers and Graphics, Vol. 7, No. 2, pp. 149-153, 1983

  Offsets of Two-Dimensional Profiles
  Wayne Tiller and Eric G. Hanson
  IEEE Computer Graphics and Applications, Vol. 4, No. 9, pp. 36-46, 1984

  Computing Offsets of B-Spline Curves
  Sabine Coquillart
  Computer-Aided Design, Vol. 19, No. 6, pp. 305-309, 1987
  /* three dimensional case */