Path: utzoo!utgpu!attcan!uunet!oddjob!ncar!ames!pasteur!ucbvax!decwrl!labrea!csli!rustcat
From: rustcat@csli.STANFORD.EDU (Vallury Prabhakar)
Newsgroups: comp.graphics
Subject: Re: Bezier Surface computations
Message-ID: <4953@csli.STANFORD.EDU>
Date: 9 Aug 88 01:13:21 GMT
References: <618@mailrus.cc.umich.edu>
Reply-To: rustcat@csli.UUCP (Vallury Prabhakar)
Distribution: comp.graphics
Organization: Stanford University
Lines: 33

In article <618@mailrus.cc.umich.edu> shane@um.cc.umich.edu (Shane Looker) writes:

# There was also a nice section on converting the functions to using only
# integer math (it implies) and using function step differences to compute
# the next value of the function.  There is an explination of how a 4 point
# Bezier curve is a cubic polynomial of the form:
#          f(u) = a[0]*u^3 + a[1]*u^2 + a[2]*u + a[3]
# 
# The real problem with this nice bit of math is that it NEVER gives a clue as
# to what the coefficients a[0]...a[3] are related to!!!  I spend over an hour
# yesterday trying to figure out how to derive these and how to write the
# necessary translation functions for x, y, and z.
# 
# The book was:
#   Computer Graphics           (how do they come up with these clever names?)
#   Hearn and Baker.            (From Illinois)
# 

[ I tried to get this through via e-mail, but it bounced back to me. ]

This is a very simple analysis covered in almost every geometric modelling
book I've come across.  

The relationship given above is the representation of a generic parametric
cubic curve.  An equivalence between this form and any spline form can be
established.  In the case of the 4-point Bezier curve, the coefficients 
may be determined using forward-vector difference methods (de Casteljau's
method).  This is clearly covered in Geometric Modelling by Michael 
Mortenson.  Send me e-mail if you still have problems figuring it out.


						-- Vallury Prabhakar
						-- rustcat@csli.stanford.edu