shane@chianti.cc.umich.edu (Shane Looker) (08/08/88)
Well, yesterday I was reading my textbook on graphics (3D stuff), and
was going through the section on Bezier lines and surfaces. I implimented
the basic functions with no problem, and as I suspected, they are real
dogs computationally.
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)
If anybody can tell me the answers to my questions, you will get a good deal
of thanks.
Shane Looker
Looker@um.cc.umich.edurustcat@csli.STANFORD.EDU (Vallury Prabhakar) (08/09/88)
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