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J-Print News wrote:
>
> I've been waiting for replies and it seems I posted this message to Warp
> instead of the board. Sorry about that, Warp.
>
> That's equations I'm looking for, not functions. Stuff like y=mx^2 +c, etc.
> I'm not so good on the terminology. Equations to work out the x, y, and z
> coordinates for Beziers, NURBS and isosurfaces is what I want. I've never
> heard of 3d beziers but I don't think that should be impossible.
Bezier curves use a set of control points that act like magnets on a
piece of metallic string. Points on the curve are defined by the
following equation (For n+1 control points):
[use a fixed-sized font]
n
---
\
P(u) = / Pi Bi,n(u) 0<= u <=1
---
i=0
Where Pi is a control point and Bi,n(u) is the Berstein polynomial,
which is given by:
i n-i
Bi,n (u) = C(n,i)u (1-u)
where C(n,i) is a coefficient given by:
n!
C(n,i) = -------
i!(n-i)!
[Ref: I. Zeid, CAD/CAM Theory and Practice, 1991. McGraw-Hill]
I don't know how NURBS work, but you should be able to find that out in
a book on computer algorithms at your local (or college) library.
And isosurfaces are any equation you want, anything is possible.
> PS. Does anyone know the equation for the Mandelbrot pattern. I think it's
> something like y=mX+c with c being the sqare root of -1 but I can't remember
> how it is used.
It's an infinite loop.
Zn = Z(n-1)^2 + c
Where Zo is 0 and C the point (in the complex plane) whose colour you
want to find out. If, after n iterations, you are converging towards 0,
the point is inside the Mandelbrot set; on the other hand if you are
diverging towards infinity, you are outside of it. Usually
image-generating programs will assign a color to the pixel based on how
many iterations it took before the point gets outside a threshold (for
the regular "ladybug" Mandelbrot set, it's a circle of radius 2)
[Ref: T. wegner & B. Tyler, Fractal Creations, 2nd ed. 1994, Waite
Group]
Hope this helps.
--
Francois Labreque | In the future, performance will be measured
flabreque | by the size of your pipe.
@ | - Dogbert, on networking
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