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.
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