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
   videotron.ca

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Thanks Francois, that's exactly what I wanted to know. Except for the NURBs.
If anyone knows the Equations for NURBS, I'd be very grateful.

Thanks,

Nekar Xenos

Francois Labreque <fla### [at] videotronca> wrote in message
news:3A683D50.644E85D5@videotron.ca...
>
>
> 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
>    videotron.ca

Post a reply to this message

J-Print News wrote:
> Thanks Francois, that's exactly what I wanted to know. Except for the
NURBs.
> If anyone knows the Equations for NURBS, I'd be very grateful.

"The NURBS Book" by Les Piegl and Wayne Tiller (New York: Springer, second
edition 1997) is the place to start, if you can find it.

--
Lance.

http://come.to/the.zone

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J-Print News wrote:

> Thanks Francois, that's exactly what I wanted to know. Except for the NURBs.
> If anyone knows the Equations for NURBS, I'd be very grateful.

NURBS = Non Uniform Rational B-Splines, aka Bezier Patches, Bicubic Patches,
etc.  It's all the same stuff.
...Chambers

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> NURBS = Non Uniform Rational B-Splines, aka Bezier Patches, Bicubic
Patches,
> etc.  It's all the same stuff.
> ...Chambers
>

Thanks. I suspected it was something like a 3d bezier. Seems like I wasn't
far off.

Nekar

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In article <3A692D00.35295741@hotmail.com>, Ben Chambers 
<bdc### [at] hotmailcom> wrote:

> NURBS = Non Uniform Rational B-Splines, aka Bezier Patches, Bicubic 
> Patches, etc.  It's all the same stuff.

Depends on what you mean by "same stuff"...they are *not* identical as 
far as I know, but they are all spline based objects. "Bezier patch" and 
"bicubic patch" seem to be used interchangeably, though. And MegaPOV 
supports a "rational bezier patch" object.

-- 
Christopher James Huff
Personal: chr### [at] maccom, http://homepage.mac.com/chrishuff/
TAG: chr### [at] tagpovrayorg, http://tag.povray.org/

<><

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Another question. What exactly is meant by the term 'patch'?

Thanks

Nekar
Chris Huff <chr### [at] maccom> wrote in message
news:chrishuff-D120DF.11515920012001@news.povray.org...
> In article <3A692D00.35295741@hotmail.com>, Ben Chambers
> <bdc### [at] hotmailcom> wrote:
>
> > NURBS = Non Uniform Rational B-Splines, aka Bezier Patches, Bicubic
> > Patches, etc.  It's all the same stuff.
>
> Depends on what you mean by "same stuff"...they are *not* identical as
> far as I know, but they are all spline based objects. "Bezier patch" and
> "bicubic patch" seem to be used interchangeably, though. And MegaPOV
> supports a "rational bezier patch" object.
>
> --
> Christopher James Huff
> Personal: chr### [at] maccom, http://homepage.mac.com/chrishuff/
> TAG: chr### [at] tagpovrayorg, http://tag.povray.org/
>
> <><

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J-Print News wrote:
> 
> I'd like to find out more in depth info about the equations for Iso
> surfaces, NURBS and beziers.
> For instance, how copuld I use  them in BASIC?

Slowly?

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"J-Print News" <vir### [at] iconcoza> wrote in message
news:3a6bd499@news.povray.org...
> Another question. What exactly is meant by the term 'patch'?

Patches are like 2D areas which can be curved in a 3D way.  Like parts of a
patchwork quilt.
Triangles and polygons could be said to be patches also I suppose, except you
can't bend a triangle physically and the usual polygon is flat too (although
points could leave the 2D plane).
Hopefully I'm not misinforming too much.

Bob H.

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J-Print News wrote in message <3a6bd499@news.povray.org>...
>Another question. What exactly is meant by the term 'patch'?


A patch object is a finite object without a well-defined interior.

--
Mark

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