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?

Thanks

Nekar

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J-Print News <vir### [at] iconcoza> 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?

All three are surfaces created by well-defined mathematical rules, and so
should be "useable" in any programming language that offers the necessary
mathematical functions.  The question, however, is what you mean by "use".
Do you want to draw representations of them on screen?  If so, do you want
to raytrace them, or tesselate them to draw wireframe images?  Do you want
to calculate properties of the surfaces?

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> All three are surfaces created by well-defined mathematical rules, and so
> should be "useable" in any programming language that offers the necessary
> mathematical functions.  The question, however, is what you mean by "use".
> Do you want to draw representations of them on screen?  If so, do you want
> to raytrace them, or tesselate them to draw wireframe images?  Do you want
> to calculate properties of the surfaces?
>

I'd like to know the mathematical functions to do all of the above.

Nekar

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J-Print News <vir### [at] iconcoza> wrote:
: I'd like to know the mathematical functions to do all of the above.

  You are asking for much. Both, raytracing and tesselating of isosurfaces
is a quite complicated process, quite impossible to explain briefly.

-- 
char*i="b[7FK@`3NB6>B:b3O6>:B:b3O6><`3:;8:6f733:>::b?7B>:>^B>C73;S1";
main(_,c,m){for(m=32;c=*i++-49;c&m?puts(""):m)for(_=(
c/4)&7;putchar(m),_--?m:(_=(1<<(c&3))-1,(m^=3)&3););}    /*- Warp -*/

Post a reply to this message

Warp <war### [at] tagpovrayorg> wrote in message news:3a66d364@news.povray.org...
> J-Print News <vir### [at] iconcoza> wrote:
> : I'd like to know the mathematical functions to do all of the above.
>
>   You are asking for much. Both, raytracing and tesselating of isosurfaces
> is a quite complicated process, quite impossible to explain briefly.
>
> --
> char*i="b[7FK@`3NB6>B:b3O6>:B:b3O6><`3:;8:6f733:>::b?7B>:>^B>C73;S1";
> main(_,c,m){for(m=32;c=*i++-49;c&m?puts(""):m)for(_=(
> c/4)&7;putchar(m),_--?m:(_=(1<<(c&3))-1,(m^=3)&3););}    /*- Warp -*/ -
Yup this looks quite complicated  :)


Not raytracing. I just need the mathematical functions to work out if a
point is on the surface.

Sorry for the misunderstanding.

Thanks

Nekar

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J-Print News <vir### [at] iconcoza> wrote:
: Not raytracing. I just need the mathematical functions to work out if a
: point is on the surface.

  That's easy.
  An isosurface is defined by a function f(x,y,z) and a threshold value 'n'
(which is constant).
  A point <x,y,z> is inside the object if the value of f at that point is
less than 'n'.
  A point <x,y,z> is outside the object if the value of f at that point is
greater than 'n'.
  A point <x,y,z> is exactly in the surface of the object if the value of
f at that point is 'n'.

-- 
char*i="b[7FK@`3NB6>B:b3O6>:B:b3O6><`3:;8:6f733:>::b?7B>:>^B>C73;S1";
main(_,c,m){for(m=32;c=*i++-49;c&m?puts(""):m)for(_=(
c/4)&7;putchar(m),_--?m:(_=(1<<(c&3))-1,(m^=3)&3););}    /*- Warp -*/

Post a reply to this message

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.

Phew! hope we find each other on this one  :)

Nekar

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.

Post a reply to this message

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

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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.

"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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