i o u

thanks,again.

peter

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On Wed, 09 Feb 2000 19:43:19 -0800, SamuelT. wrote:
>It does repeat.
>
>Ron, your' guesses were excellent! You had the basic idea down, but not
>the procedure. I like your first try at it best, although it does have a
>diagonal tendancy, and I could not remove it. Best thing about yours is
>that it doesn't repeat. I didn't know metric did that to crackle.

One thing that might remove some of it would be to use my later unit-square
patterns (Pig3, Pig3 rotate 90*y) since they don't have the left/bottom bias
of the tiles I used in the first example.  None of my attempts repeat, btw.

I have to be honest: crackle solid metric 1 has some problems.  It doesn't
seem to do very well for a certain range of numbers, but I haven't stopped
to determine what that range is.  That's why the translate 20.

Finally, I leave you all with yet another maze pattern, of sorts. This is
an adaptation of an idea I first saw on the Commodore VIC 20 years and years
ago, and is the basic idea behind my second and third tries at the maze code:

#declare S=seed(12346);#declare Y=0;#while(Y<20)#declare X=0;#while(X<50)
#if (rand(S)<.5)#debug"/"#else#debug"\\"#end#declare X=X+1;#end#debug"\n"
#declare Y=Y+1;#end

-- 
These are my opinions.  I do NOT speak for the POV-Team.
The superpatch: http://www2.fwi.com/~parkerr/superpatch/
My other stuff: http://www2.fwi.com/~parkerr/traces.html

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"SamuelT." wrote:

>     It was made with MegaPov (Superpatch works too).

Oh! See, I was looking fot this "metric" thing and could not find it. What
does it do?

--
___     ______________________________________________________
 | \     |_                 <dav### [at] faricynet> <ICQ 55354965>
 |_/avid |ontaine               http://www.faricy.net/~davidf/

"Sitting on a cornflake, waiting for the van to come" -Beatles

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On Thu, 10 Feb 2000 15:47:25 -0600, David Fontaine wrote:
>"SamuelT." wrote:
>
>>     It was made with MegaPov (Superpatch works too).
>
>Oh! See, I was looking fot this "metric" thing and could not find it. What
>does it do?

It changes the distance metric used to determine which centroid is
closest.  metric 2 is the default, which corresponds to what's known
as the 2-norm, |V|_2=sqrt(V.x^2+V.y^2+V.z^2).  This is also usually
called the Pythagorean Theorem.  metric 1 is commonly called the 
"taxicab distance" as it's the distance a taxicab would drive if it 
had to follow city streets in a standard block pattern.
It's |V|_1=V.x+V.y+V.z.  In general, POV raises each component of
the vector to the specified power, then adds them together, then 
raises them to the reciprocal power (i.e. 1/2 if the metric was 2.)

In the limit as the metric approaches infinity, the norm is equal to
whichever of V.x, V.y, or V.z is largest.  In the space between 2 and
infinity, neat things can happen.

Now ask me about form. :)

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>Now ask me about form. :)

What about it? Tell me more...

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Hey Teach,

What is form?

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On Thu, 10 Feb 2000 18:22:51 -0500, TonyB wrote:
>>Now ask me about form. :)
>
>What about it? Tell me more...

form is used to set the equation crackle uses to calculate the value
of the pattern at any given point.  The default form is <-1,1,0>, which
returns the difference between the second-closest point and the closest
point.  Form takes the distance to the closest point and multiplies it
by the first term, the distance to the second-closest point by the second
term, and the distance to the third-closest point by the third term, then
adds them all up.  Messing with form can give completely different patterns.
form <1,0,0>, for example, gives you something that looks kinda like a 
bunch of bacteria.

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ron### [at] povrayorg (Ron Parker) writes:

> On Thu, 10 Feb 2000 15:47:25 -0600, David Fontaine wrote:
> >"SamuelT." wrote:
> >Oh! See, I was looking fot this "metric" thing and could not find it. What
> >does it do?
> 
> It changes the distance metric used to determine which centroid is
> closest.  metric 2 is the default, which corresponds to what's known
> as the 2-norm, |V|_2=sqrt(V.x^2+V.y^2+V.z^2).  This is also usually
> called the Pythagorean Theorem.  metric 1 is commonly called the 
> "taxicab distance" as it's the distance a taxicab would drive if it 
> had to follow city streets in a standard block pattern.
> It's |V|_1=V.x+V.y+V.z.  

You probably meant |V|_1 = |V.x| + |V.y| + |V.z|. In Germany they call this
the Manhattan metric, because standard block pattern aren't standard
here.

Another nice metric I know is the metric of the French railway:
You always have to travel via Paris. So, the distance between two points
A and B is the distance from a A to Paris plus the distance from Paris
to B. There is an exception though: If A, B and Paris lie on a straight
line, it's the distance between A and B.

Formally:

d(A,B) = |A-B|        if P in (A,B)
         |A-P|+|P-B|  otherwise
where | | denotes the Euclidean norm.

Perhaps we should start to distinguish between "metric" and "norm".
"metric" denotes something that can be interpreted as a distance
between to points. "norm" denotes something that can be interpreted
as the distance of a point to the origin in a vector space.

These notions are interconnected though. If you have a norm | |, you can
easily create a metric d by defining d(A,B) as d(A,B):=|A-B|. However,
this doesn't work the other way round. If you are given a norm, you
can't -- generally -- define a metric such that the equation above holds.
In particular, this isn't possible for the "metric of the French railway".
(For seeing this, we must define "norm" and "metric" more precisely.
If someone really wants to know, please ask.)

Thomas

-- 
http://thomas.willhalm.de/ (includes pgp key)

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ron### [at] povrayorg (Ron Parker) writes:

> On Thu, 10 Feb 2000 18:22:51 -0500, TonyB wrote:
> >>Now ask me about form. :)
> >
> >What about it? Tell me more...
> 
> form is used to set the equation crackle uses to calculate the value
> of the pattern at any given point.  The default form is <-1,1,0>, which
> returns the difference between the second-closest point and the closest
> point.  Form takes the distance to the closest point and multiplies it
> by the first term, the distance to the second-closest point by the second
> term, and the distance to the third-closest point by the third term, then
> adds them all up.  Messing with form can give completely different patterns.
> form <1,0,0>, for example, gives you something that looks kinda like a 
> bunch of bacteria.

I assume that you know:

Steven Worley
A Cellular Texture Basis Function
Siggraph 1996

He has created some very nice textures by adding crackle patterns of 
different scale. By this, the pattern becomes some sort of fractal.

Thomas


-- 
http://thomas.willhalm.de/ (includes pgp key)

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On 11 Feb 2000 08:54:52 +0100, Thomas Willhalm wrote:
>ron### [at] povrayorg (Ron Parker) writes:
>
>> On Thu, 10 Feb 2000 15:47:25 -0600, David Fontaine wrote:
>> >"SamuelT." wrote:
>> >Oh! See, I was looking fot this "metric" thing and could not find it. What
>> >does it do?
>> 
>> It changes the distance metric used to determine which centroid is
>> closest.  metric 2 is the default, which corresponds to what's known
>> as the 2-norm, |V|_2=sqrt(V.x^2+V.y^2+V.z^2).  This is also usually
>> called the Pythagorean Theorem.  metric 1 is commonly called the 
>> "taxicab distance" as it's the distance a taxicab would drive if it 
>> had to follow city streets in a standard block pattern.
>> It's |V|_1=V.x+V.y+V.z.  
>
>You probably meant |V|_1 = |V.x| + |V.y| + |V.z|. In Germany they call this
>the Manhattan metric, because standard block pattern aren't standard
>here.

Whoops, you're right.  Thanks for the correction.

>Perhaps we should start to distinguish between "metric" and "norm".
>"metric" denotes something that can be interpreted as a distance
>between to points. "norm" denotes something that can be interpreted
>as the distance of a point to the origin in a vector space.

Yes, my description was a little sloppy there.  I was thinking of the
closest centroid as the origin in each case, but of course it isn't.

-- 
These are my opinions.  I do NOT speak for the POV-Team.
The superpatch: http://www2.fwi.com/~parkerr/superpatch/
My other stuff: http://www2.fwi.com/~parkerr/traces.html

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