Re: LOGO-L> Arcs once again

["Olga Tuzova" <olgatu@ort.spb.ru>]

George,

As always, I'm feeling as if I'm trying to catch up the running away 
train. May be, it's just the time difference. :-)
 
> Date:          Wed, 06 May 1998 14:59:57 -0400
> From:          George Mills <mills@softronix.com>
> To:            Olga Tuzova <olgatu@ort.spb.ru>
> Cc:            logo-l@gsn.org, cpthook@global2000.net
> Subject:       Re: LOGO-L> Arcs once again
> Reply-to:      George Mills <mills@softronix.com>

> Sorry Olga, I could not resist. Here is a a true Hilbert "Curve".
> 
> Note, that all I did was take someones Hilbert Curve
> http://www.xylem.demon.co.uk/hilbert.htm
> and replace FD with FDA (Forward Arc).

It's a good challenge.
I'm not well familiar with space filling curves and I'd rather go to 
the library and read something on this question, but as I haven't 
done this yet, I'm putting there the questions, for which I can't 
find answers.

The space filling curves I've seen before could fill a square and any 
iteration of the curve never leaves it. To fill a square I may just 
increase the level of iteration and don't have to change a size 
parameter.
It looks like this Hilbert curve sooner or later will cross the 
boundaries of any large square. Is it so?

Apparently, there are characteristics for the curves  which allow to 
conclude whether the curve is space filling or not. I don't know 
them. Looking through the web sites I found some images which are 
considered to be iterations for space filling curves without proofs. 
Nevertheless they look fine and I tried to design Logo codes for 
one of them (http://www.math.utk.edu/~morwen/fill.html ).

I hope, they describe the curve perfectly and I took it for granted 
that it's space filling.

And, of course, I couldn't resist the temptation to make a "convex" 
version of it. Though, in this case, I'm afraid, the curve is loosing 
it's property to fill the space. In the attachment, if it isn't lost, 
there are iterations of the "straight" (red) and "convex" (blue) 
curves.

Regards,
Olga.
-----------------------------------------
to filling.curve :size :level :p
if :level<1 [fd :size rt :p*90 fd :size stop]
filling.curve :size/2 :level-1 -:p rt :p*90
filling.curve :size/2 :level-1 :p  lt :p*90
filling.curve :size/2 :level-1 :p  rt :p*90
filling.curve :size/2 :level-1 -:p 
end

to convex.filling.curve :size :level :p
if :level<1 [fda :size rt :p*90 fda :size stop]
convex.filling.curve :size/2 :level-1 -:p rt :p*90
convex.filling.curve :size/2 :level-1 :p  lt :p*90
convex.filling.curve :size/2 :level-1 :p  rt :p*90
convex.filling.curve :size/2 :level-1 -:p 
end

to fda :size
lt 90
arc2 180 :size/2
lt 90
end
-----------------------------------------


> 
> to fda :dist
> lt 90
> arc2 180 :dist/2
> lt 90
> end
> 
> to go
> cs
> pu
> setxy 250 -250
> pd
> hilbert 20 5 45
> end
> 
> to hilbert :size :level :parity
> if :level > 0 ~
>   [
>   lt :parity*90 hilbert :size :level-1 :parity*-1
>   fda :size
>   rt :parity*90 hilbert :size :level-1 :parity
>   fda :size
>   hilbert :size :level-1 :parity rt :parity*90
>   fda :size
>   hilbert :size :level-1 :parity*-1 lt :parity*90
>   ]
> end
> 
> -- 
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