Path: utzoo!utgpu!news-server.csri.toronto.edu!dgp.toronto.edu!stam
Newsgroups: comp.graphics
From: stam@dgp.toronto.edu (Jos Stam)
Subject: Re: Need info on Peano curves
Message-ID: <1991Mar11.182809.28921@jarvis.csri.toronto.edu>
References: <1991Mar11.202541.2292@ms.uky.edu>
Date: 11 Mar 91 23:28:09 GMT
Lines: 37

hucaby@mri.uky.edu (David Hucaby) writes:

>Does anybody have references on how to generate Peano curves
>via computer? I've found the original Peano paper, where he derives
>his space-filling curve theoretically, but haven't come across
>any modern papers so far.

You can generate finer and finer approximations of the Peano curve recursively
as follows:

start with any segment: --------------------------

At each step recursively replace each segment by:

           --------
          |        |
          |        |
          |        |
  -------- -------- --------
          |        |
          |        |
          |        |
           --------

From this construction it is also easy to show that the fractal dimension
of the curve is 2.

(Think I've seen this construction in the "Science of Fractals", 
Springer-Verlag)

To draw the limit curve just fill the whole screen :-)

It can be shown that such curves cannot be bijective and continuous at the same
time, hence a line is not homeomorphic to a plane. This intuitive sound result 
is however extremely hard to prove...

Jos