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From: rich@sdcc13.UUCP (rich)
Newsgroups: net.graphics
Subject: Fractal mountains ( VERY LONG)
Message-ID: <262@sdcc13.UUCP>
Date: Sat, 15-Jun-85 12:54:16 EDT
Article-I.D.: sdcc13.262
Posted: Sat Jun 15 12:54:16 1985
Date-Received: Mon, 17-Jun-85 20:19:13 EDT
Organization: lack of
Lines: 157
Keywords: fractals


Note:
     I am posting this in response to  all  the  people  out
there  who  wrote me and said " please tell me what you find
out" when I originally asked for info on fractals.   I  hope
this  is the proper place for it. Send flames to yourselves.
This is an small portion of one of the papers I have written
since  starting  to play with fractals in March,1985. I have
actually done a lot more work with two dimensional  fractals
(reproduced  most  of  Mandelbrot's book plus more). Any way
the net seemed more interested in  mountains  so  here  they
are. They are still a bit crude, but the idea is there and I
have drawn a few this way. (They look a bit more like  rocks
than mountains right now)

     A fractal may be described as  a  self  similar  recur-
sively  defined geometric object. A fractal mountain, simply
enough, is a fractal object that  happens  to  look  like  a
mountain. The following , short, paper describes the methods
which I have used to generate such  objects.  The  paper  is
short  in  the hope of giving a general understanding of the
method. Following this will be the code,  with  all  of  the
ugly details.* The simplest way to describe a  mountain,  or
at least something that sort of looks like one is by drawing
it in a two dimensional coordinate system. This  is  simpler
to  start  with  since  it  is easier to visualize, and thus
easier to debug. So to begin with we will think of a  simple
,  recursive , subdivision of a triangle. The algorithm goes
as follows:



     1) Choose a triangle

     2) Choose a point, with a high probability of being
        inside the current triangle.

     3)  Create a new triangle by connecting the  new  point
     to the vertices of side A.

     4) Recurse on this new triangle to make new smaller
        triangles on it.

     5) Choose a point, with a high probability of being
        inside the current triangle.

     6)  Create a new triangle by connecting the new point
        to the vertices of side B.

_________________________
* For the most part things that I have done here,  have
been  done  simply  because  they have seemed like they
would work. And so it goes.










                           - 2 -


     7) Recurse on this new triangle to make new smaller
        triangles on it.

     8) Choose a point, with a high probability of being
        inside the current triangle.

     9)  Create a new triangle by connecting the  new  point
     to the vertices of side C.

     10) Recurse on this new triangle to make new smaller
         triangles on it.





     This simple algorithm actually  works.  It  is  just  a
shame  that it is not a "fractal" algorithm. A fractal form,
though seemingly random, contains no random points. This  is
easily  over  come though.  Simply define the new point as a
function of the old points  and  you  have  two  dimensional
fractal mountains.

     Now the problem of  creating  a  more  realistic  three
dimensional  image.  The  first consideration went using the
above algorithm to create a series of slices  with  a  given
thickness  in  three  dimensions.  I would then "glue" these
slices together in three space. The triangles that were ori-
ginally  used to define each slice get progressively smaller
as you approach the viewpoint. This method did not work  out
too well.

     The next seems to be the best method that I have  stum-
bled across, in three space:



     1) Choose a four faced pyramid. { A 90x does just fine }

     2) Subdivide face A a into three new faces by choosing a
        new point by means of some function.

     3) Recurse on the algorithm.

     4) Subdivide face B a into three new faces by choosing a
        new point by means of some function.

     5) Recurse on the algorithm.

     6) Subdivide face C a into three new faces by choosing a
        new point by means of some function.

     7) Recurse on the algorithm.









                           - 3 -


     8) Subdivide face D a into three new faces by choosing a
        new point by means of some function.

     9) Recurse on the algorithm.




     In all of these methods almost any parameterizing func-
tion  will  do  in choosing the new points. Though different
functions leave you with very differing functions.
   I hope that you have enjoyed this.

   -rich stewart

     On a note of  personal  interested.  I  graduate  March
1986.  I  am interested in living off of drawing pretty pic-
tures on computers.  Does anyone out there do this  or  know
of what type of places pay people for this.

{ Lucas films I am out here!!!  , hint hint}



ihnp4--        decvax--       akgua----dcdwest---somewhere--
ucbvax-------- sdcsvax -- sdcc3 --rich