iles@hplabs.UUCP (06/10/85)
Also check out (in the how to create pretty pictures catagory) the work coming out of Stony Brook in "experimental mathematics." It requires a moderate amount of understanding of the functions of a complex variable, but not too much, and you do get some very pretty pictures for minimal coding effort... Dan Lieman hplabs!iles
rich@sdcc13.UUCP (rich) (06/15/85)
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 --richrivero@kovacs.UUCP (Michael Foster Rivero) (06/19/85)
--------------------------------------- In regards to Rich Stewart's excerpted paper on fractal subdivision. > 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. > > 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. > > Close examination of the above will reveal that all three of the smaller triangles share a common side with the larger triangles. A simple diagram would be.... /|\ / | \ / | \ / | \ / | \ / | \ / ,' ', \ / ,' ', \ / ,' ', \ / ,' ' ,\ /,'__________________'\ Its main drawback is that even at the limits of recursion, there will always be triangles with one of the vertecies of the starting figure, albeit very long and very thin. This will produce images with long thin triangular artifacts. A better approach, (and the one I use) displaces the center of each vertex by a random amount. The new triangles are built from one existing vertex, and two displaced centerpoints, thus.... /\ / \ / \ / \ /________\ /\ /\ / \ / \ / \ / \ / \ / \ /________\/________\ As you can see, each triangle retains a fairly equalateral shape, and decreases in size in direct proportion to the level of recursion. By seeding the random displacement generator with a combination of all 6 end verticies, sides of adjoining triangles will produce identical midpoint displacements, eliminating "holes" in the final fractal form. This allows one to apply a psuedo-fractal surface to a predefined form. Sorry Rich! You're busted! Michael Rivero On a note of personal interest. I cut my graphics teeth with NASA. I already make a living off of drawing pretty pic- tures on computers. ( Lucas films, I will see you at SIGGRAPH!!!)
rich@sdcc12.UUCP (rich) (06/25/85)
In article <241@kovacs.UUCP> rivero@kovacs.UUCP (Michael Foster Rivero) writes: > > Sorry Rich! You're busted! > > > Michael Rivero > naw, I appreciate the help. thanks, rich ps. i am trying to get to the letters, I havent had the time yet. >
fournier@Navajo.ARPA (06/27/85)
*** REPLACE THIS LINE WITH YOUR FRACTAL *** Indeed the triangular subdivision you mention is better than the earlier one. It still has a big drawback though, in that the subtriangles created will be recursively subdivided without any correlation with the adjoining subtriangles. If you want to approximate fractional Brownian motion (one of the random fractal processes), this is incorrect since the 2-D version is non-Markovian, meaning that one side of the surface cannot be computed knowing only the boundary. A better subdivision scheme can be found in the Fournier/Fussell/Carpenter paper in CACM, June 1982, or the Piper/Fournier Siggraph paper of 1984. Of course a rejoinder by B. Mandelbrot and a reply can be found in the following CACM issue of August (or July). Several people are currently working on new schemes to generate 2-D approximations of fBm. There will be a tutorial at Siggraph on fractals, with Loren Carpenter and Benoit Mandelbrot together again for the first time. This should be fun! If anybody is interested in 3-D fractal (or approximations thereof), I can If anybody is interested in C code to generate 3-D fractal (and the code is short indeed, but no display is included), mail to me. It will take a few days, since I just got here and I have to bring most everything up anew. Remember that 3-D here means the process generate a volume, not a 2-D surface to be inbedded in a 3-D space.