Paul Bourke : Fractal Landscapes
Introduction
Fractal  landscapes  are  often  generated  using  a  technique  called  spatial
subdivision. For magical  reasons this results  in surfaces that  are similar in
appearance to the earths terrain. 

The idea  behind  spatial  subdivision is quite simple. Consider a square on the 
x-y plane,

   (1) split the square up into a 2x2 grid 
   (2) vertically perturb each of the 5 new vertices by a random amount 
   (3) repeat this process for each new square decreasing the pertubation each 
       iteration.
	   
               

The controls normally available when generating such landscapes are: 

   * A  seed for  the random  number generator.  This starts  the rundom  number
   generator and means that the  same landscape can be recreated  by remembering
   only one number. 

   *  A  roughness  parameter.  This  is  normally  the  factor  by  which   the
   perturbations are  reduced on  each iteration.  A factor  of 2  is the  usual
   default, lower values result in a rougher terrain, higher values result in  a
   smoother surface. 

   *  The  initial perturbation  amount.  This set  the  overall height  of  the
   landscape. 

   *  Initial points.  It is  often desirable  to specify  some initial  points,
   normally on the corners of the initial rectangles. This provides some  degree
   of control over the macro appearance of the landscape. 

   * Sea level. This  "flood" the terrain to  a particular level simulating  the
   water level. 

   * Colour ramp. This is used for  shading of the terrain surface based on  the
   height. Normally two or three colours are defined for particular heights, the
   surface at other heights is linearly interpolated from these points. 

   * Number of iterations. This results in the density of the mesh that  results
   from the iteration process.
   

FracHill

An application that creates fractal landscapes has been written by myself called
FracHill. It fully implements fractal terrain generation including control  over
(x,y)  range,  sea colour,  background  colour, terrain  colour  ramp, lighting,
rendering options,  camera attributes,  grid density,  9 initial  points, height
variation, and roughness. 

This  application  runs on  any  Macintosh computer  with  colour QuickDraw.  It
supports image  saving to  PICT files  and colour  printing through the standard
Apple printing mechanism. 

It also has the ability to  geometrically morph between any two terrain  models,
images  or  models  can  be automatically  exported  and  formed  into animation
sequences. 

FracHill was  primarily written  as a  creator of  terrain models  for other  3D
modelling and  rendering packages  and thus  provides the  ability to export the
land surface  in a  number of  CAD formats  so that  it can  be imported into 3D
modelling packages. 

At the time of writing it exports to the following file formats 

DXF
Siper3D
Radiance
RayShade
POV-Ray
points

The following shows a  terrain surface at various  grid resolutions from 2x2  to
32x32. 

                   
                   
                   
                   

In  addition to  wireframe views,  FracHill performs  other types  of rendering
including
          hiddenline
		  coloured
		  shaded
		  
Some examples directly from FracHill are shown below 

                   

A well known artifact (bug) with  terrains generated this way is the  appearance
of "seams" or "creases". These generally  occur along the edges of the  geometry
associated with  early iterations.  A clear  example is  shown below, notice the
crease in the bottom left part of the terrain highlighted by the shadow zone:

                

References

Heinz-Otto and Dietmar Saupe, The Science of Fractal Images, Springer-Verlag

The Synthesis and  Rendering or  Eroded Fractal Terrains, F.Kenton Musgrave,
Craig E Kolb, Robert S. Mace, IEEE Computer Graphics & Applications