Fractal Landscapes 1

Fractal Landscapes

Realistic landscapes can be generated with the midpoint displacement algorithm (see Section 9.6 in PJS). In one type, one begins with a flat square grid and an arbitrary (positive) displacement of the four corners. Then at each stage the midpoints of these corners and the centre point are displaced by an amount = avg + scale*random where avg is the average value of the adjacent corners, scale decreases at each stage (say, by a factor of 2) and random is a random number. At the next stage, the midpoints between these new points and their centres are displaced. This continues until the midpoints are spaced at every grid point. Different choices for the distribution of the random numbers, the scaling, the amplitude of the scaling, and the type of grid (eg., square, triangular, etc), will affect the appearance of the resulting landscape. See the Fractal Landscapes Variation page for examples.

For example, the following images show the successive stages of the midpoint displacement algorithm for generating fractal landscapes (or more precisely, fractal mountains) where the initial grid is 65x65 square and the initial displacements of the four corners was zero. Click on the image for a larger view.

These were drawn with a MATLAB program, which you can download here.

See the Fractal Landscapes Variation page for examples of variations.

See also The Science of Fractal Images, edited by H.-O. Peitgen and D. Saupe, Chapters 1 and 2 for more detailed discussions.