Section: User Commands (1)
Updated: 25 October 1991Index
Return to Main Contents
ppmforge - fractal forgeries of clouds, planets, and starry skies
generates three kinds of ``random fractal forgeries,'' the term coined
by Richard F. Voss of the IBM Thomas J. Watson Research Center for
seemingly realistic pictures of natural objects generated by simple
algorithms embodying randomness and fractal self-similarity. The
techniques used by
are essentially those
given by Voss, particularly the technique of spectral synthesis
explained in more detail by Dietmar Saupe.
The program generates two varieties of pictures: planets and clouds,
which are just different renderings of data generated in an identical
manner, illustrating the unity of the fractal structure of these very
different objects. A third type of picture, a starry sky, is
synthesised directly from pseudorandom numbers.
The generation of planets or clouds begins with the preparation of an
array of random data in the frequency domain. The size of this
array, the ``mesh size,'' can be set with the
option; the larger the mesh the more realistic the pictures but the
calculation time and memory requirement increases as the square of the
mesh size. The fractal dimension, which you can specify with the
option, determines the roughness of the terrain on the planet or the
scale of detail in the clouds. As the fractal dimension is increased,
more high frequency components are added into the random mesh.
Once the mesh is generated, an inverse two dimensional Fourier
transform is performed upon it. This converts the original random
frequency domain data into spatial amplitudes. We scale the real
components that result from the Fourier transform into numbers from 0
to 1 associated with each point on the mesh. You can further
modify this number by applying a ``power law scale'' to it with the
option. Unity scale
leaves the numbers unmodified; a power scale of 0.5 takes the square
root of the numbers in the mesh, while a power scale of 3 replaces the
numbers in the mesh with their cubes. Power law scaling is best
envisioned by thinking of the data as representing the elevation of
terrain; powers less than 1 yield landscapes with vertical scarps that
look like glacially-carved valleys; powers greater than one make
fairy-castle spires (which require large mesh sizes and high
resolution for best results).
After these calculations, we have a array of the specified size
containing numbers that range from 0 to 1. The pixmaps are generated as
A colour map is created that ranges from pure blue to white by
increasing admixture (desaturation) of blue with white. Numbers less
than 0.5 are coloured blue, numbers between 0.5 and 1.0 are coloured
with corresponding levels of white, with 1.0 being pure white.
The mesh is projected onto a sphere. Values less than 0.5 are treated
as water and values between 0.5 and 1.0 as land. The water areas are
coloured based upon the water depth, and land based on its elevation.
The random depth data are used to create clouds over the oceans. An
atmosphere approximately like the Earth's is simulated; its light
absorption is calculated to create a blue cast around the limb of the
planet. A function that rises from 0 to 1 based on latitude is
modulated by the local elevation to generate polar ice caps--high
altitude terrain carries glaciers farther from the pole. Based on the
position of the star with respect to the observer, the apparent colour
of each pixel of the planet is calculated by ray-tracing from the star
to the planet to the observer and applying a lighting model that sums
ambient light and diffuse reflection (for most planets ambient light
is zero, as their primary star is the only source of illumination).
Additional random data are used to generate stars around the planet.
A sequence of pseudorandom numbers is used to generate stars with a
user specified density.
Cloud pictures always contain 256 or fewer colours and may be
displayed on most colour mapped devices without further processing.
Planet pictures often contain tens of thousands of colours which
must be compressed with
before encoding in a colour mapped format. If the display resolution is
generally produces better looking planets.
tends to create discrete colour bands, particularly in the oceans,
which are unrealistic and distracting. The number of colours in starry
sky pictures generated with the
option depends on the value specified for
Small values limit the colour temperature distribution of the stars
and reduce the number of colours in the image.
is set to 0, none of the stars will be coloured and the resulting
image will never contain more than 256 colours.
Night sky pictures with many different star colours often look
best when colour compressed by
settings of 63, 31, or 15 with
to reduce the number of colours in the picture to 256 or fewer.
Generate clouds. A pixmap of fractal clouds is generated. Selecting clouds
sets the default for fractal dimension to 2.15 and power scale factor
- -dimension dimen
Sets the fractal dimension to the specified
which may be any floating point value between 0 and 3. Higher fractal
dimensions create more ``chaotic'' images, which require higher
resolution output and a larger FFT mesh size to look good. If no
dimension is specified, 2.4 is used when generating planets and 2.15
- -glaciers level
The floating point
setting controls the extent to which terrain elevation causes ice to
appear at lower latitudes. The default value of 0.75 makes the polar
caps extend toward the equator across high terrain and forms glaciers
in the highest mountains, as on Earth. Higher values make ice sheets
that cover more and more of the land surface, simulating planets in the
midst of an ice age. Lower values tend to be boring, resulting in
unrealistic geometrically-precise ice cap boundaries.
- -hour hour
When generating a planet,
is used as the ``hour angle at the central meridian.'' If you specify
for example, the planet will be fully illuminated, corresponding to
high noon at the longitude at the centre of the screen. You can
specify any floating point value between 0 and 24 for
but values which place most of the planet in darkness (0 to 4 and 20
to 24) result in crescents which, while pretty, don't give you many
illuminated pixels for the amount of computing that's required. If no
option is specified, a random hour angle is chosen, biased so that
only 25% of the images generated will be crescents.
- -ice level
Sets the extent of the polar ice caps to the given floating point
The default level of 0.4 produces ice caps similar to those of the Earth.
Smaller values reduce the amount of ice, while larger
settings create more prominent ice caps. Sufficiently large values,
such as 100 or more, in conjunction with small settings for
(try 0.1) create ``ice balls'' like Europa.
- -inclination|-tilt angle
The inclination angle of the planet with regard to its primary star is
which can be any floating point value from -90 to 90. The inclination
angle can be thought of as specifying, in degrees, the ``season'' the
planet is presently experiencing or, more precisely, the latitude at
which the star transits the zenith at local noon. If 0, the planet
is at equinox; the star is directly overhead at the equator.
Positive values represent summer in the northern hemisphere, negative
values summer in the southern hemisphere. The Earth's inclination
angle, for example, is about 23.5 at the June solstice, 0 at the
equinoxes in March and September, and -23.5 at the December solstice.
If no inclination angle is specified, a random value between -21.6 and
21.6 degrees is chosen.
- -mesh size
A mesh of
size by size
will be used for the fast Fourier transform (FFT). Note that memory
requirements and computation speed increase as the square of
if you double the mesh size, the program will use four times the
memory and run four times as long. The default mesh is 256x256, which
produces reasonably good looking pictures while using half a megabyte
for the 256x256 array of single precision complex numbers
required by the FFT. On machines with limited memory capacity, you
may have to reduce the mesh size to avoid running out of RAM.
Increasing the mesh size produces better looking pictures; the
difference becomes particularly noticeable when generating high
resolution images with relatively high fractal dimensions (between 2.2
A starry sky is generated. The stars are created by the same algorithm
used for the stars that surround planet pictures, but the output
consists exclusively of stars.
- -power factor
Sets the ``power factor'' used to scale elevations synthesised from
the FFT to
which can be any floating point number greater than zero. If no
factor is specified a default of 1.2 is used if a planet is being
generated, or 0.75 if clouds are selected by the
option. The result of the FFT image synthesis is an array of elevation
values between 0 and 1. A non-unity power factor exponentiates each
of these elevations to the specified power. For example, a power
factor of 2 squares each value, while a power factor of 0.5 replaces
each with its square root. (Note that exponentiating values between 0
and 1 yields values that remain within that range.) Power factors
less than 1 emphasise large-scale elevation changes at the expense of
small variations. Power factors greater than 1 increase the roughness
of the terrain and, like high fractal dimensions, may require a larger
FFT mesh size and/or higher screen resolution to look good.
- -saturation sat
Controls the degree of colour saturation of the stars that surround planet
pictures and fill starry skies created with the
option. The default value of 125 creates stars which resemble the sky
as seen by the human eye from Earth's surface. Stars are dim; only
the brightest activate the cones in the human retina, causing colour
to be perceived. Higher values of
approximate the appearance of stars from Earth orbit, where better
dark adaptation, absence of skyglow, and the concentration of light
from a given star onto a smaller area of the retina thanks to the lack
of atmospheric turbulence enhances the perception of colour. Values
greater than 250 create ``science fiction'' skies that, while pretty,
don't occur in this universe.
Thanks to the inverse square law combined with Nature's love of
mediocrity, there are many, many dim stars for every bright one.
This population relationship is accurately reflected in the skies
Dim, low mass stars live much longer than bright massive stars,
consequently there are many reddish stars for every blue giant. This
relationship is preserved by
You can reverse the proportion, simulating the sky as seen in a starburst
galaxy, by specifying a negative
- -seed num
Sets the seed for the random number generator to the integer
The seed used to create each picture is displayed on standard output (unless
suppressed with the
option). Pictures generated with the same seed will be identical. If no
is specified, a random seed derived from the date and time will be
chosen. Specifying an explicit seed allows you to re-render a picture
you particularly like at a higher resolution or with different viewing
- -stars fraction
Specifies the percentage of pixels, in tenths of a percent, which will
appear as stars, either surrounding a planet or filling the entire
is specified. The default
- -xsize|-width width
Sets the width of the generated image to
pixels. The default width is 256 pixels. Images must be at least as
wide as they are high; if a width less than the height is specified,
it will be increased to equal the height. If you must have a long
skinny pixmap, make a square one with
to extract a portion of the shape and size you require.
- -ysize|-height height
Sets the height of the generated image to
pixels. The default height is 256 pixels. If the height specified
exceeds the width, the width will be increased to equal the height.
All flags can be abbreviated to their shortest unique prefix.
The algorithms require the output pixmap to be at least as wide as it
is high, and the width to be an even number of pixels. These
constraints are enforced by increasing the size of the requested
pixmap if necessary.
You may have to reduce the FFT mesh size on machines with 16 bit
integers and segmented pointer architectures.
Voss, Richard F., ``Random Fractal Forgeries,'' in Earnshaw
et. al., Fundamental Algorithms for Computer Graphics, Berlin:
Peitgen, H.-O., and Saupe, D. eds., The Science Of Fractal Images,
New York: Springer Verlag, 1988.
Avenue des Champs-Montants 14b
- 038/33 88 15
- 038/33 76 33
Permission to use, copy, modify, and distribute this software and its
documentation for any purpose and without fee is hereby granted,
without any conditions or restrictions. This software is provided ``as
is'' without express or implied warranty.
If you like this kind of stuff, you may also enjoy ``James Gleick's
Chaos--The Software'' for MS-DOS, available for $59.95 from your
local software store or directly from Autodesk, Inc., Attn: Science
Series, 2320 Marinship Way, Sausalito, CA 94965, USA. Telephone:
(800) 688-2344 toll-free or, outside the U.S. (415) 332-2344 Ext
4886. Fax: (415) 289-4718. ``Chaos--The Software'' includes a more
comprehensive fractal forgery generator which creates
three-dimensional landscapes as well as clouds and planets, plus five
more modules which explore other aspects of Chaos. The user guide of
more than 200 pages includes an introduction by James Gleick and
detailed explanations by Rudy Rucker of the mathematics and algorithms
used by each program.
- SEE ALSO