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ppppppppmmmmffffoooorrrrggggeeee((((1111)))) XXXXEEEENNNNIIIIXXXX SSSSyyyysssstttteeeemmmm VVVV ((((22225555 OOOOccccttttoooobbbbeeeerrrr 1111999999991111)))) ppppppppmmmmffffoooorrrrggggeeee((((1111)))) NNNNAAAAMMMMEEEE ppmforge - fractal forgeries of clouds, planets, and starry skies SSSSYYYYNNNNOOOOPPPPSSSSIIIISSSS ppppppppmmmmffffoooorrrrggggeeee [----cccclllloooouuuuddddssss] [----nnnniiiigggghhhhtttt] [----ddddiiiimmmmeeeennnnssssiiiioooonnnn _d_i_m_e_n] [----hhhhoooouuuurrrr _h_o_u_r] [----iiiinnnncccclllliiiinnnnaaaattttiiiioooonnnn||||----ttttiiiilllltttt _a_n_g_l_e] [----mmmmeeeesssshhhh _s_i_z_e] [----ppppoooowwwweeeerrrr _f_a_c_t_o_r] [----ggggllllaaaacccciiiieeeerrrrssss _l_e_v_e_l] [----iiiicccceeee _l_e_v_e_l] [----ssssaaaattttuuuurrrraaaattttiiiioooonnnn _s_a_t] [----sssseeeeeeeedddd _s_e_e_d] [----ssssttttaaaarrrrssss _f_r_a_c_t_i_o_n] [----xxxxssssiiiizzzzeeee||||----wwwwiiiiddddtttthhhh _w_i_d_t_h] [----yyyyssssiiiizzzzeeee||||----hhhheeeeiiiigggghhhhtttt _h_e_i_g_h_t] DDDDEEEESSSSCCCCRRRRIIIIPPPPTTTTIIIIOOOONNNN ppppppppmmmmffffoooorrrrggggeeee generates three kinds of ``random fractal for- geries,'' the term coined by Richard F. Voss of the IBM Tho- mas J. Watson Research Center for seemingly realistic pic- tures of natural objects generated by simple algorithms em- bodying randomness and fractal self-similarity. The tech- niques used by ppppppppmmmmffffoooorrrrggggeeee are essentially those given by Voss[1], particularly the technique of spectral synthesis explained in more detail by Dietmar Saupe[2]. The program generates two varieties of pictures: planets and clouds, which are just different renderings of data generat- ed 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 prepara- tion of an array of random data in the frequency domain. The size of this array, the ``mesh size,'' can be set with the ----mmmmeeeesssshhhh option; the larger the mesh the more realistic the pictures but the calculation time and memory requirement in- creases as the square of the mesh size. The fractal dimen- sion, which you can specify with the ----ddddiiiimmmmeeeennnnssssiiiioooonnnn 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 ampli- tudes. 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 ----ppppoooowwwweeeerrrr op- tion. 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 Page 1 (printed 9/7/93) ppppppppmmmmffffoooorrrrggggeeee((((1111)))) XXXXEEEENNNNIIIIXXXX SSSSyyyysssstttteeeemmmm VVVV ((((22225555 OOOOccccttttoooobbbbeeeerrrr 1111999999991111)))) ppppppppmmmmffffoooorrrrggggeeee((((1111)))) 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 follows: CCCClllloooouuuuddddssss 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. PPPPllllaaaannnneeeetttt 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 ap- proximately 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 modu- lated 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 ap- parent colour of each pixel of the planet is cal- culated 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 pri- mary star is the only source of illumination). Additional random data are used to generate stars around the planet. NNNNiiiigggghhhhtttt A sequence of pseudorandom numbers is used to gen- erate 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 ppppppppmmmmqqqquuuuaaaannnntttt or ppppppppmmmmddddiiiitttthhhheeeerrrr before encoding in a colour mapped format. If the display resolution is high enough, ppppppppmmmmddddiiiitttthhhheeeerrrr generally pro- duces better looking planets. ppppppppmmmmqqqquuuuaaaannnntttt tends to create discrete colour bands, particularly in the oceans, which are unrealistic and distracting. The number of colours in star- ry sky pictures generated with the ----nnnniiiigggghhhhtttt option depends on Page 2 (printed 9/7/93) ppppppppmmmmffffoooorrrrggggeeee((((1111)))) XXXXEEEENNNNIIIIXXXX SSSSyyyysssstttteeeemmmm VVVV ((((22225555 OOOOccccttttoooobbbbeeeerrrr 1111999999991111)))) ppppppppmmmmffffoooorrrrggggeeee((((1111)))) the value specified for ----ssssaaaattttuuuurrrraaaattttiiiioooonnnn. Small values limit the colour temperature distribution of the stars and reduce the number of colours in the image. If the ----ssssaaaattttuuuurrrraaaattttiiiioooonnnn 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 ppppnnnnmmmmddddeeeepppptttthhhh rather than ppppppppmmmmqqqquuuuaaaannnntttt or ppppppppmmmmddddiiiitttthhhheeeerrrr. Try _n_e_w_m_a_x_v_a_l settings of 63, 31, or 15 with ppppnnnnmmmmddddeeeepppptttthhhh to reduce the number of colours in the picture to 256 or fewer. OOOOPPPPTTTTIIIIOOOONNNNSSSS ----cccclllloooouuuuddddssss Generate clouds. A pixmap of fractal clouds is generated. Selecting clouds sets the default for fractal dimension to 2.15 and power scale factor to 0.75. ----ddddiiiimmmmeeeennnnssssiiiioooonnnn _d_i_m_e_n Sets the fractal dimension to the specified _d_i_m_e_n, which may be any floating point value between 0 and 3. Higher fractal dimensions create more ``chaotic'' images, which require higher resolu- tion 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 for clouds. ----ggggllllaaaacccciiiieeeerrrrssss _l_e_v_e_l The floating point _l_e_v_e_l setting controls the ex- tent to which terrain elevation causes ice to ap- pear at lower latitudes. The default value of 0.75 makes the polar caps extend toward the equa- tor 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. ----hhhhoooouuuurrrr _h_o_u_r When generating a planet, _h_o_u_r is used as the ``hour angle at the central meridian.'' If you specify ----hhhhoooouuuurrrr 11112222, 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 _h_o_u_r, 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 ----hhhhoooouuuurrrr option is specified, a random hour angle is chosen, biased so that only Page 3 (printed 9/7/93) ppppppppmmmmffffoooorrrrggggeeee((((1111)))) XXXXEEEENNNNIIIIXXXX SSSSyyyysssstttteeeemmmm VVVV ((((22225555 OOOOccccttttoooobbbbeeeerrrr 1111999999991111)))) ppppppppmmmmffffoooorrrrggggeeee((((1111)))) 25% of the images generated will be crescents. ----iiiicccceeee _l_e_v_e_l Sets the extent of the polar ice caps to the given floating point _l_e_v_e_l. The default level of 0.4 produces ice caps similar to those of the Earth. Smaller values reduce the amount of ice, while larger ----iiiicccceeee settings create more prominent ice caps. Sufficiently large values, such as 100 or more, in conjunction with small settings for ---- ggggllllaaaacccciiiieeeerrrrssss (try 0.1) create ``ice balls'' like Euro- pa. ----iiiinnnncccclllliiiinnnnaaaattttiiiioooonnnn||||----ttttiiiilllltttt _a_n_g_l_e The inclination angle of the planet with regard to its primary star is set to _a_n_g_l_e, which can be any floating point value from -90 to 90. The inclina- tion angle can be thought of as specifying, in de- grees, the ``season'' the planet is presently ex- periencing 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, nega- tive 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. ----mmmmeeeesssshhhh _s_i_z_e A mesh of _s_i_z_e by _s_i_z_e will be used for the fast Fourier transform (FFT). Note that memory re- quirements and computation speed increase as the square of _s_i_z_e; 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 capaci- ty, you may have to reduce the mesh size to avoid running out of RAM. Increasing the mesh size pro- duces better looking pictures; the difference be- comes particularly noticeable when generating high resolution images with relatively high fractal di- mensions (between 2.2 and 3). ----nnnniiiigggghhhhtttt A starry sky is generated. The stars are created by the same algorithm used for the stars that sur- Page 4 (printed 9/7/93) ppppppppmmmmffffoooorrrrggggeeee((((1111)))) XXXXEEEENNNNIIIIXXXX SSSSyyyysssstttteeeemmmm VVVV ((((22225555 OOOOccccttttoooobbbbeeeerrrr 1111999999991111)))) ppppppppmmmmffffoooorrrrggggeeee((((1111)))) round planet pictures, but the output consists ex- clusively of stars. ----ppppoooowwwweeeerrrr _f_a_c_t_o_r Sets the ``power factor'' used to scale elevations synthesised from the FFT to _f_a_c_t_o_r, 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 ----cccclllloooouuuuddddssss 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 fac- tors 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. ----ssssaaaattttuuuurrrraaaattttiiiioooonnnn _s_a_t Controls the degree of colour saturation of the stars that surround planet pictures and fill star- ry skies created with the ----nnnniiiigggghhhhtttt option. The de- fault value of 125 creates stars which resemble the sky as seen by the human eye from Earth's sur- face. Stars are dim; only the brightest activate the cones in the human retina, causing colour to be perceived. Higher values of _s_a_t 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 created by ppppppppmmmmffffoooorrrrggggeeee. Dim, low mass stars live much longer than bright massive stars, consequent- ly there are many reddish stars for every blue gi- ant. This relationship is preserved by ppppppppmmmmffffoooorrrrggggeeee. You can reverse the proportion, simulating the sky Page 5 (printed 9/7/93) ppppppppmmmmffffoooorrrrggggeeee((((1111)))) XXXXEEEENNNNIIIIXXXX SSSSyyyysssstttteeeemmmm VVVV ((((22225555 OOOOccccttttoooobbbbeeeerrrr 1111999999991111)))) ppppppppmmmmffffoooorrrrggggeeee((((1111)))) as seen in a starburst galaxy, by specifying a negative _s_a_t value. ----sssseeeeeeeedddd _n_u_m Sets the seed for the random number generator to the integer _n_u_m. The seed used to create each picture is displayed on standard output (unless suppressed with the ----qqqquuuuiiiieeeetttt option). Pictures gen- erated with the same seed will be identical. If no ----sssseeeeeeeedddd 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 parameters. ----ssssttttaaaarrrrssss _f_r_a_c_t_i_o_n Specifies the percentage of pixels, in tenths of a percent, which will appear as stars, either sur- rounding a planet or filling the entire frame if ----nnnniiiigggghhhhtttt is specified. The default _f_r_a_c_t_i_o_n is 100. ----xxxxssssiiiizzzzeeee||||----wwwwiiiiddddtttthhhh _w_i_d_t_h Sets the width of the generated image to _w_i_d_t_h 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 ppppppppmmmmffffoooorrrrggggeeee, then use ppppnnnnmmmmccccuuuutttt to extract a portion of the shape and size you require. ----yyyyssssiiiizzzzeeee||||----hhhheeeeiiiigggghhhhtttt _h_e_i_g_h_t Sets the height of the generated image to _h_e_i_g_h_t 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 pre- fix. BBBBUUUUGGGGSSSS 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. SSSSEEEEEEEE AAAALLLLSSSSOOOO ppppnnnnmmmmccccuuuutttt(1), ppppnnnnmmmmddddeeeepppptttthhhh(1), ppppppppmmmmddddiiiitttthhhheeeerrrr(1), ppppppppmmmmqqqquuuuaaaannnntttt(1), ppppppppmmmm(5) [1] Voss, Richard F., ``Random Fractal Forgeries,'' in Page 6 (printed 9/7/93) ppppppppmmmmffffoooorrrrggggeeee((((1111)))) XXXXEEEENNNNIIIIXXXX SSSSyyyysssstttteeeemmmm VVVV ((((22225555 OOOOccccttttoooobbbbeeeerrrr 1111999999991111)))) ppppppppmmmmffffoooorrrrggggeeee((((1111)))) Earnshaw et. al., Fundamental Algorithms for Computer Graphics, Berlin: Springer-Verlag, 1985. [2] Peitgen, H.-O., and Saupe, D. eds., The Science Of Fractal Images, New York: Springer Verlag, 1988. AAAAUUUUTTTTHHHHOOOORRRR John Walker Autodesk SA Avenue des Champs-Montants 14b CH-2074 MARIN Suisse/Schweiz/Svizzera/Svizra/Switzerland Usenet: kelvin@Autodesk.com Fax: 038/33 88 15 Voice: 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 restric- tions. This software is provided ``as is'' without express or implied warranty. PPPPLLLLUUUUGGGGWWWWAAAARRRREEEE!!!! 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 introduc- tion by James Gleick and detailed explanations by Rudy Ruck- er of the mathematics and algorithms used by each program. Page 7 (printed 9/7/93)