Hiram... Did you ever get an answer? I too am interested in plotting orbits. Kent Watring ----- Original Message ----- From: Hiram Berry To: Fractint and General Fractals Discussion Sent: Monday, March 01, 2004 7:03 PM Subject: [Fractint] Orbit Fractals with an Arbitrary Formula Is it possible to get fractint to plot these and if so how? Fractint seems to have specialized orbit types like the Henon attractor, but I want to specify the formula. IOW the "formula" type iterates over the screen pixels in an orderly gridlike fashion, but I want to go from pixel to pixel by iterating Pixel_new = F (Pixel_old) and mark the pixel each time. I found this process that seems to generate continuous(with sufficient iteration) fractal curves that look like wads of string with knots in random places (with greater iteration count, the large-scale length of the curve grows, but less than proportional to the number of iterations, and backfilling occurs at the same time at seemingly random locations so the curve is probably dense): x(n) = summation[m=1 to n]( sin(f(n)) * n^-a) y(n) = summation[m=1 to n]( cos(f(n)) *n^-a) where a is a constant between 0 and 1. One f that gives a fractal curve (briefly described above) is sqr, while f=log gives a smooth spiral. I'd like to plot these with fractint which has deep zooming capabilities, instead of the program I used which does not, and which failed at around 40000 iterated points. I know about the orbit screen in fractint, but it seems to me that if I use that with a "formula" type it will run the hundreds of thousands of iterations for every single pixel of the screen until the maximum iteration count is hit. That would take much too long. Is there some arcane .par magic to get fractint to display an orbit generated from only one point instead of the main screen? Hiram Berry ------------------------------------------------------------------------------ _______________________________________________ Fractint mailing list Fractint@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint

Hiram Berry wrote:

Is it possible to get fractint to plot these and if so how?

Apologies to you and Kent Watring for not answering at all, the postings of both of you somehow "evaporated" from my memory! There are, IMHO, two possible approaches to rendering orbit formulas, that is, something like x(n+1) = f(x(n),y(n)), y(n+1) = g(x(n),y(n)). Method A: Convert the orbit formula into an escape time one by checking if one of the orbit points has hit a pixel. One only needs to specify a small epsilon around each pixel and bail out when an orbit point falls within the limit. The last lines of such formula would look like ... z = x + flip(y) |z-pixel| < eps } There's a drawback, though. Orbit fractals normally need thousands of plotted points to produce a decent image - and we would have to run the whole orbit *for each pixel* just waiting for the pixel to get hit once! And there would be no shortcuts to speed up rendering - we'd need to check every single pixel! Talk about patience :-) Method B: As you have suggested - using the Orbit Window. (*Not* the orbit display accessed by pressing <o> while the fractal is running - this one displays the orbits of every pixel of the screen. The Orbit Window is opened with either the <o> key after the fractal has finished or by using <Ctrl>+<o> while the fractal is running.) BTW one should set a small view window for the main display via the <v>-screen to let the Orbit Window have the whole screen's "real estate" (and it's a good idea to get acquainted with the OW controls, which are different from those of the main fractal display). Considerations for the formula itself: Besides of a big number of points to plot (easy, just crank up the iteration count on the <x>-screen) and switching off periodicity-checking (just to be on the safe side), we need a formula which hands the orbit point coordinates back to Fractint and which doesn't bail out early - again easy, put those coordinates into the variable "z" and close the formula with a final check that never is false, like ... variable-name == variable-name } And that's all! Two pars + formulas appended (...your formula indeed produces strange and beautiful images). Regards, Gerald ;----------------------- PARs following ----------------------- Lauwerier { ;Press <Ctrl>+<o> after loading ; reset=2003 type=formula formulafile=gkd.frm formulaname=oLauwerier corners=-2/2/-1.5/1.5 params=3/2/0.1/0.1 float=y maxiter=100000000 periodicity=0 viewwindows=4.2/0.75/yes/0/0 colors=@firestrm.map } Hiram { ;Press <Ctrl>+<o> after loading, <,> to zoom out! ; reset=2003 type=formula formulafile=gkd.frm formulaname=oHiram function=sqr corners=-2/2/-1.5/1.5 params=0.5/0 float=y maxiter=10000000 periodicity=0 viewwindows=4.2/0.75/yes/0/0 colors=@firestrm.map } frm:oLauwerier {;from "Algorithmen fuer Chaos und Fraktale", ;p.131 ;--------------- ; p1real = a ; p1imag = b ; p2real = x(0) ; p2imag = y(0) ;--------------- a = real(p1), b = imag(p1) x = real(p2), y = imag(p2): tmp = y y = x + y*(a*x-1-b*y) x = tmp z = x + flip(y) tmp == tmp } frm:oHiram { ;------------------- ; p1real = a (0<a<1) ;------------------- a = real(p1) x = y = n = 0: n = n + 1 x = x + sin(fn1(n))/n^a y = y + cos(fn1(n))/n^a z = x + flip(y) n == n } ;------------------------ End of PARs -------------------------