skull fractal & XaoS
recently ive been developing my port of XaoS fractal zoomer with skull fractal added to it. but... direct usage of skull fractal formula had given me quite poor effect, XaoS turned to be verrrry slow. and that was unacceptable ! to make it fast again i had to play with mathematics to get equivalent but faster to calculate skull fractal formula. ive started with skull fractal formula
in computations, the most expensive part of this formula are hiperbolic functions (for further details on calculations of hiperbolic functions with complex arguments go here). after some time i came up with formula of form
now we have to do less trigonometric calculations. what is most important, sine and cosine functions of the same real number can be evaluated in just one step using fpu instruction fsincos. using good implementation we can use this trick also with hiperbolic functions. but its not enough, because we can still do some further simplifications. which are related with arguments of used trigonometric functions. as we can see above, we have addition or subtraction of two complex numbers defined in skull fractal formula. in our case it is easy to show that these two operations can be expressed as :
there cis(a)=cos(a)+isin(a). After shown operations my XaoS port rendered skull fractal much faster.
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