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User Formulas

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Zoltán Kovács edited this page Jan 4, 2020 · 16 revisions

XaoS supports entering your own custom formulas:

  1. Select User formula from the Fractal menu to define a custom fractal formula.
  2. (Optional) Select User initialization to change the sequence starting point z0. By default z0 is set to 0.

User formulas should be interpreted as functions in the form Z(n+1) = formula (Z(n), Z(n-1), C). Here Z(0) is the same as z0 above.

Variables

In user formulas, as shown above, three variables are available:

variable meaning
Z current sequence point Z(n)
C current plane 'point'
P previous sequence point Z(n-1)
N current iteration number n (beginning with 1)

Constants

Format for complex numbers is {re;im}, e.g. {3;2} is complex number 3+2i. You must write constants for the coordinates. Note that you need to use a semicolon. Also, parameters in functions must be separated by semicolons (see below).

The number {0;1} can also be written as I. If you need to express a complex number with a non-constant, split it to real and imaginary parts like this: (RE(Z)+IM(Z)*I)^2+C.

Real numbers can be used as usual.

Functions

Available functions are listed below:

Basic operations

+, -, *, /, ^

The multiplication sign may be omitted in most cases.

Real and imaginary part

  • RE(z) - real part of z
  • IM(z) - imaginary part of z

Power functions

  • POWI(z;n) - power of z with integer exponent n (fastest)
  • POWD(z;n) - power of z with real exponent n
  • POWDC(z;n) - power of z with complex exponent n
  • POW(z;n) - same as powd(z;n) The symbol ^ can also be used (see above) without explicitly defining the number set for the exponent.

Basic trigonometry

  • SIN(z), COS(z), TAN(z), COT(z), ASIN(z), ACOS(z), ATAN(z), ACOT(z), SINH(z), COSH(z), TANH(z), COTH(z) can be used for computing trigonometrical functions

Exponential and logarithmic functions

  • EXP(z), LOG(z) - exponential and natural logarithm of z
  • LOG2(z), LOG10(z) - logarithm of z in base 2 and 10
  • LOGN(n;z) - logarithm of z in base n (where n is integer)
  • LOGCN(c;z) - logarithm of z in base c (where c is complex)
  • SQRT(z) - square root of z

Other functions

  • RAND(max) - random real number in range [0, max)
  • ABS(z) - absolute value of z

Examples

  • Z^2+C and Z^3+C produce the quadratic and cubic Mandelbrot sets. Z^2.5+C produces another fractal "between them".
  • (RE(Z)+IM(Z)I)^2+C also produces the quadratic Mandelbrot set.
  • POWD(Z;2)+C creates the quadratic Mandelbrot set. A minor change in the formula, by using POWDC(Z;{2;0.1})+C is equivalent to Z^{2;0.1}+C.
  • (ABS(RE(Z))+I*ABS(IM(Z)))^2+C produces the Burning Ship fractal.
  • Z^2+C*(-1)^N expresses the iteration (((z0^2-z0)^2+z0)^2-z0)^2+....

Notes

  • The entered formula is case insensitive. But it will always be rewritten in capitalized letters if the parsing was successful.
  • Handling user formulas is still in experimental phase. Please expect random crashes or malfunctioning in some cases.
  • Fast computation of user formulas is based on Mateusz Malczak's SFFE library.
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