You can always introduce lighting-effect in the colouring, but it usually looks best when the iterations are towards a super-attracting cycle.
(if no Mandelbrot set appears you must go to the next number).Ī Julia set (or rather, the domain outside the Julia set) is composed of a number of so-called Fatou domains (each containing a critical point), and the "Julia" program draws them all. I both cases you make a choice by setting the parameter critical point to one of the numbers 1, 2, 3. The following are all the pairs lying symmetrically around the x-axis. Namely in this way: The first consists of two points having the smallest and the largest x-value (if there is a difference in the x-values). In the opposite case, we have ordered some pairs of critical points, so that we also here only need to make a single choice. If the first row contains at least two numbers more than the second, infinity is a critical point, and this is used as the one of the two points, so that we in this case only have to choose one critical point. All the critical points are calculated automatically by the program, and they are lying symmetrically around the x-axis (because the coefficients of our rational function are real). The Mandelbrot set is constructed on the basis of two so-called critical points (= the solutions to the equation f'(z) = 0). You should also make the boundary thinner, when you draw a large picture. Before you make a picture in large size, you must set the number width to the width (in pixels) of the picture, in order to adjust the numbers in the calculations of the lighting-effect. The light is also dependent on two angles. The closeness of the colours and the intensity of the lighting-effect are determined by two numbers of density which you must carefully adjust.
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The colour scale is displaced by a pc displace. This number and the maximum iteration number must match each other, otherwise there will be flaws in the picture. In this case a parameter bailout, determining the distance from the cycle that stops the iteration, is required. I the second case, that is, when the first row does not contain at least two numbers more than the second, the iterations are towards cycles of finite points. In the first case the iterations are towards infinity, and in this case you can, by setting the formula parameter light to a number different from 0, introduce lighting-effect in the Mandelbrot set: The Mandelbrot set is drawn in two different ways, depending on whether or not the first row contains at least two numbers more than the second. This consists of a "Mandelbrot" program and of one or two "Julia" programs. When RatioUF is activated, an Ultra Fractal program with the name Ratio is produced (you must alter its name if you want to save it). We get the usual Mandelbrot set if we write: There must not be other spaces than the space between the two rows, and you must press Enter after the last row. document with the name "function" in the folder with the program, and write the coefficients of the rational function in precisely this way: A colour scale is imported by clicking on the folder in Layer Properties/Outside.
For the moment you can use the enclosed pictures of colour scales. The colouring algorithm consists of three programs: Gradient is the primary, Image is for the use of an imported picture of a cyclic colour scale, and FieldLines is for drawing of field lines, this technique demands two imported colour scales. Put the program RatioUF, the file texfil and the colouring algorithm RatioCol in the folder where you have your Ultra Fractal formula programs. As a polynomial is given by a row of numbers (the coefficients, in order of increasing exponents), a rational function is given by two rows of numbers, for instance 1 0 -1 and 0 1 -0.01 0.005.
More precisely the iteration formula is z → f(z) + c, where f(z) is a rational function, that is, a function of the form f(z) = p(z)/q(z), where p(z) and q(z) are polynomials. This program can draw the Mandelbrot set and the Julia sets for any rational function. Programs for Ultra Fractal inspired of the technique of
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