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Computer Graphics
Many programs are used to create the Mandelbrot set and other fractals. These programs use a range of
algorithms to conclude the color of individual pixels and achieve well-organized computation.
One of the uncomplicated algorithms to create an illustration or representation of the Mandelbrot set is
the Escape time algorithm. In the plot area, for each x, y point, a repeated and continuous calculation is
carried out. In addition, the escape time algorithm is based on the behavior of that calculation, and a
color is chosen for that pixel.
The x and y positions of each point are used as starting or initial values in an iterating or repeating
calculation. The outcome of every repeated calculation is used as the starting values for the next value.
The values are checked for all iterations, in order to know whether they have reached a critical escape
condition or a bailout situation. If that specific condition is reached, then the calculation is stopped, the
pixel is drawn, and the next x, y point is checked. For a number of starting values, the escape occurs
swiftly, only after a small number of iterations. For starting values, which are very close to each other,
but not in the set, possibly will take hundreds or thousands of iterations to escape.
Escape never occurs for the values within the Mandelbrot set.
Only the programmer or the user must choose how much iteration or depth is required
in order to examine iterations. The higher the maximum number of iterations, the more
detailed information emerges in the final image. However, it will take longer time to
calculate the fractal image
It is important to know that the escape conditions can be easy or difficult. The reason is that no number
with a real or imaginary part greater than 2 can be a part of the set. The main objective of a general
bailout is to escape, when any of the co-efficients goes beyond 2.
The swiftness of values reaching the escape point is represented by the color of each point. Regularly
the color black is used to explain values that fail to escape before the iteration limit. In addition, the
brighter colors gradually are used for points that escape. This provides a visual demonstration of how
many cycles were necessary before reaching the escape condition.
The escape time algorithm is popular for its simple calculations. However, it creates groups of colors,
which can detract from an image's artistic value. This can be enhanced with the help of an algorithm
known as Normalized Iteration Count. The normal iteration count provides a certain level of transition
of colors between iterations. The algorithm connects a real number ν with each value of z, by using the
connection of the iteration number with the potential function. This function is given by the following
formula:
log z n
φ ( ) =z lim
n → ∞ P n
In this formula, z n is the value after n iterations and P is the power for which, z is raised to the
Mandelbrot set equation (z n+1 = z nP + c, P is generally 2).
If a large bailout radius N (Ex: 10100) is chosen, we have that
log z n = log(N )
P n P v (≈ )
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