Fractals/Hopalong

Hopalong orbit-fractals.

=Name=
 * the Martin-Attractors: were discovered by Barry Martin
 * Hopalong: an image is built of points hopping along on an elliptical path starting from one point in the center.

=History=
 * Hopalong orbits were discovered by Barry Martin from the Aston University, Birmingham.
 * A.K. Dewdney presented the Hopalongs in the magazine "Scientific American" (1986)

=Algorithm= An image is calculated using three parameters: a, b and c.

$$x_{n + 1} = y_n - sign(x_n)*sqrt|b*x_n - c|$$ $$y_{n + 1} = a - x_n$$

Where
 * a, b and c are constants which, by default are initialised to -55, -1 and -42 respectively
 * ABS is the absolute value function
 * SIGN(x) is the same as x/ABS(x). If x>0 then SIGN(x) = 1, if x<0 then SIGN(x) = −1 and if x = 0 then the result of SIGN(x) is zero, too.
 * The color is changed every 2000 dots, until the 10 predefined colors are used, then the colors are repeated.

Devaney's program

INPUT num INPUT a, b, c x = 0 y = 0 PLOT(x, y) FOR i = 1 TO num xx = y - SIGN(x) * [ABS(b*x - c)]^0.5 yy = a - x x = xx y = yy

=Implementations=
 * openprocessing: Hopalong Attractor by Naoki Tsutae
 * Attractors from pyviz

From Barry Martin, here with code for two variants from François Pacull:

OpenProcessing code by Naoki Tsutae:

=References=