Draw some Barry Martin Attractors. More...

Detailed Description

Draw some Barry Martin Attractors.

Author: Mitch Richling https://www.mitchr.me

Standards: C++20

See also: https://www.mitchr.me/SS/barrymartin/index.html

Copyright

Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:

Redistributions of source code must retain the above copyright notice, this list of conditions, and the following disclaimer.
Redistributions in binary form must reproduce the above copyright notice, this list of conditions, and the following disclaimer in the documentation and/or other materials provided with the distribution.
Neither the name of the copyright holder nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

Details

Barry Martin's "Hopalong" orbit fractals are a family of discrete-time dynamical systems:

Classic Barry Martin fractal:
\[ \begin{align*} x_{n+1} & = y_n - \mathrm{sgn}(x_n) \cdot \sqrt{\vert b x_n-c\vert} \\ y_{n+1} & = a-x_n \end{align*} \]
Positive Barry Martin fractal:
\[ \begin{align*} x_{n+1} & = y_n + \mathrm{sgn}(x_n) \cdot \sqrt{\vert b x_n-c\vert} \\ y_{n+1} & = a-x_n \end{align*} \]
Additive Barry Martin fractal:
\[ \begin{align*} x_{n+1} & = y_n + \sqrt{\vert b x_n-c\vert} \\ y_{n+1} & = a-x_n \end{align*} \]
Sinusoidal Barry Martin fractal:
\[ \begin{align*} x_{n+1} & = y_n + \sin{(b x_n-c)} \\ y_{n+1} & = a-x_n \end{align*} \]

All of these maps, and others, may be constructed from the following system as special cases:

\[ \begin{align*} x_{n+1} & = y_n+d \cdot \mathrm{ssgn}(x_n) \cdot \left(f \cdot \sqrt{\vert b x_n-c\vert} + g \cdot \sin{(b x_n-c)} + h \cdot \vert b x_n-c\vert \right) \\ y_{n+1} & = a-x_n \end{align*} \]

Where

\[ \mathrm{ssgn}(v) = \cases{ s & $v\lt 0$ \cr 1 & $v\ge 0$ } \]

\[ \mathrm{sgn}(v) = \cases{ -1 & $v\lt 0$ \cr +1 & $v\ge 0$ } \]

Note the simplified form of the $\mathrm{sgn}(v)$ function used – the standard definition may be used for nearly identical results.

Some special cases:

When $ d=-1, s=-1, f=1, g=0, h=0 $, the map becomes the "Classic Barry Martin fractal"
When $ d=1, s=-1, f=1, g=0, h=0 $, the map becomes the "Positive Barry Martin fractal"
When $ d=1, s=1, f=1, g=0, h=0 $, the map becomes the "Additive Barry Martin fractal"
When $ d=1, s=1, f=0, g=1, h=0, c=0 $, the map becomes the "Sinusoidal Barry Martin fractal"
When $ d=1, s=1, f=0, g=0, h=1, c=0 $, the map becomes the "The Gingerbread Man":
\[ \begin{align*} x_{n+1} & = y_n + \sin{(b x_n)} \\ y_{n+1} & = a-x_n \end{align*} \]
When $ d=1, s=1, f=0, g=0, h=1, c\ne0 $, the map becomes the "The Shifted Gingerbread Man":
\[ \begin{align*} x_{n+1} & = y_n + \sin{(b x_n-c)} \\ y_{n+1} & = a-x_n \end{align*} \]

Definition in file barrymartin.cpp.