apollony.cpp File Reference

Draw the Apollony Gasket via an ifs. More...

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Detailed Description

Draw the Apollony Gasket via an ifs.

Author

Mitch Richling https://www.mitchr.me

Keywords

apollony fractal gasket

Standards

C++20

See also

https://www.mitchr.me/SS/AGasket/index.html

Copyright

Copyright (c) 1988-2015,2017, Mitchell Jay Richling https://www.mitchr.me All rights reserved.

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

1. Redistributions of source code must retain the above copyright notice, this list of conditions, and the following disclaimer.
2. 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.
3. 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

Pick an initial \(z\in\mathbb{C}\), say \(\frac{1}{10}+\frac{2}{10}i\). Then iterate. For each iteration let the next $z$ value be \(f_n(z)\) where \(n\) is selected at random. If we plot the \(z\) points, we obtain the classical Apollony Gasket.

\[\begin{array}{rcl} f_1(z) &=& f(z) \\ f_2(z) &=& \frac{-1 + i\sqrt{3}}{2f(z)} \\ f_3(z) &=& \frac{-1 - i\sqrt{3}}{2f(z)} \\ \end{array}\]

Where

\[ f(z) = \frac{3}{1-z+\sqrt{3}} - \frac{1+\sqrt{3}}{2+\sqrt{3}} \]

Definition in file apollony.cpp.