Moebius modifier

From Fractal Wiki
Jump to navigation Jump to search

Moebius modifier for escape time fractals

Wraps (either left or right or both) inverting Y, as if on an infinitely tall Moebius strip cylinder. Can be applied after each iteration step to various escape time fractals (Mandelbrot, Burning Ship, etc). Invented by Hiato on fractalforums.com in 2013.

Modifier function

   moebius (x + i y) =
     | x < -a = (x + 2 a) - i y
     | -a < x < a = x + i y
     | a < x = (x - 2 a) - i y

Modifier function (perturbed)

Z = X + i Y high precision reference, z = x + i y low precision delta (small); may need custom glitch detection when z is on the other side of the fold to Z.

   P(moebius)(X + i Y, x + i y) := moebius((X + x) + i (Y + y)) - moebius(X + i Y) =
     | X < -a
       | X + x  < -a = x - i y
       | -a < X + x < a = (x - 2 a) + i (2 Y + y)
       | a < X + x = (x - 4 a) - i y
     | -a < X && X < a
       | X + x < -a = (x + 2 a) - i (2 Y + y)
       | -a < X + x < a = x + i y
       | a <  X + x = (x - 2 a) - i (2 Y + y)
     | a < X
       | X + x < -a = (x + 4 a) - i y
       | -a < X + x < a = (x + 2 a) + i (2 Y + y)
       | a < X + x = x - i y

References