Möbius Turn
w = (z − i · exp(i · t · 0.4))/(z + i)
Open in the app The dials and keys named below are the app's.
One zero, one pole
(z − i · exp(i · t · 0.4))/(z + i) is a ratio of two straight-line expressions in z, a Möbius transformation. The denominator vanishes at z = −i, a simple pole fixed at the near edge of the plot. The numerator vanishes at i · exp(i · t · 0.4), and since that factor has modulus 1 the zero rides the unit circle. It starts at i and comes back to it every 2π/0.4 ≈ 15.7 seconds at Speed 1. So one dimple circles a standing spire.
Circles of Apollonius
The height is a ratio of two distances, |w| = |z − a|/|z + i| with a the moving zero. Points twice as far from a as from −i have |w| = 2, and the set of them is a circle. Every brightness ring is one of these circles of Apollonius. The exception is |w| = 1, the points equally far from both, which is the straight perpendicular bisector of the two.
At t = 0, with the zero at i and the pole at −i, that line is the real axis, and the sheet sits there at a flat log(2) ≈ 0.693. Far from both points the ratio tends to 1, so the corners of the plot settle near the same level.
The moment it flattens
After π/0.4 ≈ 7.85 seconds the zero has gone half way round and reached −i, the pole. Numerator and denominator are then the same expression, they cancel, and w = 1 at every point at once. The sheet is level at log(2) ≈ 0.693 and a single tone, with no zero and no pole anywhere. A moment later the two separate again and the landscape grows back. Every Möbius transformation collapses like this when its determinant reaches zero, which for this one happens once per revolution.
Try
- Pause near the flat moment: the whole plane goes one colour.
- Hold the zero still with (z − i)/(z + i), the Cayley transform, which carries the upper half-plane onto the unit disc.
- Move the pole out of the way with (z − i · exp(i · t · 0.4))/(z − 2): the spire sits on the real axis. The zero keeps circling, the two never meet, and the sheet never flattens.
- Turn the zero the other way with (z − i · exp(−i · t · 0.4))/(z + i), and the dimple circles clockwise instead.
- Raise Iterations to 20 so the map is fed its own output over and over. The picture becomes a map of the orbits, the points that escape falling to the outside tone and the rest standing on a flat plateau.