# Möbius Turn

Complex · w = f(z, t)

`w = (z − i · exp(i · t · 0.4))/(z + i)`

[Open in the app](https://www.wavelace.com/app#p=56) · [This page](https://www.wavelace.com/presets/mobius-turn)

### 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.

### Read more

- [Möbius transformation](https://en.wikipedia.org/wiki/M%C3%B6bius_transformation)
- [Cayley transform](https://en.wikipedia.org/wiki/Cayley_transform)
- [Circles of Apollonius](https://en.wikipedia.org/wiki/Circles_of_Apollonius)
- [Domain coloring](https://en.wikipedia.org/wiki/Domain_coloring)