# Breathing Ring

Curve · (x, y, z) = f(u, t)

`(cos(u) · (3 + sin(6u + t)),  0.6 · sin(6u + t),  sin(u) · (3 + sin(6u + t)))`

[Open in the app](https://www.wavelace.com/app#p=46) · [This page](https://www.wavelace.com/presets/breathing-ring)

### What it draws

The first and third expressions are `cos(u)` and `sin(u)` times the same bracket, so they place the point at angle `u` around a ring at distance `3 + sin(6u + t)` from the axis. That is a circle of radius `3` with a ripple of amplitude `1` laid over it, six waves to the turn, giving a distance that runs between `2` and `4`.

The second expression, `0.6·sin(6u + t)`, is the height, and the same ripple drives it. The ring therefore rises and falls between `±0.6` six times as well. Six is a whole number of waves per turn, which is why the ends meet.

### One cone, one travelling ripple

Because a single quantity sets both the distance and the height, the two are locked together: every point of the curve obeys `height = 0.6 · (distance − 3)`. That is the equation of a cone about the vertical axis, and the curve never leaves it, whatever the clock does. The ring is highest exactly where it is widest and lowest where it is narrowest. It is not a wave on a flat circle but a closed path wound over a fixed cone.

The clock adds a phase to the ripple and nothing else. A crest sits where `6u + t` is fixed, so as `t` grows the crest slides backwards around the ring at `1/6` of a radian per second at `Speed` 1. A full circuit takes `12π ≈ 37.7` seconds. The six crests are alike, so the picture repeats after each one has taken its neighbour's place, every `2π ≈ 6.28` seconds.

### Try

- `Top`: from above, only the ripple in distance shows, a six-lobed rosette between `2` and `4`.
- `Front`: the ring edge on, where the rise and fall of `±0.6` is the whole story.
- Change every `6u` to `9u`: nine lobes, still meeting at the ends, and a slower drift of `1/9` of a radian per second.
- Change both `3 + sin(6u + t)` to `1 + sin(6u + t)`: the distance now runs from `0` to `2` and the ring pinches through the axis six times, a rosette of six petals.
- Change `0.6` to `−0.6`: the cone turns over, and the ring is now lowest where it is widest.

### Read more

- [Sine wave](https://en.wikipedia.org/wiki/Sine_wave)
- [Phase velocity](https://en.wikipedia.org/wiki/Phase_velocity)
- [Cone](https://en.wikipedia.org/wiki/Cone)
- [Rotational symmetry](https://en.wikipedia.org/wiki/Rotational_symmetry)