# Spherical Spiral

Curve · (x, y, z) = f(u, t)

`(3 · cos(u/2 − π/2) · cos(8u + t),  3 · sin(u/2 − π/2),  3 · cos(u/2 − π/2) · sin(8u + t))`

[Open in the app](https://www.wavelace.com/app#p=41) · [This page](https://www.wavelace.com/presets/spherical-spiral)

### What it draws

Call `u/2 − π/2` the latitude. The second expression, `3sin(u/2 − π/2)`, is the height, and the distance from the vertical axis is `3cos(u/2 − π/2)`. That distance is split into `x` and `z` by the angle `8u + t`, the longitude. Squared and added, the three coordinates come to `9` at every `u`: the curve lies exactly on a sphere of radius `3`.

At `Turns of u` 1 the latitude runs from `−π/2` to `π/2`, pole to pole once, while the longitude turns eight times. That is eight loops, from the bottom of the sphere to the top. They crowd near the poles, because a circle of latitude is only `2π·3cos` of the latitude around: `18.85` at the equator, nothing at a pole. The clock adds to the longitude and to nothing else. So the whole spiral turns rigidly about the vertical axis, a full turn every `2π ≈ 6.28` seconds at Speed 1.

### Longitude against latitude

Since the latitude is `u/2 − π/2`, the longitude `8u + t` is `16` times the latitude plus a constant. Longitude and latitude therefore advance in fixed proportion. A curve on a sphere with that property is a *Clelia curve*, and one whose ratio is `2` or more is called a spherical spiral. It is not the rhumb line, or loxodrome, which crosses every meridian at the same angle. A rhumb line winds around a pole infinitely often and never arrives, while this curve reaches both poles after its eight loops.

### History

The Italian mathematician Guido Grandi named these curves *clelie* in a book of 1728. He dedicated it to Clelia Grillo Borromeo, a Genoese noblewoman and mathematician who had founded her own academy nine years before.

### Try

- `Turns of u` to 2: the latitude carries on past the north pole and back down, so a second set of eight loops returns to the south pole and the curve closes on itself.
- Open the winding out. Change both `8u` to `4u`: four loops instead of eight, and the sphere shows through.
- `Top`: from above, the distance from the axis swells from nothing at the south pole to `3` at the equator and shrinks back, so the loops read as a flat spiral out and in again.
- Turn on `trace`: the pen climbs the whole way from the south pole to the north, one loop at a time.

### Read more

- [Clelia curve](https://en.wikipedia.org/wiki/Clelia_curve)
- [Rhumb line](https://en.wikipedia.org/wiki/Rhumb_line)
- [Spherical coordinate system](https://en.wikipedia.org/wiki/Spherical_coordinate_system)
- [Spiral](https://en.wikipedia.org/wiki/Spiral)