# Hypotrochoid 7/5

Curve · (x, y, z) = f(u, t)

`(−2/5 · cos(u) + h · cos(2u/7),  0,  −2/5 · sin(u) + h · sin(2u/7))`

[Open in the app](https://www.wavelace.com/app#p=167) · [This page](https://www.wavelace.com/presets/hypotrochoid-7-5)

### What it draws

The same rule as a Spirograph: a circle of radius `b` rolls on a fixed circle of radius 1, and a pen `h` from its centre draws. Here `b = 7/5`, so the rolling circle is the larger. The fixed circle sits inside it, touching, and the big one rolls round it.

The first term of each expression is the rolling circle's centre, at radius `|1 − b| = 0.4`. The minus sign in `−2/5` puts it on the far side. The second term is the pen, turning `2/7` as fast, now the same way round.

### Two circles, five lobes

Written this way the curve is two circular motions added, a small one of radius 0.4 and a large one of radius `h`. That is the deferent and epicycle of Ptolemy's astronomy.

After 7 turns of `u` the pen has turned twice, and the curve closes. That is why the preset opens with `Turns of u` at 7. Both turn the same way, so the symmetry is their difference, `7 − 2 = 5`.

At the opening `h = 0.5` the curve swings between 0.1 and 0.9 from the centre. At `h = b = 1.4` the pen is on the rim. It stops each time the rim meets the fixed circle, at radius 1, and there the curve has 5 cusps.

### Try

- Drag `h` to 0.4. The two radii are equal, and every lobe passes through the centre.
- Drag `h` to 1.4. The pen is on the rim, and the curve comes to 5 cusps between radii 1 and 1.8.
- Drag `h` to 0.1. A ring between 0.3 and 0.5, the centre's circle barely disturbed.
- Change `2u/7` to `3u/7` in both expressions. The pen now turns 3 times in 7 turns, and the symmetry becomes `7 − 3 = 4`.

### Read more

- [Hypotrochoid](https://en.wikipedia.org/wiki/Hypotrochoid)
- [Deferent and epicycle](https://en.wikipedia.org/wiki/Deferent_and_epicycle)
- [Epitrochoid](https://en.wikipedia.org/wiki/Epitrochoid)
- [Roulette (curve)](https://en.wikipedia.org/wiki/Roulette_(curve))