# Hypotrochoid 3/4

Curve · (x, y, z) = f(u, t)

`(1/4 · cos(u) + h · cos(u/3),  0,  1/4 · sin(u) − h · sin(u/3))`

[Open in the app](https://www.wavelace.com/app#p=168) · [This page](https://www.wavelace.com/presets/hypotrochoid-3-4)

### What it draws

A circle of radius `b = 3/4` rolls inside a fixed circle of radius 1. A pen `h` from its centre draws the curve, flat in the floor's plane.

The first term of each expression is the rolling circle's centre, running round at radius `1 − b = 1/4`. The second is the pen about it, turning the other way at `(1 − b)/b = 1/3` of the speed.

### Four lobes

After 3 turns of `u` the pen has turned once the other way, and the curve closes. That is why the preset opens with `Turns of u` at 3. The two counts add to the symmetry, `3 + 1 = 4`.

At the opening `h = 0.5` the curve swings between 0.25 and 0.75 from the centre, in four lobes that cross one another.

### From star to square

At `h = b` the pen is on the rim and stops each time the rim meets the fixed circle: the curve becomes a four-pointed star with its points at radius 1.

Far past that, the pen's own circle of radius `h` dominates. It turns once in the 3 turns of `u`. The centre's small circle turns 3 times the other way, and the difference of 4 adds a fourfold wobble. At `h = 2.3` the radius only swings between 2.05 and 2.55, and the curve bends the same way everywhere: a rounded square.

### Try

- Replace `h` with `3/4` in both expressions. The pen is on the rim, and the lobes close into a four-pointed star.
- Drag `h` to 2.3. The rounded square.
- Drag `h` to 1.2. Between the two, the sides of the square bend inward between its corners.
- Drag `h` to 0.1. The centre's small circle, taken three times with a slight wobble.

### Read more

- [Hypotrochoid](https://en.wikipedia.org/wiki/Hypotrochoid)
- [Spirograph](https://en.wikipedia.org/wiki/Spirograph)
- [Roulette (curve)](https://en.wikipedia.org/wiki/Roulette_(curve))