# Pulsing Cardioid

Polar · r = f(θ, t)

`r = (1 + cos(θ)) · (1 + 0.3 · sin(4t))`

[Open in the app](https://www.wavelace.com/app#p=26) · [This page](https://www.wavelace.com/presets/pulsing-cardioid)

### What it draws

The formula is a product of one factor in `θ` and one in `t`, and the two never mix. `1 + cos(θ)` is the shape: `r = 2` straight ahead, `r = 1` at a quarter turn, and `r = 0` at `θ = π`, where the curve runs into the origin and turns round on itself. That single point is the *cusp*, and the heart it makes is the *cardioid*. At this size it encloses an area of `3π/2 ≈ 4.71`, and its perimeter is exactly `8`.

`1 + 0.3·sin(4t)` is the pulse. It swings between `0.7` and `1.3` and multiplies every radius by the same amount, so the heart never changes shape. It only grows and shrinks by three tenths either way, one beat every `π/2 ≈ 1.57` seconds at `Speed` 1.

### Where the shape comes from

Roll a circle on the outside of another circle of the same size and mark one point of the rolling rim: the path it traces is this curve, the epicycloid with a single cusp. The cusp is the instant the marked point touches the fixed circle and its speed drops to zero. The same outline turns up as the bright caustic inside a mug, when the lamp sits on the rim itself and the light reflected off the far wall gathers on the surface. It is also the pickup pattern of a directional microphone, where `1 + cos(θ)` is the sensitivity at an angle and the cusp is the deaf spot behind it.

### History

Johann Castillon gave the curve its name in 1741, from the Greek for heart. It had been drawn well before that as a member of the limaçon family, which Étienne Pascal studied in the 1630s.

### Try

- `Top` in the deck: the heart flat on, cusp pointing back.
- Stop the pulse with `1 + cos(θ)` and the plain cardioid stays put.
- Push the first factor past a balance, `(1 + 1.5·cos(θ)) · (1 + 0.3·sin(4t))`: `r` now goes negative around the cusp and the curve grows an inner loop. That is a limaçon.
- Pull it back to `(1 + 0.5·cos(θ)) · (1 + 0.3·sin(4t))`: no cusp and no loop, just a dented oval.
- Beat it faster with `(1 + cos(θ)) · (1 + 0.3·sin(12t))`, three pulses in the time of one.

### Read more

- [Cardioid](https://en.wikipedia.org/wiki/Cardioid)
- [Limaçon](https://en.wikipedia.org/wiki/Lima%C3%A7on)
- [Epicycloid](https://en.wikipedia.org/wiki/Epicycloid)
- [Caustic (optics)](https://en.wikipedia.org/wiki/Caustic_(optics))