Pulsing Cardioid
r = (1 + cos(θ)) · (1 + 0.3 · sin(4t))
Open in the app The dials and keys named below are the app's.
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.