The heart curve: equations for a heart shape

(x² + y² − 1)³ = x²y³ starts from the unit circle, x² + y² = 1, and bends it. Where y is positive, the right side is positive, so x² + y² must be slightly more than 1: the top bulges out into two lobes. Where y is negative, x² + y² must be less than 1 and the bottom is pulled in, meeting itself in a cusp at (0, −1).

The curve is easy to type but hard for many graphers, because near the cusp the left side barely changes sign. This calculator’s certified mode proves which pixels the curve passes through, so the point of the heart is drawn sharp.

The smoothest heart is x = 16 sin³t, y = 13 cos t − 5 cos 2t − 2 cos 3t − cos 4t for 0 ≤ t ≤ 2π. The sin³ term gives the width, and the sum of cosines is a short Fourier series that sculpts the dip at the top and the point at the bottom. Dividing both by 8, as in the equation above, makes it about the same size as the others.

Try removing the cos 4t term, or changing 5 to 4: each coefficient controls one feature of the outline.

x² + (y − |x|^(2/3))² = 1 is a circle whose centre rides up and down with |x|^(2/3), which pinches the top into a dimple. The polar cardioid r = 1 − sin θ is the oldest of the four: a curve traced by a point on a circle rolling around another circle of the same size. It looks more like an apple than a heart, but it is where the name comes from (cardia is Greek for heart).