The Heart Curve

A heart curve is any mathematical equation that produces a heart shape when plotted. Few equations produce recognizable heart-shaped curve. The most famous one is defined by the following implicit equation.

$$\left(x^2 + y^2 - 1\right)^3 - x^2 y^3 = 0$$

But, What Is an Implicit Equation?

Most familiar curves are written explicitly — $y$ as a function of $x$:

$$y = \sin(x) \qquad y = x^2 \qquad y = \sqrt{1 - x^2}$$

On the other hand, an implicit equation is defined differently. We write a function $F(x, y)$ and take all points where it equals zero:

$$F(x, y) = 0$$

The curve is the zero set of $F$ — every $(x, y)$ satisfying that condition. This is more general than the explicit form. An explicit function can only produce one $y$ value per $x$, so it cannot represent closed loops or curves that fold back on themselves.

Implicit equations have no such restriction, which is why the heart shape is impossible to capture in a single explicit formula.

The circle is the canonical example: $x^2 + y^2 - r^2 = 0$. The heart curve generalises this idea dramatically.

Anatomy of the Equation

The equation has two sides that are worth examining separately.

$$\left(x^2 + y^2 - 1\right)^3 = x^2 y^3$$

Left side: $\left(x^2 + y^2 - 1\right)^3$

The expression $x^2 + y^2 - 1$ is zero exactly on the unit circle. Cubing it preserves the sign and greatly amplifies values away from the unit circle — the expression is strongly negative inside the circle and strongly positive outside. At the unit circle itself, both sides must be zero, so the curve is anchored to it.

Right side: $x^2 y^3$

This term breaks the circular symmetry in a carefully controlled way. Note that $x^2$ is always non-negative, and $y^3$ is positive above the $x$-axis and negative below it. So the right side is:

• zero on the $y$-axis ($x = 0$) and on the $x$-axis ($y = 0$)
• positive in the upper half-plane ($y > 0$)
• negative in the lower half-plane ($y < 0$)

When the right side is positive (upper half), the left side must also be positive, which pushes the zero set outside the unit circle — the lobes of the heart bulge outward. When the right side is negative (lower half), the left side must be negative, pushing the zero set inside the unit circle — the bottom of the heart pulls inward to a point.

Symmetry

The heart curve has bilateral symmetry about the $y$-axis. Replacing $x$ with $-x$:

$$\left((-x)^2 + y^2 - 1\right)^3 - (-x)^2 y^3 = \left(x^2 + y^2 - 1\right)^3 - x^2 y^3$$

The equation is unchanged — every point $(x, y)$ on the curve has a mirror image $(-x, y)$ also on the curve. This is why the two lobes are identical.

There is no symmetry about the $x$-axis. The upper half (lobes) and lower half (cusp) behave fundamentally differently, which is exactly what makes the shape work.

The Cusp at the Bottom

At the origin $(0, 0)$:

$$\left(0 + 0 - 1\right)^3 - 0 = -1 \neq 0$$

So the origin is not on the curve. The pointed tip at the bottom is a cusp — a singular point where the curve comes to a sharp corner rather than a smooth arc. It occurs at $(0, -1)$. Substituting:

$$\left(0 + 1 - 1\right)^3 - 0 = 0 \checkmark$$

At this point, the gradient of $F(x, y)$ vanishes — both partial derivatives are zero — which is the precise condition for a singular point. Near $(0, -1)$, the curve does not have a well-defined tangent line, giving the characteristic sharp cusp.

The Two Lobes

The top of the heart has two bumps separated by an inward dent at $(0, 1)$. Substituting:

$$\left(0 + 1 - 1\right)^3 - 0 = 0 \checkmark$$

So $(0, 1)$ is on the curve. This is also a singular point — another cusp — where the curve folds back and the two lobes meet. The overall shape is bounded, with the curve fitting roughly inside the square $[-1.2, 1.2] \times [-1.2, 1.2]$.

Conclusion

That’s it! When we plot an equation on a graph, we can discover familiar and interesting shapes. The heart curve mentioned in this article is one of them. So, send it to your family, nalentine, and friends to show your love mathematically.