Mathwords logoMathwords

Heart Curve — Definition, Formula & Examples

A heart curve is a mathematical curve whose shape resembles a heart (♥). Several different equations produce heart-shaped graphs, with the most common versions written in polar, parametric, or implicit Cartesian form.

A heart curve typically refers to one of several algebraic or transcendental curves that produce a heart-like shape when plotted. A widely used implicit Cartesian form is (x2+y2−1)3−x2y3=0(x^2 + y^2 - 1)^3 - x^2 y^3 = 0. In polar coordinates, a cardioid r=1−sin⁡θr = 1 - \sin\theta produces a rounded heart shape, while parametric forms allow finer control over the curve's proportions.

Key Formula

(x2+y2−1)3=x2 y3\left(x^2 + y^2 - 1\right)^3 = x^2\,y^3
Where:
  • xx = Horizontal coordinate in the Cartesian plane
  • yy = Vertical coordinate in the Cartesian plane

How It Works

To graph a heart curve, choose the equation form that suits your tools. For a graphing calculator or Desmos, the implicit equation (x2+y2−1)3=x2y3(x^2 + y^2 - 1)^3 = x^2 y^3 is simplest — just type it in. For parametric plotting, use x=16sin⁡3tx = 16\sin^3 t and y=13cos⁡t−5cos⁡2t−2cos⁡3t−cos⁡4ty = 13\cos t - 5\cos 2t - 2\cos 3t - \cos 4t with tt running from 00 to 2π2\pi. The polar cardioid r=a(1−sin⁡θ)r = a(1 - \sin\theta) gives a rounder shape and is commonly studied in precalculus and calculus courses. Each version produces a slightly different heart silhouette, so experimenting with constants changes the proportions.

Worked Example

Problem: Verify that the point (0, 1) lies on the heart curve (x2+y2−1)3=x2y3(x^2 + y^2 - 1)^3 = x^2 y^3.
Step 1: Substitute x=0x = 0 and y=1y = 1 into the left side of the equation.
(02+12−1)3=(0)3=0\left(0^2 + 1^2 - 1\right)^3 = (0)^3 = 0
Step 2: Substitute x=0x = 0 and y=1y = 1 into the right side of the equation.
02⋅13=00^2 \cdot 1^3 = 0
Step 3: Compare both sides. Since 0=00 = 0, the equation is satisfied.
0=0✓0 = 0 \checkmark
Answer: Yes, (0, 1) lies on the heart curve because both sides equal 0.

Another Example

Problem: Plot five points on the parametric heart curve x=16sin⁡3tx = 16\sin^3 t, y=13cos⁡t−5cos⁡2t−2cos⁡3t−cos⁡4ty = 13\cos t - 5\cos 2t - 2\cos 3t - \cos 4t for t=0,π2,π,3π2,2πt = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi.
t = 0: Compute xx and yy when t=0t = 0.
x=16sin⁡30=0,y=13−5−2−1=5x = 16\sin^3 0 = 0,\quad y = 13 - 5 - 2 - 1 = 5
t = π/2: At t=π2t = \frac{\pi}{2}, sin⁡t=1\sin t = 1 and cosines of multiples of π2\frac{\pi}{2} are 0,−1,0,10, -1, 0, 1.
x=16(1)3=16,y=0−5(−1)−0−1=4x = 16(1)^3 = 16,\quad y = 0 - 5(-1) - 0 - 1 = 4
t = π: At t=πt = \pi, sin⁡t=0\sin t = 0.
x=0,y=13(−1)−5(1)−2(−1)−1(1)=−17x = 0,\quad y = 13(-1) - 5(1) - 2(-1) - 1(1) = -17
t = 3π/2: By symmetry with t=π2t = \frac{\pi}{2}, sin⁡t=−1\sin t = -1.
x=16(−1)3=−16,y=0+5−0−1=4x = 16(-1)^3 = -16,\quad y = 0 + 5 - 0 - 1 = 4
Answer: Key points: (0, 5) top center, (16, 4) right bump, (0, −17) bottom tip, (−16, 4) left bump. Connecting these traces a heart shape.

Why It Matters

Heart curves appear in precalculus and calculus courses whenever students study polar coordinates, parametric equations, or implicit relations. Plotting them builds fluency with graphing tools and reinforces how different coordinate systems describe the same shape. They also show up in computer graphics and generative art, where designers use mathematical curves to create smooth, scalable shapes.

Common Mistakes

Mistake: Using (x2+y2−1)3=x2y3(x^2 + y^2 - 1)^3 = x^2 y^3 and expecting the heart to point right or left.
Correction: This particular equation produces a heart that points downward (tip at the bottom). Rotate by swapping xx and yy if you want a sideways heart.
Mistake: Confusing the cardioid r=1−cos⁡θr = 1 - \cos\theta with the pointed heart curve.
Correction: A cardioid is smooth and rounded with no cusp at the bottom. The implicit or parametric heart curves give the classic pointed Valentine shape.