Heart Curve
The parametric valentine x = 16sin³t with a four-term cosine partner — mathematics’ most shared curve.
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Formulas and key values
- Parametric equations
- AreaIn the graph ≈ 565.5
- Width
t runs from 0 to 2π; s scales the whole heart (s = 1 below).
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Start the video01 · Definition
What is the Heart Curve?
The most popular “heart curve” is the parametric pair x = 16sin³t, y = 13cos t − 5cos 2t − 2cos 3t − cos 4t. The cubed sine flattens the sides into lobes, while the four cosine harmonics sculpt the top notch and the bottom point — a tiny, hand-tuned Fourier series that happens to spell affection.
It is far from the only heart in mathematics: the implicit sextic (x² + y² − 1)³ = x²y³ draws one, and the cardioid is its polar cousin. But this version, popularized by graphing calculators and search-engine easter eggs, is the one most people have actually plotted — and a perfect first exercise in reading parametric equations term by term.
02 · Properties
Key properties
- Symmetric about the y-axis; the bottom point sits at (0, −17), the top notch at (0, 5).
- Before scaling: width 32 (x = ±16 at t = ±π/2), 22 from the notch to the tip, and about 28.9 in full height (the lobes peak at y ≈ 11.92).
- The y-series is a four-harmonic cosine sum — a hand-crafted Fourier sketch.
- The implicit heart (x² + y² − 1)³ = x²y³ is an algebraic alternative of degree six.
- Related: the cardioid gets its name from the same Greek word for heart.
03 · Questions
Heart Curve: common questions
- What is the equation of the Heart Curve?
- Parametric equations: x = 16 sin^3 t, y = 13 cos t - 5 cos 2t - 2 cos 3t - cos 4t. t runs from 0 to 2π; s scales the whole heart (s = 1 below).
- What is the area of the Heart Curve?
- Area: A = 180π ≈ 565.5. t runs from 0 to 2π; s scales the whole heart (s = 1 below).
04 · Keep exploring