Heart Curve
Parametric curvesThe parametric valentine x = 16sin³t with a four-term cosine partner — mathematics’ most shared curve.
Adjust parameters
Equation
Open the Creator with a ready-made animation brief for this curve — review the scene plan and Manim code before rendering.
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.
Key properties
- Symmetric about the y-axis; the bottom point sits at (0, −17), the top notch at (0, 5).
- Width 32 (from x = ±16 at t = ±π/2) and height 22 before scaling.
- 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.
Where you'll see it
Graphing calculator art, search-engine plotting easter eggs, Valentine’s coding demos, and parametric-equations lessons.