Functions
AI Assistant

Heart Curve

Draw a heart with parametric equations — math is beautiful

This red heart is drawn entirely by mathematics. The parametric equations use sine and cosine to trace the heart shape as the parameter t goes from 0 to 2π:

x(t) = 16 sin³(t)
y(t) = 13 cos(t) − 5 cos(2t) − 2 cos(3t) − cos(4t)

These aren't random numbers — each cosine term sculpts a different feature of the heart shape. Math can create stunning visual art. Ask the AI to show you butterflies, flowers, or spirals too.

Graph

FAQ

How does this equation make a heart shape?
The x equation uses sin³(t) to create the symmetric left-right lobes. The y equation combines multiple cosine terms at different frequencies — each one refines the shape. cos(t) gives the basic top, cos(2t) carves the dip, and cos(3t) and cos(4t) sharpen the bottom point.
What are parametric equations?
Instead of y = f(x), parametric equations define x and y separately as functions of a parameter t. As t increases, the point (x(t), y(t)) traces a curve. This lets you draw shapes that aren't functions — like circles and hearts that fail the vertical line test.
Can math draw other shapes?
Absolutely! A butterfly curve, rose curves (r = cos(nθ) in polar), Lissajous figures, spirals (Archimedean, logarithmic), and even approximations of any shape using Fourier series. Math is one of the most powerful drawing tools.
How can I modify the heart?
Multiply the equations by a constant to change size. Add offsets to shift position. Multiply x by a factor to make it wider or narrower. Change the coefficients in the y equation to alter the heart's proportions.

Related Topics

Parametric EquationsSine & CosineThe Unit Circle