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.

Related Topics

Parametric EquationsSine & CosineThe Unit Circle

Suggested Lessons

Slope Quadratic Equations Unit Circle Word Problems
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.
What can it graph?
It can plot explicit, implicit, and parametric functions, add points and geometry, and animate sliders on the same graph.
Can I use voice or a photo?
Yes. You can talk to the tutor, upload a worksheet or handwritten problem, and let the graph update from that input.
Will it explain the steps?
Yes. The AI explains what it is drawing and why, so you see the answer on the graph instead of getting only a final number.