Famous curves
Some curves are worth knowing for their beauty, their history, or the surprising maths inside them. Each page gives the equations to type, the story behind the curve and the properties that make it interesting.
Curves in the plane
Several equations that draw a heart, and why they work.
r = 1 + cos θ: a circle rolling around a circle.
r = cos(kθ): k petals for odd k, 2k for even.
Temple Fay’s transcendental butterfly from 1989.
The figure-eight curve and the infinity symbol.
r = aθ, r = e^(bθ) and r² = θ: three kinds of spiral.
The path of a point on a rolling wheel.
(sin at, sin bt): patterns from two vibrations.
x^(2/3) + y^(2/3) = 1: a star with four cusps.
x³ + y³ = 3axy: a leaf with an asymptote.
Iterate z² + c, and an endlessly detailed shape appears.
Curves and surfaces in 3D
Parametric, polar and implicit equations all work in the same calculator, in 2D and 3D.