Parametric equation grapher

A parametric curve gives x and y as functions of a third variable, usually t for time. Type the pair (x(t), y(t)) and the calculator traces the curve, including loops and self-crossings that no function y = f(x) can draw.

Open in calculator

A Lissajous figure

(3sin(3t), 3sin(4t))

and the unit circle

(cos(t), sin(t))

How to use the parametric equation grapher

1
Type the coordinates as a pair in t: (cos(t), sin(t)).
2
t runs from 0 to 2π by default. Change the range in the row’s settings, for example −4π to 4π for longer curves.
3
Add a third coordinate for a 3D curve: (cos(t), sin(t), t/4) in 3D mode.
4
Use sliders for parameters, and select the curve to see where it crosses the axes and itself.

What parametric equations are

Instead of y as a function of x, both coordinates depend on a parameter: x = x(t), y = y(t). As t increases the point moves along the curve, so the equations describe not just a shape but a journey along it, with direction and speed.

This is how motion is naturally described. A ball thrown at speed v and angle α follows

(vcos(α)t, vsin(α)t−4.9t²)

where t is the time in seconds.

Eliminating the parameter

Sometimes t can be removed to find an equation in x and y. From x = cos t and y = sin t, cos²t + sin²t = 1 gives x² + y² = 1, the unit circle. From x = t, y = t², substituting gives y = x². Many parametric curves, like the cycloid and Lissajous figures, have no tidy Cartesian form, which is exactly why parametric form is useful.