# Parametric Equations of Curves

A pair of equations

are parametric equations for a curve. You graph the curve by plugging values of t into x and y, then plotting the points as usual.

Example. The parametric equations

represent a circle of radius 1 centered at the origin.

You can sometimes recover the x-y equation of a parametric curve by eliminating t from the parametric equations. In this case,

Notice that the graph of a circle is not the graph of a function. Parametric equations can represent more general curves than function graphs can.

Example. The parametric equations

represent a spiral.

This is not the graph of a function .

Example. Find the x-y equation for

Notice that

So

This is the standard form for the equation of an ellipse.

Example. The parametric equations

represent a hypocycloid of four cusps.

In this case, it would be difficult to eliminate t to obtain an x-y equation.

Example. To parametrize a curve means to obtain parametric equations for the curve.

If you have x-y equations in which x or y is solved for, it's easy. For example, to parametrize , set . Then . A parametrization is given by

There are infinitely many ways to parametrize a curve. For example,

is another parametrization of .

To parametrize , set . Then , so

is a parametrization of . (This is how you can graph x-in-terms-of-y equations on your calculator.)

Example. If and are points, the segment from to may be parametrized by

Notice that when , , and when , .

If you allow t to range from to , this gives the line through and .

For example, the line through and is

An analogous result holds for lines in 3 dimensions (or in any number of dimensions).

Example. Find parametric equations for the Folium of Descartes:

Set . Then

Therefore, . A parametrization is given by

The first and second derivatives give information about the shape of a curve. Here's how to find the derivatives for a parametric curve.

First,

by the Chain Rule. Solving for gives

To find the second derivative, I differentiate the first derivative.

Since will come out in terms of t, I want to be sure to differentiate with respect to t. Use the Chain Rule again:

That is,

Example. Find and for

First,

So

Next,

So

Example. At what points on the curve

is the tangent line horizontal?

Find the derivative:

for and for .

When , and . When , and . There are horizontal tangents are and at .

Example. Find the equation of the tangent line to the curve

at the point corresponding to .

When , and . The tangent line is