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

Send comments about this page to: Bruce.Ikenaga@millersville.edu.

Copyright 2013 by Bruce Ikenaga