# 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 