## Converting Parametric Curves to Cartesian

Parametric equations define a surface or a curve. If the equations define a curve in theplane, thenandare expressed as functions ofTo convert the parametric equations to a single cartesian equation that relatesandwe must eliminate the parameterfrom the two equations. For example, ifandthenandso

The parametric equationbecomes the single cartesian equation

Example: From the parametric equationsfind a cartesian equation that relatesand

Because there areandterms in the parametric equations, we look for an equation that relatesandWe can rearrangeto giveand to givehence Expanding and simplifying gives

Example: From the parametric equationsfind a cartesian equation that relatesand

We can makethe subject of the first equation and substitute it into the second.

There is a slight complication hanging over from the parametric equation which is not visible in the cartesian equations.since we must be taking the square root of a non negative number, andIf we consider the cartesian equation in isolation, we can substitute any value ofWe must have the condition inherited from the parametric equations that

Example: From the parametric equationsfind a cartesian equation that relatesand

Because there areandterms in the parametric equations, we look for an equation that relatesandWe can rearrangeto giveand to givehence Expanding and simplifying gives