Converting Parametric Curves to Cartesian
Parametric equations define a surface or a curve. If the equations define a curve in the
plane, then
and
are expressed as functions of
To convert the parametric equations to a single cartesian equation that relatesand
we must eliminate the parameter
from the two equations. For example, if
and
then
and
so
The parametric equation
becomes the single cartesian equation
Example: From the parametric equations
find a cartesian equation that relatesand
Because there are
and
terms in the parametric equations, we look for an equation that relates
and
We can rearrange
to give
and 
to give
hence 
Expanding and simplifying gives
Example: From the parametric equations
find 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, and
If we consider the cartesian equation in isolation, we can substitute any value of
We must have the condition inherited from the parametric equations that
Example: From the parametric equations
find a cartesian equation that relatesand
Because there areandterms in the parametric equations, we look for an equation that relates
andWe can rearrange
to give
and 
to give
hence 
Expanding and simplifying gives