Parametric Coordinates – Converting Between Rectangular or Cartesian and Parametric Form
Parametric equations define a curve in terms of some third quantity. The
and
coordinates are expressed in terms of this quantity, called a parameter. For example the line
which is written in cartesian coordinates may be written in parametric form as
where
is the parameter.. Notice that thecoordinate here is always one more than thecoordinate, reflecting that for the line
we add one to thevalue to obtain thevalue.
We can convert a cartesian equation to parametric form or parametric to cartesian form.
To convert parametric to cartesian form, you can invert one or both of the equations
to makeoror both the subject. If only one is inverted, say
to obtain
we can substitute this into the equation
to obtain
which now givesas a function of
If both are inverted then we have
where
and
are distinct functions. This equation my be rearranged, if possible or desirable to giveas a function of
which is often the desired form.
Example: Convert
into cartesian form.
and
The simplest method is to makethe subject into the second equation and substitute this into the first.
the above equation may be inverted to makethe subject, obtaining
To convert cartesian into parametric form we have to introduce a parameter. For example
Let
and
and the the curve becomes the parametric coordinates
