## Parametric Coordinates – Converting Between Rectangular or Cartesian and Parametric Form

Parametric equations define a curve in terms of some third quantity. Theandcoordinates are expressed in terms of this quantity, called a parameter. For example the linewhich is written in cartesian coordinates may be written in parametric form aswhereis the parameter.. Notice that thecoordinate here is always one more than thecoordinate, reflecting that for the linewe 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 equationsto makeoror both the subject. If only one is inverted, sayto obtainwe can substitute this into the equationto obtainwhich now givesas a function ofIf both are inverted then we havewhereandare distinct functions. This equation my be rearranged, if possible or desirable to giveas a function ofwhich is often the desired form.

Example: Convertinto cartesian form.

andThe 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

Letandand the the curve becomes the parametric coordinates