## 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 the coordinate here is always one more than the coordinate, reflecting that for the line we add one to the value to obtain the value.

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 make or or both the subject. If only one is inverted, say to obtain we can substitute this into the equation to obtain which now gives as 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 give as a function of which is often the desired form.

Example: Convert into cartesian form. and The simplest method is to make the subject into the second equation and substitute this into the first. the above equation may be inverted to make the 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  