A Level Maths Notes: FP1 – 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
