Equations written in terms of \(x\) and \(y\) are known as Cartesian equations.

For example, the general Cartesian equation for a circle with radius \(r\) and center at the origin is:

\(x^2+y^2=r^2\)

Equations written in terms of another parameter (e.g. \(t\)) are known as parametric equations.

For example, the general parametric equation for the above circle is:

\(x=r\cos{t}, y=r\sin{t}\)

Conversion from parametric to Cartesian

Substitution can be used to convert equations from parametric to Cartesian form by eliminating the parameter.

If the parametric equations involves trigonometric functions, then trigonometric identities can be used to convert from parametric to Cartesian form.

Domain and range

For parametric equations \(x=p(t), y=q(t)\) with Cartesian equation \(y=f(x)\):

• The domain of \(f(x)\) is the range of \(p(t)\);


• The range of \(f(x)\) is the range of \(q(t)\).