Langara College - Department of Mathematics and Statistics

## Parametric Equations

Many interesting curves are not function graphs with y=f(x), but can be described by specifying both x and y as functions of a third variable (called a parameter). For example the circle x^2+y^2=1 can be described "parametrically" with x=cos(t) and y=sin(t). The path of any moving point can be described this way with the parameter, t, being the time.

There is a fair amount of precalculus material on parametric equations.

We can use calculus to express the slope of the curve in terms of the derivatives of x(t) and y(t).
In fact, by the chain rule, dy/dx = (dy/dt)/(dx/dt). This can be used to help sketch the curve and to find exact locations of 'critical points' where the tangent is horizontal or vertical.
(See, for example,  this page from the University of Tennessee's Visual Calculus site )

We can also express the area inside a closed loop of the curve by means of a definite integral.
(See Area Inside a Parametric Curve(MathCad+QT@odu) )

If you have come across good web-based illustrations of these or any other calculus applications to  parametric curves, please do let us know and we will add them here.