Parametric Equations

A set of equations linked by one or more independent variables (called the parameters).

So instead of y as a function of x, we can have both x and y as functions of a third variable t called the "parameter", which is often time.

Example, here are two functions linked by the parameter "t":

Parametric Equations
  • x = cos(t)
  • y = sin(t)

As t goes from 0 to 2π the x and y values make a circle!

at t=0: x=1 and y=0 (the right side of the circle)
at t=π/2: x=0 and y=1 (the top of the circle)
at t=π: x=−1 and y=0 (the left side of the circle)
etc.

It really does work!

Try it yourself with pencil and paper (and a calculator), using values of t from 0 to 2π in steps of 0.2: 0, 0.2, ... 6

Parametric Equations

Example:

This curve is made using:

  • x = a cos(t)
  • y = a sin(2t) / 2

Example: Flying bird

Imagine a bird flying through the sky. It is speeding up and at the same time dipping down and up in a curve. As time "t" goes by we can say:

  • It's forward motion is speeding up, example: x(t) = t + 0.2t2
  • It's height follows a curved path, example: y(t) = 10 + cos(t)

This is a natural way to describe it's motion, and could be harder if we tried to have it in purely y = f(x) style.

Some Shapes as Parametric Equations


Circle:
x = r cos(t) + h
y = r sin(t) + k
Ellipse:
x = a cos(t) + h
y = b sin(t) + k
Parabola:
x = 2pt + h
y = pt2 + k
Hyperbola:
x = a sec(t) + h
y = b tan(t) + k

Where we use these functions:

 

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