Cycloid

In the previous section, curves were defined by providing a relation between x and y [i.e., y = f(x) or x = g(y)]. Another method, which can be used to describe curves, is by means of the parametric equations.

Plot of x = 2t² and y = 2 + 4t

Some curves (e.g., cycloid, which is traced by rolling a circle along a flat surface) are easier to describe using a pair of parametric equations. The coordinates x and y of the curve are given using a third variable t, such as

     x = f(t) and y = g(t)

where t is referred to as the parameter. Hence, for a given value of t, a point (x,y) is determined. For example, let t be the time while x and y are the positions of a particle. The parametric equations then describe the path of the particle at different times.

The curve shown on the left is defined using the following parametric equations:

     x = 2t²
     y = 2 + 4t

for 0 ≤ t ≤ 2. As shown in the table, once the value of t is given, the coordinates are set.