The Parameter Equation in 2D

By a plain curve described using a parameter equation, one means that as both the x- and the y-coordinate of the points of the curve can be expressed in terms of a real parameter u, as illustrated below:

Parameter equation in 2D

For the parameter u, there is always a range needed. Here, you can find for instance the image of the curve with parameter equation x(u) = 4*sin(4*u) and y(u) = 3*sin(5*u):

Lissajous image

As range for the parameter u, the interval (0, 2*pi) was given. The curve that we obtain is known as a Lissajous image. Lissajous images arise in physics as the orbit of a particle that is influenced by two perpendicular harmonical vibrations.