Parametric Equations -- from Wolfram MathWorld

Parametric equations are a set of equations that express a set of quantities as explicit functions of a number of independent variables, known as "parameters." For example, while the equation of a circle in Cartesian coordinates can be given by , one set of parametric equations for the circle are given by

illustrated above. Note that parametric representations are generally nonunique, so the same quantities may be expressed by a number of different parameterizations. A single parameter is usually represented with the parameter , while the symbols and are commonly used for parametric equations in two parameters.

Parametric equations provide a convenient way to represent curves and surfaces, as implemented, for example, in the Wolfram Language commands ParametricPlot[x, y, t, t1, t2] and ParametricPlot3D[x, y, z, u, u1, u2, v, v1, v2]. Unsurprisingly, curves and surfaces obtained by way of parametric equation representations are known as parametric curves and parametric surfaces, respectively.