Parametric Equations

Parametric plot of the circle

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 r^2=x^2+y^2, 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 t, while the symbols u and v 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.

See also

Parameter, Parameterization, Parametric Curve, Parametric Surface Explore this topic in the MathWorld classroom

Portions of this entry contributed by Christopher Stover

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Cite this as:

Stover, Christopher and Weisstein, Eric W. "Parametric Equations." From MathWorld--A Wolfram Web Resource.

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