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In mathematics, a parametric equation explicitly relates two or more variables in terms of one or more independent parameters. Abstractly, a relation is given in the form of an equation, and it is shown also to be the image of a functions from, say, Rⁿ. It is therefore somewhat more accurately defined as a parametric representation. See also parameter, parametrization, regular parametric representation.

For example, the simplest equation for a parabola,

,

can be parametrized by using a free parameter , and setting

.

Although the preceding example appears somewhat trivial, consider the following parametrization of a circle of radius :

.

Finally, there are certain geometric forms which are nearly impossible to describe as a single equation but have very elegant expressions in parametric form:

which describe a three-dimensional curve, the helix, which has a radius of a and rises by units per turn. (Note that the equations are identical in the plane to those for a circle; in fact, a helix is just "a circle whose ends don't have the same z-value".)

Such expressions as the one above are commonly written as

This way of expressing curves is practical as well as efficient; for example, one can integrate and differentiate such curves termwise. Thus, one can describe the velocity of a particle following such a parametrized path as:

and the acceleration as:

In general, a parametric curve is a function of one independent parameter (usually denoted ). Parametrized surfaces, of great use in such vector calculusMultivariate calculus Vector calculus is a field of mathematics concerned with multivariate real analysis of vectors in 2 or more dimensions. It consists of a suite of formulas and problem solving techniques very useful for engineering and physics. We con applications as Stokes' theoremStokes' theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. It is named after Sir George Gabriel Stokes ( 1819- 1903). Let M be an oriented piecewise smooth m, are functions of two parameters, most commonly or .

An example of a parametrized surface is the (capless) cylinderThe word cylinder has several meanings. For the geometric object, see Cylinder (geometry . For the engine component, see Cylinder (engine . In firearms the cylinder is the rotating device that contains the firing chambers of a revolver. The phonograph cyl given by

The fact that this represents a cylinder is evident when one considers the equation as representing a circle in the plane, which is then allowed to take on arbitrary values of z.