# Parametric equation

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Graph of a butterfly curve, a parametric equation discovered by Temple H. Fay
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,

y = x2,

can be parametrized by using a free parameter t, and setting

x = t, y = t2.

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

xacos(t), yasin(t).

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

xacos(t)
yasin(t)
zbt

which describe a three-dimensional curve, the helix, which has a radius of a and rises by 2πb 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

r(t) ≡ (x(t), y(t), z(t)) = (acos(t), asin(t), bt).

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:

v(t) ≡ r′(t) = (x′(t), y′(t), z′(t)) = (−asin(t), acos(t), b)

and the acceleration as:

a(t) ≡ r′′(t) = (x′′(t), y′′(t), z′′(t)) = (−acos(t), −asin(t), 0).

In general, a parametric curve is a function of one independent parameter (usually denoted t). Parametrized surfaces, of great use in such vector calculus applications as Stokes' theorem, are functions of two parameters, most commonly (s, t) or (u,v).

An example of a parametrized surface is the (capless) cylinder given by

r(u,v) ≡ (x(u,v), y(u,v), z(u,v)) = (acos(u), asin(u), v).

Considering the equation as representing a circle in the plane, it is evident that this represents a cylinder. It is then allowed to take on arbitrary values of z.

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