Parametric equation

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,

<math>y=x^2 \;<math>,

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

<math>x=t,y=t^2 \;<math>.

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

<math>x\equiv a\cos t,y\equiv a\sin t<math>.

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

<math>x\equiv a\cos t<math>

<math>y\equiv a\sin t<math>

<math>z\equiv bt<math>

which describe a three-dimensional curve, the helix, which has a radius of a and rises by <math>2 \pi b<math> 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

<math>r(t)\equiv(x(t),y(t),z(t))=(a\cos t,a\sin t, bt)<math>

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:

<math>v(t)\equiv r'(t)=(x'(t),y'(t),z'(t))=(-a\sin t,a\cos t, b)<math>

and the acceleration as:

<math>a(t)\equiv r''(t)=(x''(t),y''(t),z''(t))=(-a\cos t,-a\sin t, 0)<math>

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

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

<math>r(u,v)\equiv(x(u,v),y(u,v),z(u,v))=(a\cos u,a\sin u, v)<math>

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.