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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,
- <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.
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