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,

y=x^2 \;

can be parametrized by using a free parameter

t

, and setting

x=t,y=t^2 \;

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

a

x\equiv a\cos t,y\equiv a\sin 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:

x\equiv a\cos t

y\equiv a\sin t

z\equiv bt

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

2 \pi b

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

and the acceleration as:

a(t)\equiv r

(t)=(x(t),y(t),z(t))=(-a\cos t,-a\sin 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)

(u,v)

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

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

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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