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,
- ,
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 calculus applications as
Stokes' theorem, are functions of two parameters, most commonly or .
An example of a parametrized surface is the (capless) cylinder 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 values of
z.