Definitions

# Parametric equation

In mathematics, parametric equations are a method of defining a curve. A simple kinematical example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion.

Abstractly, a relation is given in the form of an equation, and it is shown also to be the image of functions from items such as Rn. It is therefore somewhat more accurately defined as a parametric representation. It is part of regular parametric representation.

## Examples

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:

$x = a cos\left(t\right),$
$y = a sin\left(t\right),$

Parametric equations are convenient for describing curves in higher-dimensional spaces. For example:

$x = a cos\left(t\right),$
$y = a sin\left(t\right),$
$z = bt,$

describes 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

$r\left(t\right) = \left(x\left(t\right), y\left(t\right), z\left(t\right)\right) = \left(a cos\left(t\right), a sin\left(t\right), b t\right),$

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:

$v\left(t\right) = r\text{'}\left(t\right) = \left(x\text{'}\left(t\right), y\text{'}\left(t\right), z\text{'}\left(t\right)\right) = \left(-a sin\left(t\right), a cos\left(t\right), b\right),$

and the acceleration as:

$a\left(t\right) = r$(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). For the corresponding concept with two (or more) independent parameters, see Parametric surface.

## Conversion from two parametric equations to a single equation

Converting a set of parametric equations to a single equation involves solving one of the equations (usually the simplest of the two) for the parameter. Then the solution of the parameter is substituted into the remaining equation, and the resulting equation is usually simplified. It should be noted that the parameter is never present when the equation is in singular form (i.e., it must "cancel out" during conversion). Or, the process put simply: the simultaneous equations need to be solved for the parameter, and the result will be one equation. Additional steps need to be performed if there are restrictions on the value of the parameter.