Parametric equation

A parametric equation is an equation where the coordinates are expressed in terms of a, usually represented with $t$ . The classic example is the equation of the unit circle,

$\begin{align}x&=\cos(t)\\y&=\sin(t)\end{align}$

Parametric equations are commonly used in physics to model the trajectory of an object, with time as the parameter. They are also used in multivariable calculus to create curves and surfaces.

Properties and applications

Parametric equations are often represented in terms of a radius vector $R$ , reducing the need for multiple equations. For example, the unit circle can also be represented as

$\vec R(t)=\cos(t)\mathbf{\hat i}+\sin(t)\mathbf{\hat j}$

with $\mathbf{\hat i},\mathbf{\hat j}$ being the unit vectors in the $x, y$ directions respectively.

A three-dimensional parametric surface

Surfaces can also be parameterized by using two parameters, often represented by $u$ and $t$ . For example the surface on the right is known as a catenoid, and has the equation

$\vec R(t)=\cosh(t)\cos(u)\mathbf{\hat i}+\cosh(t)\sin(u)\mathbf{\hat j}+t\mathbf{\hat k}$

Another common parametric surface is the unit sphere, which has the equation

$\vec R(t)=\sin(t)\cos(u)\mathbf{\hat i}+\sin(t)\sin(u)\mathbf{\hat j}+\cos(t)\mathbf{\hat k}$

Calculus with parametric curves

Derivatives

The derivative of a parametric function with respect to $x$ is equal to

$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}},\ \frac{dx}{dt}\ne0$

The second derivative is equal to

$\frac{d^2y}{dx^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right)=\dfrac{\dfrac{d}{dt}\left(\dfrac{dy}{dx}\right)}{\dfrac{dx}{dt}}$

Arc length

See also: Arc length function

The arc length of a parametric curve is equal to

$\begin{align}\int\limits_a^b\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}dt\end{align}$

or

$\begin{align}\int\limits_a^b\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2+\left(\frac{dz}{dt}\right)^2}dt\end{align}$

in $\mathbb R^{3}$ .

Area

The area between a parametric curve and the x-axis is equal to

$\int\limits_a^b y(t)x'(t)dt$

provided $x'(t)$ is never 0.