Properties and applications
Parametric equations are often represented in terms of a radius vector , reducing the need for multiple equations. For example, the unit circle can also be represented as
with being the unit vectors in the directions respectively.
Surfaces can also be parameterized by using two parameters, often represented by and . For example the surface on the right is known as a catenoid, and has the equation
Another common parametric surface is the unit sphere, which has the equation
Calculus with parametric curves
The derivative of a parametric function with respect to is equal to
The second derivative is equal to
The arc length of a parametric curve is equal to
The area between a parametric curve and the x-axis is equal to
provided is never 0.