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

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 , 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.

A three-dimensional parametric surface

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

Derivatives

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

The second derivative is equal to

Arc length

The arc length of a parametric curve is equal to

or

in .

Area

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

provided is never 0.

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