Arc Length

Arc length is defined as the length along a curve,

(1)

where is a differential displacement vector along a curve . For example, for a circle of radius , the arc length between two points with angles and (measured in radians) is simply

(2)

Defining the line element , parameterizing the curve in terms of a parameter , and noting that is simply the magnitude of the velocity with which the end of the radius vector moves gives

(3)

In polar coordinates,

(4)

so

			(5)
			(6)

In Cartesian coordinates,

			(7)
			(8)
			(9)
			(10)

Therefore, if the curve is written

(11)

then

(12)

If the curve is instead written

(13)

then

(14)

In three dimensions,

(15)

so

(16)

The arc length of the polar curve is given by

(17)