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Date: 10-6-2019
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Date: 5-9-2019
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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) |
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(6) |
In Cartesian coordinates,
(7) |
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(8) |
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(9) |
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(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) |
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