Arc Length
المؤلف:
المرجع الالكتروني للمعلوماتيه
المصدر:
المرجع الالكتروني للمعلوماتيه
الجزء والصفحة:
...
29-9-2018
2550
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
In Cartesian coordinates,
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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