Arc Length
Arc length is defined as the length along a curve,
| [画像: s=int_gamma|dl|, ] |
(1)
|
where dl is a differential displacement vector along a curve gamma. For example, for a circle of radius r, the arc length between two points with angles theta_1 and theta_2 (measured in radians) is simply
| s=r|theta_2-theta_1|. |
(2)
|
Defining the line element ds^2=|dl|^2, parameterizing the curve in terms of a parameter t, and noting that ds/dt is simply the magnitude of the velocity with which the end of the radius vector r moves gives
so
Therefore, if the curve is written
| r(x)=xx^^+f(x)y^^, |
(11)
|
then
If the curve is instead written
| r(t)=x(t)x^^+y(t)y^^, |
(13)
|
then
In three dimensions,
| r(t)=x(t)x^^+y(t)y^^+z(t)z^^, |
(15)
|
so
The arc length of the polar curve r=r(theta) is given by
See also
Curvature, Geodesic, Normal Vector, Radius of Curvature, Radius of Torsion, Speed, Surface Area, Tangential Angle, Tangent Vector, Torsion, Velocity Explore this topic in the MathWorld classroomExplore with Wolfram|Alpha
More things to try:
Cite this as:
Weisstein, Eric W. "Arc Length." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ArcLength.html