Arc length is defined as the length along a curve,
![\begin{displaymath}
s \equiv \int_a^b\vert d{\boldsymbol{\ell}}\vert.
\end{displaymath}](a_1334.gif) |
(1) |
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
![\begin{displaymath}
s = \int_a^b ds = \int_a^b{{ds\over dt}\,dt} = \int_a^b\vert{\bf r}'(t)\vert\,dt.
\end{displaymath}](a_1338.gif) |
(2) |
In Polar Coordinates,
![\begin{displaymath}
d\boldsymbol{\ell}= {\bf\hat r} \,dr + r {\bf\hat\theta}\,d\...
...er d\theta}
{\bf\hat r} + r {\bf\hat\theta}}\right)\,d\theta,
\end{displaymath}](a_1339.gif) |
(3) |
so
In Cartesian Coordinates,
Therefore, if the curve is written
![\begin{displaymath}
{\bf r}(x) = x {\bf\hat x}+f(x){\bf\hat y},
\end{displaymath}](a_1347.gif) |
(8) |
then
![\begin{displaymath}
s = \int_a^b \sqrt{1+f'^2(x)} \,dx.
\end{displaymath}](a_1348.gif) |
(9) |
If the curve is instead written
![\begin{displaymath}
{\bf r}(t) = x(t) {\bf\hat x}+ y(t) {\bf\hat y},
\end{displaymath}](a_1349.gif) |
(10) |
then
![\begin{displaymath}
s = \int_a^b \sqrt{x'^2(t)+y'^2(t)}\, dt.
\end{displaymath}](a_1350.gif) |
(11) |
Or, in three dimensions,
![\begin{displaymath}
{\bf r}(t) = x(t){\bf\hat x}+ y(t){\bf\hat y} + z(t){\bf\hat z},
\end{displaymath}](a_1351.gif) |
(12) |
so
![\begin{displaymath}
s = \int_a^b \sqrt{x'^2(t)+y'^2(t)+z'^2(t)}\,dt.
\end{displaymath}](a_1352.gif) |
(13) |
See also Curvature, Geodesic, Normal Vector, Radius of Curvature, Radius of Torsion,
Speed, Surface Area, Tangential Angle, Tangent Vector, Torsion (Differential Geometry),
Velocity
© 1996-9 Eric W. Weisstein
1999-05-25