Arc Length

Arc length is defined as the length along a curve,

$$s \equiv \int_a^b\vert d{\boldsymbol{\ell}}\vert.$$ (1)

Defining the line element $ds^2\equiv \vert d{\boldsymbol{\ell}}\vert^2$, 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 ${\bf r}$ moves gives

$$s = \int_a^b ds = \int_a^b{{ds\over dt}\,dt} = \int_a^b\vert{\bf r}'(t)\vert\,dt.$$ (2)

In Polar Coordinates,

$$d\boldsymbol{\ell}= {\bf\hat r} \,dr + r {\bf\hat\theta}\,d\... ...er d\theta} {\bf\hat r} + r {\bf\hat\theta}}\right)\,d\theta,$$ (3)

so

$$ds = \vert d\boldsymbol{\ell}\vert = \sqrt{r^2 + \left({dr\over d\theta}\right)^2} \,d\theta$$ (4)

$$s = \int \vert d\boldsymbol{\ell}\vert = \int_{\theta_1}^{\theta_2}{\sqrt{r^2+\left({dr\over d\theta}\right)^2}\,d\theta}.$$ (5)

In Cartesian Coordinates,

$$d\boldsymbol{\ell} = x {\bf\hat x} + y {\bf\hat y}$$ (6)

$$ds = \sqrt{dx^2+dy^2} = \sqrt{\left({dy \over dx}\right)^2 +1}\,dx.$$ (7)

Therefore, if the curve is written

$${\bf r}(x) = x {\bf\hat x}+f(x){\bf\hat y},$$ (8)

then

$$s = \int_a^b \sqrt{1+f'^2(x)} \,dx.$$ (9)

If the curve is instead written

$${\bf r}(t) = x(t) {\bf\hat x}+ y(t) {\bf\hat y},$$ (10)

then

$$s = \int_a^b \sqrt{x'^2(t)+y'^2(t)}\, dt.$$ (11)

Or, in three dimensions,

$${\bf r}(t) = x(t){\bf\hat x}+ y(t){\bf\hat y} + z(t){\bf\hat z},$$ (12)

so

$$s = \int_a^b \sqrt{x'^2(t)+y'^2(t)+z'^2(t)}\,dt.$$ (13)

See also Curvature, Geodesic, Normal Vector, Radius of Curvature, Radius of Torsion, Speed, Surface Area, Tangential Angle, Tangent Vector, Torsion (Differential Geometry), Velocity