Let be any curve and . We define the arc length function from which will denote by by
Since is, in general, continuous, the function is only and
If we assume is regular, i.e. for any , then by the Inverse Function Theorem is differentiable increasing open function. Then if we put
then is a diffeomorphism between two open intervals. Let be the inverse diffeomorphism and let be the re-parametrization of given by
Then this new curve satisfies
and thus for any . It follows from the above that any regular curve admits a re-parametrization by arc length. So we have
Definition. A curve is said to be parametrized by arc length if for all .
Roughly speaking, instead of using a time variable to parametrize a curve we use its arc length . The new equation for the curve in terms of is called the curve parametrized by arc length.
Example (logarithmic spiral). The curve given by
with and is called the logarithmic spiral. The case and is showed below.
The arc length of the logarithmic spiral is given by
Solving this equation gives
Thus the equation for parametrized by arc length is given by
A lot of formulas become simpler when your curve is parametrized by arc length. For example, the curvature is calculated by taking the derivative of the tangent vector. When the curve is parametrized by arc length, the tangent vector has constant length one. This implies that the derivative of the tangent vector is always normal to the tangent vector, which is necessary to find the curvature.