Parameterizing a curve by arc length

To parameterize a curve by arc length, the procedure is

  1. Find the arc length $ s(t)$ . If the curve is regular then $ s(t)$ is a monotonically increasing function.
  2. Calculate the inverse of the arc length $ t(s)$ . This is another monotonically increasing function.
  3. Set the new parameter $ u = t(s)$ . Now $ \alpha(u)$ has arc length parameterization.
Intuitively, if the curve if ``moving'' quickly in some region, then its arc length will increase rapidly and the inverse $ t(s)$ will increase proportionally slowly. In the parameterized curve $ \alpha(u)$ these two factors will cancel out and the curve will increase in length uniformly. What is the velocity of $ \alpha(u)$ ? We have

$\displaystyle \alpha'(u)$ $\displaystyle = \ensuremath{\frac{d}{d s}}\alpha(t(s))$ (2.1)
  $\displaystyle = \alpha'(t) t'(s)$ (2.2)
  $\displaystyle = \alpha'(t) / s'(t)$ (2.3)
  $\displaystyle = \alpha'(t) / \vert \alpha'(t) \vert,$ (2.4)

where we have used the fact that $ \ensuremath{\frac{d}{d s}}t(s) = 1 / \ensuremath{\frac{d}{d t}} s(t)$ when $ s$ and $ t$ are inverses of each other, and the definition of arc length in equation (2.4). Hence we have found that the velocity $ \vert\alpha'(u)\vert$ is one for an arc length parameterized curve.

Copyright © 2005 Adrian Secord. All rights reserved.