Summary: 1. Arclength reparameterization.
Suppose I is an interval and
r : I Rn
is a curve in Rn
whose speed is never zero. Suppose t0 I and let
|v|() d for I.
Then is strictly increasing with range some interval H and
(t) = |v|(t) for t I.
: H I
be the function which is inverse to . Then
((t)) = t for t I and ((s)) = s for s H.
From the chain rule we obtain
for t I and (s) =