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1. Arclength reparameterization. Suppose I is an interval and
 

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
(t) =
t
t0
|v|() d for I.
Then is strictly increasing with range some interval H and
(t) = |v|(t) for t I.
Let
: 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
(t) =
1
((t))
for t I and (s) =

  

Source: Allard, William K. - Department of Mathematics, Duke University

 

Collections: Mathematics