# Geometry Question

• February 8th 2011, 09:54 AM
maximus101
Geometry Question
If C is the curve parameterised by
t à (cost, sint, cosht):[-pi,pi] à reals ^3
Reparameterise C by arclength?
• February 8th 2011, 02:34 PM
zzzoak
$
s=\int _{- \pi}^t \sqrt{x'^2t)+y'^2(t)+z'^2(t)}dt
$

$
s=\int _{- \pi}^t \sqrt{sin^2 p+cos^2 p+sinh^2 p} \; dp=\int _{- \pi}^t \sqrt{1+sinh^2 p} \; dp=\int _{- \pi}^t \; cosh (p) \; dp=
$

$
=sinh(t)-sinh(- \pi).
$

Now we need to express t as function of s

$
t=f(s)
$

and the curve becomes

$
( \; cos(f(s)), \; sin(f(s)), \; cosh(f(s)) \; ).
$