If C is the curve parameterised by
t à (cost, sint, cosht):[-pi,pi] à reals ^3
Reparameterise C by arclength?
$\displaystyle
s=\int _{- \pi}^t \sqrt{x'^2t)+y'^2(t)+z'^2(t)}dt
$
$\displaystyle
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=
$
$\displaystyle
=sinh(t)-sinh(- \pi).
$
Now we need to express t as function of s
$\displaystyle
t=f(s)
$
and the curve becomes
$\displaystyle
( \; cos(f(s)), \; sin(f(s)), \; cosh(f(s)) \; ).
$