1. ## Arc Length Parametrization

Don't need too much help here, just a bit of clarification.

Suppose I have to find the arclength parametrization of t --> (cost,sint,cosht) : [-pi,pi] --> R^3 .

I take the norm of the derivitive of the function which ends up as cosh(t) and integrate it. Do I integrate cosh(s)ds (random dummy variable) with upper limit t and lower limit 0, or upper limit t and lower limit -pi ?

Unfortunately our notes don't explain the choice of limits.

Thanks.

2. It doesn't matter. Your lower limit determines where you start measuring the arclength but any starting point will give you an arclength parameterization.

Specifically, $\displaystyle s= \int_0^s |\vec{T}|dt$ gives a parameterization with s= 0 at (1, 0, 1), where t= 0. That way, values of t in $\displaystyle [-\pi, 0)$ will have negative values of s and values of t in $\displaystyle (0, \pi]$ will have positive values of s.

If you use, instead, $\displaystyle s= \int_{-\pi}^s|\vec{T}|dt$ then s= 0 will correspond to the point (-1, 0, cosh(-1)) where $\displaystyle t= -\pi$ and all points in $\displaystyle [-\pi, \pi]$ will correspond to positive values of s.

(You do have "t" and "s" reversed in your integrals. If "s" represents arc length, your limits of integration should be "s" not t.)