- #1

- 160

- 0

## Homework Statement

Let ##\alpha(s)## and ##\beta(s)## be two unit speed curves and assume that ##\kappa_{\alpha}(s)=\kappa_{\beta}(s)## and ##\tau_{\alpha}(s)=\tau_{\beta}(s)##, where ##\kappa## and ##\tau## are respectively the curvature and torsion. Let ##J(s) = T_{\alpha}(s)\dot\ T_{\beta}(s)+N_{\alpha}(s) \dot\ N_{\beta}(s) +B_{\alpha}(s) \dot\ B_{\beta}(s).##

Show that:

##J(0)=3## and ##J(s)=3## implies that the Frenet frames of ##\alpha## and ##\beta## agree at ##s##

##J'(s) = 0## and ##\alpha(s) = \beta(s)## for all ##s##.

## Homework Equations

Frenet frames

## The Attempt at a Solution

For the first question, I know that the Frenet frame vectors $T,B,N$ are unit vectors but how can I formally prove the given statement?

I know for the second statement that ##J'(s) = 0## everywhere implies that the Frenet frames agree and since ##\alpha## and ##\beta## are unit speed, they are equal to the integral of their tangent vector. Also, how can I formally prove this second part?