# Unit speed curves

## 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##.

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?

Dick
Homework Helper

## 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##.

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?

For any two vectors v.w=|v||w|cos(θ). If v and w are unit vectors then that's less than one unless the vectors are parallel. So?

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Dick -

If they are unit vectors and are parallel then there dot product is 1.

Dick
Homework Helper
Dick -

If they are unit vectors and are parallel then there dot product is 1.

The point is the dot product is ONLY 1 if they are parallel, otherwise it's less. You have three of them summing to 3. So?

Since they sum to 3 they are parallel hence their dot product equals 1.

Is it really that simple or am I missing something?

Dick
Homework Helper
Since they sum to 3 they are parallel hence their dot product equals 1.

Is it really that simple or am I missing something?

No, I think it's really that simple.

Thanks Dick!

For D(s), is it essentially the same thing?

Dick
Homework Helper
Thanks Dick!

For D(s), is it essentially the same thing?

What's D(s)? Look, if J(s)=3 then all three Frenet frame vectors must be equal in ##\alpha## and ##\beta##, yes? Am I missing something?

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Opps, I meant J(s), sorry about that. But yes they will be equal, thanks again!