Let $\alpha(t)$ and $\beta(t)$ be two different curves. $\alpha'(t)=T(t)-k(t)T(t)\lambda(t)$?

Question

I can't understand a step in a solution:

Let $\alpha(t)$ and $\beta(t)$ be two different curves. Suppose that $\beta(t)=\alpha(t)+\lambda(t)N(t)$.

Then:

$$\beta'(t)=\alpha'(t)+\lambda'(t)N(t)+\lambda(t)N'(t)=T(t)-k(t)T(t)\lambda(t)+\lambda'(t)N(t)-\lambda(t)\tau(t)B(t).$$

How they arrive to $\alpha'(t)=T(t)-k(t)T(t)\lambda(t)$? I suppose that they are writing $T_{\alpha(t)}(t)$ in $\beta(t)$ frame, or something similar, right?

Thanks for your time.

Answer 1

Just a direct application of arc rate. See Frenet -Serret relations, for arc rate of normal. Adopt the same $TNB$ vector handedness of forming a right or a left handed system.(RH parallel Bertrand above LH parallel below).