Special curves- tangent vector of one is collinear with binormal vector of another curve

Question

Is there some special name for curves such that a tangent vector of one curve is colinear with a binormal vector of another curve?

Answer 1

I might be wrong, but it seems unlikely these pairs of curves have a name, since they are quite abundant. If $B_\alpha(s)$ is the binormal vector field along any curve $\alpha$, then the curve $\beta(s) = \int B(s)\,ds$ (componentwise integration) is a curve with tangent vector $T_\beta(s) = B_\alpha(s)$.

Note: there is no pair of curves such that the binormal lines on one curve equal tangent lines of the other.

Take two spaces curves $\alpha,\beta\colon I \subset \mathbb{R}\to \mathbb{E}^3$ and let $s$ be a unit speed parameter for $\alpha$. Assume that the binormal lines of $\alpha$ at $\alpha(s)$ equals the tangent line of $\beta$ at corresponding points $\beta(s)$.

Every point $\beta(s)$ lies on a binormal line of $\alpha$, hence there is a function $\lambda\colon I \to \mathbb{R}$ such that $$ \beta(s) = \alpha(s) + \lambda(s) B_\alpha(s). $$ Deriving and using Frenet gives $$ \beta'(s) = T_\alpha(s) + \lambda'(s) B_\alpha(s) - \lambda(s) \tau(s)N_\alpha(s). $$ Now take the inner product with $T_\alpha(s)$. Since $\beta'(s)$ is parallel with $T_\beta(s) = \pm B_\alpha(s)$, we get $0 = 1$. Contradiction.