Prove that certain parameterization is one to one

Question

Let $\alpha:I\longrightarrow\mathbb{R}^2$ be a one to one $\mathcal{C}^{\infty}$ curve parameterized by arc length. Im considering the function $$F(s,t)=\alpha(s)+tN(s)$$ where $\{T(s),N(s)\}$ is the Frenet dihedron. I want to find $U\subset\mathbb{R}^2$ such that $F|_U$ is a parameterization (injective and injective diferential). It's straight foward to compute the diferential and see that a necessary condition is $1\neq tk(s)$, where $k(s)$ is the curvature, but I haven't managed to prove that $U=\{(s,t):tk(s)\neq 1 \}$ or anything similar does what I want.