Show that a "smooth" curve intersects boundary of small enough ball at two points

Question

Given a "smooth" (non selfintersecting) curve $f:[0,1] ->$ $\mathbb{R}^2$. $t\in (0,1)$ I want to show that for small enough $r$, there are exactly two points $t_1,t_2$ for which $$\| f(t_i)- f(t) \|_2 =r $$ and further $t_1<t<t_2$.

Not sure how much smoothness is needed.