In the plane equipped with an orthonormal basis, let us consider the two points $A$ and $B$ whose coordinates are $(-2,0)$ and $(1,1)$, respectively. Is there a path from $A$ to $B$ (i.e. a continuous map $\gamma : [0,1] \to {\mathbb R}^2$, with $\gamma(0)=A, \gamma(1)=B$) such that the sets $I,\rho (I), \rho^2 (I)$ and $\rho ^3(I)$ are all disjoint, where $\rho$ is the rotation whose center is the origin and whose angle is $\frac{\pi}{2}$, and $I$ is the image of the path : $I=\gamma([0,1])$.
Update 14:30 Now that I know that such paths do exist (see carlop’s answer), I ask : what is the minimal length of a path meeting those constraints ?