Suppose a finite set of plane curves parameterized by $t$: $p_i(t)=(x_i(t),y_i(t)), i=1...N$, satisfy
- boundedness: $|p_i(t)|\leq 1\ \ \forall i$
- smoothness: $p_i(t)$ differentiable to any order, $\forall i$
- having non-postive curvature: $(\dot{x}_i\ddot{y}_i-\dot{y}_i\ddot{x}_i)\leq 0\ \ \forall i,t$
Question: is it possible that the superposition of all these curves $p(t)=p_1(t)+...+p_N(t)$ produces a curve that locally forms a counterclockwise loop? For example, a particular $p_i(t)$ may look like this
Note that it is always "turning right", a result of non-positive curvature. The questioned shape of a $p(t)$ may look like this:
What I already know is that $p(t)$ may have strictly positive curvature at some particular $t$, but I could not prove or disprove the existence of a counterclockwise "loop", which is more difficult to construct using "always-turning-right" curves.