TITLE: Short arc of small curvature whose every planar shadow crosses itself
QUESTION [14 upvotes]: Let $\cal{A}$ denote the family of all twice-differentiable simple open arcs $A$ in $\mathbb{R}^3$ satisfying the following properties:
(1) at each point of every arc $A\in\cal{A}$ the curvature of $A$ is at most $1$;
(2) no perpendicular projection of $A$ to any plane is an injection.

Question 1. What is the infimum of the length $|A|$ of $A\in\cal{A}$ ?
Question 2. Is there an arc $A_{min}$ of minimum length among all arcs in $\cal{A}$? If so, is it unique (up to isometry)?
Remark. This question is related to the notion of rope knots, see: Hans Stricker, Some questions about ideal knots

REPLY [2 votes]: This doesn't seem close to tight, but there's an easy lower bound of $\pi$. If you have a curve shorter than $\pi$, choose any projection direction perpendicular to the curve at its midpoint. The resulting projected curve will be monotonic and therefore non-self-intersecting.