The lost in a forest problem is famous, specifically see this problem. The problem takes place in the plane. Consider the following 3d version. There is a point $p \in \mathbb{R}^3$ and a plane $P$ which is unit distance from $p$. My question: What is the minimum length (rectifiable) curve $C$ in $\mathbb{R}^3$ which starts at $p$, and such that all rotations of $C$ about $p$ intersect $P$? (Of course, rotations are elements of $SO(3)$.)
I suspect a minimum length curve actually exists, because we can bound it from above. For example, let $r>1$ and consider the curve consisting of a radius of length $r$ from $p$ together with two circles $C_1, C_2$ of radii $r$ centered at $p$ such that $C_1$ intersects $C_2$ at a right angle. For sufficiently large $r>1$, it is clear that all rotations of this curve intersect $P$.
My second question: What about the problem in $\mathbb{R}^3$ in which we replace the plane $P$ with a line $\ell$ unit distance from $p$. This is exactly the lost in a forest problem but in $\mathbb{R}^3$. Is there a minimum length curve for this problem, and if so what is it? I suspect that there does not exist a rectifiable curve $C$ starting at $p$ such that all rotations of it intersect $\ell$. Can this be proven (or disproven)?
I am not sure if these problems have been studied (I know related problems have been). If so, can anybody point me to a reference? Thank you.