Suppose Mario is walking on the surface of a planet. If he starts walking from a known location, in a fixed direction, for a predetermined distance, how quickly can we determine where he will stop?
More formally, suppose we are given a convex polytope $P$ in 3-space, a starting point $s$ on the surface of $P$, a direction vector $v$ (in the plane of some facet containing $p$), and a distance $\ell$. How quickly can we determine which facet of $P$ Mario will stop inside? (As a technical point, assume that if Mario walks into a vertex of $P$, he immediately explodes; fortunately, this almost never happens.)
Or if you prefer: suppose we are given the polytope $P$, the source point $s$, and the direction vector $v$ in advance. After preprocessing, how quickly can we answer the question for a given distance $\ell$?
It's easy to simply trace Mario's footsteps, especially if $P$ has only triangular facets. Whenever Mario enters a facet through one of its edges, we can determine in $O(1)$ time which of the other two edges he must leave through. Although the running time of this algorithm is only linear in the number of edge-crossings, it's unbounded as a function of the input size, because the distance $\ell$ could be arbitrarily larger than the diameter of $P$. Can we do better?
(In practice, the path length isn't actually unbounded; there is a global upper bound in terms of the number of bits needed to represent the input. But insisting on integer inputs raises some rather nasty numerical issues — How do we compute exactly where to stop? — so let's stick to real inputs and exact real arithmetic.)
Is anything nontrivial known about the complexity of this problem?
Update: In light of julkiewicz's comment, it seems clear that a real-RAM running time bounded purely in terms of $n$ (the complexity of the polytope) is impossible. Consider the special case of a two-sided unit square $[0,1]^2$, with Mario starting at $(0,1/2)$ and walking in direction $(1,0)$. Mario will stop on the front or the back of the square depending on the parity of the integer $\lfloor \ell \rfloor$. We can't compute the floor function in constant time on the real RAM, unless we're happy equating PSPACE and P. But we can compute $\lfloor \ell \rfloor$ in $O(\log \ell)$ time by exponential search, which is an exponential improvement over the naive algorithm. Is time polynomial in $n$ and $\log \ell$ always achievable?