A closed quasigeodesic is a closed curve on the surface of a polyhedron with at most $180^\circ$ of surface on both sides at all points; such curves can be locally unfolded straight. In 1949, Pogorelov proved that every convex polyhedron has at least three (non-self-intersecting) closed quasigeodesics, but the proof relies on a nonconstructive topological argument. We present the first finite algorithm to find a closed quasigeodesic on a given convex polyhedron, which is the first positive progress on a 1990 open problem by O'Rourke and Wyman. The algorithm establishes for the first time a quasipolynomial upper bound on the total number of visits to faces (number of line segments), namely, $O\left(\frac{n \, L^3}{\epsilon^2 \, \ell^3}\right)$ where $n$ is the number of vertices of the polyhedron, $\epsilon$ is the minimum curvature of a vertex, $L$ is the length of the longest edge, and $\ell$ is the smallest distance within a face between a vertex and a nonincident edge (minimum feature size of any face). On the real RAM, the algorithm's running time is also pseudopolynomial, namely $O\left(\frac{n \, L^3}{\epsilon^2 \, \ell^3} \log n\right)$. On a word RAM, the running time grows to $O\left(b^2 \cdot \frac{n^8 \log n}{\epsilon^8} \cdot \frac{L^{21}}{\ell^{21}}\cdot 2^{O(|\Lambda|)}\right)$, where $|\Lambda|$ is the number of distinct edge lengths in the polyhedron, assuming its intrinsic or extrinsic geometry is given by rational coordinates each with at most $b$ bits. This time bound remains pseudopolynomial for polyhedra with $O(\log n)$ distinct edges lengths, but is exponential in the worst case. Along the way, we introduce the expression RAM model of computation, formalizing a connection between the real RAM and word RAM hinted at by past work on exact geometric computation.

翻译：封闭准测地线是多面体表面上的一条闭合曲线，在任意点两侧的表面角均不超过$180^\circ$；此类曲线可局部展开为直线。1949年，波戈列洛夫证明每个凸多面体至少存在三条（非自交）封闭准测地线，但该证明依赖于非构造性的拓扑论证。我们提出了首个在给定凸多面体上寻找封闭准测地线的有限算法，这是对O'Rourke与Wyman于1990年提出的开放问题的首个积极进展。该算法首次建立了访问面片总次数（线段数量）的拟多项式上界，即$O\left(\frac{n \, L^3}{\epsilon^2 \, \ell^3}\right)$，其中$n$为多面体顶点数，$\epsilon$为顶点最小曲率，$L$为最长边长度，$\ell$为面内顶点与非相邻边之间的最小距离（任意面的最小特征尺寸）。在实数RAM模型上，算法运行时间同样为伪多项式，即$O\left(\frac{n \, L^3}{\epsilon^2 \, \ell^3} \log n\right)$。在字RAM模型上，运行时间增至$O\left(b^2 \cdot \frac{n^8 \log n}{\epsilon^8} \cdot \frac{L^{21}}{\ell^{21}}\cdot 2^{O(|\Lambda|)}\right)$，其中$|\Lambda|$为多面体中不同边长的数量，假设其内在或外在几何由每个至多$b$比特的有理坐标给出。对于边长相异数量为$O(\log n)$的多面体，该时间复杂度仍保持伪多项式性质，但在最坏情况下呈指数增长。在此过程中，我们引入了表达式RAM计算模型，将实数RAM与字RAM之间的联系形式化——该联系此前在精确几何计算研究中已有所暗示。