平行切片中多面体间的多面体重构
Reconstructing a Polyhedron between Polygons in Parallel Slices
摘要：
给定两个$n$ -顶点多边形，$P=(p_1, \ldots, p_n)$位于$z=0$的$xy$-平面中，而$P'=(p'_1, \ldots, p'_n)$位于$z=1$的$xy$ -平面中，带状曲面是与连接$P$和$P'$的环同胚的三角化曲面，使得三角化的边集包含所有$i =1, \ldots, n$的连接$p_i$和$p'_i$的顶点不相交路径$π_i$.然后，该表面由带组成，其中$i$ th带在$π_i$和$π_{i+1}$之间。我们给出了一个多项式时间算法，找到一个带状表面没有斯坦纳点，如果存在。我们探讨带状表面和线性变形之间的连接，在变形中的时间对应于$z$方向。特别是，我们表明，如果$P$和$P'$是凸的和线性变形从$P$ $P'$（它移动$i$第顶点上的一条直线从$p_i$到$p'_i$）保持平面在任何时候，那么有一个带状表面没有斯坦纳点。
英文摘要：
Given two $n$-vertex polygons, $P=(p_1, \ldots, p_n)$ lying in the $xy$-plane at $z=0$, and $P'=(p'_1, \ldots, p'_n)$ lying in the $xy$-plane at $z=1$, a banded surface is a triangulated surface homeomorphic to an annulus connecting $P$ and $P'$ such that the triangulation's edge set contains vertex disjoint paths $π_i$ connecting $p_i$ to $p'_i$ for all $i =1, \ldots, n$. The surface then consists of bands, where the $i$th band goes between $π_i$ and $π_{i+1}$. We give a polynomial-time algorithm to find a banded surface without Steiner points if one exists. We explore connections between banded surfaces and linear morphs, where time in the morph corresponds to the $z$ direction. In particular, we show that if $P$ and $P'$ are convex and the linear morph from $P$ to $P'$ (which moves the $i$th vertex on a straight line from $p_i$ to $p'_i$) remains planar at all times, then there is a banded surface without Steiner points.
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