论文标题
为每个身体剪裁：重塑凸Polyhedra
Tailoring for Every Body: Reshaping Convex Polyhedra
论文作者
论文摘要
鉴于任何两个凸Polyhedra p和Q，我们证明是我们的主要结果之一，即P的表面可以通过有限的“定制”步骤将P重塑为Q的同型。每个裁缝都会围绕一个顶点围绕一个digon切开，并缝合二分法关闭。该结果的一个措辞是，如果可以通过一系列带有飞机的切片从p中“雕刻” Q，那么Q可以从P中量身定制。并且在某种意义上，量身定制的量化远非雕刻P Pollohedra量身定制，因为Polyhedra可以通过雕刻的Polhedra量身定制。近似于S的同型近似。因此，可以将p“鞭打”到一个球体。 另一个主要结果可以实现相同的重塑，但是通过切除更复杂的形状，我们称为“波峰”，每个都封闭一个顶点。逆转Digon-Tailoring或crest-tailoring可以通过切割Q并插入和密封表面贴片来扩大P内部P内的任何Q。 这些结果的一个令人惊讶的推论是，对于q的一个子集，我们可以将Q切成碎片，并将其粘贴到P.的等轴测子集上。这可以看作是“展开” Q的一种形式。 我们所有的证明都是建设性的，并导致多项式算法。
Given any two convex polyhedra P and Q, we prove as one of our main results that the surface of P can be reshaped to a homothet of Q by a finite sequence of "tailoring" steps. Each tailoring excises a digon surrounding a single vertex and sutures the digon closed. One phrasing of this result is that, if Q can be "sculpted" from P by a series of slices with planes, then Q can be tailored from P. And there is a sense in which tailoring is finer than sculpting in that P may be tailored to polyhedra that are not achievable by sculpting P. It is an easy corollary that, if S is the surface of any convex body, then any convex polyhedron P may be tailored to approximate a homothet of S as closely as desired. So P can be "whittled" to e.g., a sphere S. Another main result achieves the same reshaping, but by excising more complicated shapes we call "crests," still each enclosing one vertex. Reversing either digon-tailoring or crest-tailoring leads to proofs that any Q inside P can be enlarged to P by cutting Q and inserting and sealing surface patches. One surprising corollary of these results is that, for Q a subset of P, we can cut-up Q into pieces and paste them non-overlapping onto an isometric subset of P. This can be viewed as a form of "unfolding" Q onto P. All our proofs are constructive, and lead to polynomial-time algorithms.
