Any closed manifold of genus $g$ can be cut open to form a topological disk and then mapped to a regular polygon with $4g$ sides. This construction is called the canonical polygonal schema of the manifold, and is a key ingredient for many applications in graphics and engineering, where a parameterization between two shapes with same topology is often needed. The sides of the $4g-$ gon define on the manifold a system of loops, which all intersect at a single point and are disjoint elsewhere. Computing a shortest system of loops of this kind is NP-hard. A computationally tractable alternative consists of computing a set of shortest loops that are not fully disjoint in polynomial time using the greedy homotopy basis algorithm proposed by Erickson and Whittlesey and then detach them in post processing via mesh refinement. Despite this operation is conceptually simple, known refinement strategies do not scale well for high genus shapes, triggering a mesh growth that may exceed the amount of memory available in modern computers, leading to failures. In this article we study various local refinement operators to detach cycles in a system of loops, and show that there are important differences between them, both in terms of mesh complexity and preservation of the original surface. We ultimately propose two novel refinement approaches: the former greatly reduces the number of new elements in the mesh, possibly at the cost of a deviation from the input geometry. The latter allows to trade mesh complexity for geometric accuracy, bounding deviation from the input surface. Both strategies are trivial to implement, and experiments confirm that they allow to realize canonical polygonal schemas even for extremely high genus shapes where previous methods fail.

中文翻译：

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可扩展的网格优化，用于极高属形状的规范多边形模式。

属的任何封闭流形 $ g $ 可以切开以形成拓扑磁盘，然后将其映射到具有 $ 4克双方。这种构造称为规范多边形图式它是流形的一部分，并且是许多图形和工程应用中的关键要素，在这些应用中，经常需要在具有相同拓扑的两个形状之间进行参数化。侧面$ 4克-$ gon在流形上定义一个循环系统，所有循环系统都在一个点处相交，而在其他地方不相交。计算这种最短的循环系统是NP-hard。一种易于计算的替代方案包括使用Erickson和Whittlesey提出的贪婪同伦基础算法计算一组在多项式时间内不完全不相交的最短循环，然后在后处理中通过网格细化分离它们。尽管此操作在概念上很简单，但是已知的优化策略无法很好地缩放高属形状，从而触发网格增长，可能超过现代计算机中可用的内存量，从而导致失败。在本文中，我们研究了各种局部细化运算符以分离循环系统中的循环，并表明它们之间存在重要区别，无论是网格复杂性还是原始表面的保留。我们最终提出了两种新颖的细化方法：前一种方法大大减少了网格中新元素的数量，可能以偏离输入几何形状为代价。后者允许将网格复杂度换为几何精度，以及与输入表面的边界偏差。两种策略的实现都很简单，而且实验证实，即使对于先前方法失败的极高属形状，它们也可以实现规范的多边形模式。与输入表面的边界偏差。两种策略的实现都很简单，而且实验证实，即使对于先前方法失败的极高属形状，它们也可以实现规范的多边形模式。与输入表面的边界偏差。两种策略的实现都很简单，而且实验证实，即使对于先前方法失败的极高属形状，它们也可以实现规范的多边形模式。