应用同伦群计算表示复杂拓扑闭曲面的方形图  

The Square Drawing Representative for Closed Surfaces with Complicated Topology by Use of Computation of Homotopy Group

作　　者：蒙毅琦 张家玲[1] 杨彦敏 MENG Yi-Qi;ZHANG Jia-ling;YANG Yan-min(Faculty of Science,Kunming University of Science and Technology,Kunming 650500,China)

机构地区：[1]昆明理工大学理学院,云南昆明650500

出　　处：《软件导刊》2021年第9期228-232,共5页Software Guide

基　　金：云南省教育厅科研基金项目(2021J0054);云南省科技厅科研基金项目(202001AU070041);昆明理工大学基金项目(KKZ3202007049)。

摘　　要：讨论具有复杂拓扑结构的封闭曲面方形图表示算法,通过应用同伦群生成元的计算方法,简化封闭曲面拓扑结构。由于曲面的一阶同伦群生成元集合也是其割图的同伦群生成元集合,因此沿着封闭曲面的割图切割曲面可简化曲面的拓扑复杂度。进一步以调和1-形式作为上同调群的生成元,并通过调和1-形式的积分确定方形边长,最后根据给定曲面的组合结构布置方形。将适用于平面对象的算法推广到适用于三维高亏格封闭曲面的情形,给出这类曲面的方形图表示方法,并通过测试不同曲面结果证明该方法的稳定性。This paper discusses an algorithm of square drawing representatives for closed combinatorial surfaces with complicated topology.We simplify topologically complex of a closed surface by use of the computation of homotopy group generators.Since the set of the first homotopy group generators of a surface is also the set of the homotopy group generators of its cut graph,Cutting a closed surface along its cut graph can simplify topologically complex of the surface.Furtherly,we compute the harmonic 1-forms as the representative of the cohomology group generators and determine the edge lengths of squares by the integral of harmonic 1-forms along edges.Finally,we lay out squares according to the combinatorial structure of a given combinatorial surface.By generalizing algorithms applied in the planar case into 3D closed surfaces with high genus,this paper gives out a method of the square drawing representative for closed surfaces with complicated topology and also analyzes that the proposed method is robust by tests on different objects.

关 键 词：同伦群生成元 上同调群生成元 调和1-形式 方形图表示 

分 类 号：TP317.4[自动化与计算机技术—计算机软件与理论]