用平面六边形割角的细分曲面生成方法  

A Method of Generating Subdivision Surfaces with Planar Hexagon Corner-Cutting

作　　者：韩臻[1] HAN Zhen(School of Computer Science&Technology,Beijing Jiaotong University,Beijing 100044)

机构地区：[1]北京交通大学计算机科学与技术学院,北京100044

出　　处：《计算机学报》2024年第7期1615-1639,共25页Chinese Journal of Computers

基　　金：中央高校基本科研业务费专项资金(No.2022XKRC012)资助。

摘　　要：平面六边形是一种更契合自然并具有独特优点的多边形,找到一种方便实用的平面六边形细分格式是细分曲面研究领域中的基础工作之一。本文提出了一种平面六边形细分方法,对任意初始凸多面体网格,只用平面六边形进行割角细分,可以使得细分曲面光滑、保凸、具有插值性质,且细分过程中新增的面片都是平面六边形.我们将该方法简称为平面蜂窝细分.论文给出了平面蜂窝细分的拟线性四点格式几何规则及其细分矩阵,其几何意义直观,相应的算法简单、可行且数值稳定.论文分析了细分曲面的收敛性和光滑性,给出了C光滑性条件及其证明.为了提高细分曲面的光滑性,传统的方法是使边数不同的多边形面片按等比例收缩,该方法对于平面蜂窝细分没有预期效果并且无法处理三角形特殊面;论文给出了一种新方法,获得的割角参数使得细分曲面更加光滑,且可以统一处理三角形、四边形等特殊面,从而避免特殊点/面带来的局限性.论文还提出了平面蜂窝细分方法的一种推广和一种自然边界处理方法,并讨论了平直边界和退化情形下尖锐特征生成的方式,可用于细分任意可定向初始网格.文中给出了一些细分曲面的例子,并与经典的细分方法进行了比较,验证了新方法的有效性和具有的优点.Research on subdivision surfaces has achieved significant results and has been applied in various fields,such as smooth surface modeling in animation and architectural geometric design.Among them,constructing various feasible subdivision schemes is a fundamental work in the research of subdivision surfaces.Hexagon is a polygon that is more compatible with a nature and elegant appearance.Some scholars have proposed subdivision schemes based on hexagons.But,these hexagon-based subdivision schemes either have poor smoothness and computational complexity or produce non-coplanar vertices in the facet.There are also problems of discontinuity,self-intersection,or inability to subdivide at extraordinary points/faces,especially triangular facets.However,the planarity of the mesh faces is as important as smoothness in some applications,for example in architecture.It is worth looking forward to a convenient planar hexagonal subdivision scheme.On the other hand,the construction of subdivision curves can be perfectly summarized as a sequence of corner cutting operations,which is geometrically intuitive,flexible,convex-preserving,and computationally stable.The non-uniform corner cutting subdivision curve method proposed by Gregory and Qu(referred to as the G-Q algorithm)is a classic method.With this method,for any initial polyline on the plane,by sequentially cutting all the corners formed by the polyline(the cutting parameters satisfy the corresponding smoothness conditions),a Cl smooth limit curve can be obtained.Each segment of the initial polyline has a point preserved on the limit curve,and the tangent line of the limit curve at that point is the initial polyline at that point.This method can be extended to describe the generation of spatial surfaces as follows:for any convex polyhedron in space,by sequentially cutting all the corners of the polyhedron,a C'smooth limit surface can be obtained.Each initial face has a point preserved on the limit surface,and the tangent plane of the limit surface at that point is the initia

关 键 词：细分曲面 六边形网格 平面六边形 保凸 割角 割角参数 特殊点/面 

分 类 号：TP391[自动化与计算机技术—计算机应用技术]