General Planar Quadrilateral Mesh Design Using Conjugate Direction Field

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General Planar Quadrilateral Mesh Design Using Conjugate Direction Field

Yang Liu Weiwei Xu Jun Wangy Lifeng Zhuz Baining Guo Falai Cheny Guoping Wangz

Microsoft Research Asia yUniversity of Science and Technology of China zPeking University









Abstract

We present a novel method to approximate a freeform shape with

a planar quadrilateral (PQ) mesh for modeling architectural glass

structures. Our method is based on the study of conjugate direction

fields (CDF) which allow the presence of k=4(k 2 Z) singular-

ities. Starting with a triangle discretization of a freeform shape,

we first compute an as smooth as possible conjugate direction field

satisfying the user’s directional and angular constraints, then ap-

ply mixed-integer quadrangulation and planarization techniques to

generate a PQ mesh which approximates the input shape faithfully.

We demonstrate that our method is effective and robust on various

3D models.

CR Categories: I.3.5 [Computer Graphics]: Computational Ge-

ometry and Object Modeling—Geometric algorithms, languages,

and systems;

Keywords: planar quadrilateral mesh, conjugate direction field,

architectural geometry

1 Introduction

Planar quadrilateral (PQ) meshes are essential in architectural ge-

ometry for discretizing a freeform architectural structure with pla-

nar quad faces [Glymph et al. 2004; Liu et al. 2006; Pottmann

et al. 2007b], and the study of PQ meshes is now an important

topic of discrete differential geometry [Pottmann andWallner 2008;

Bobenko and Suris 2008]. Its continuous counterpart, in differen-

tial geometry, is the conjugate curve network [Sauer 1970; Liu et al.

2006; Bobenko and Suris 2008], which is defined to be two fami-

lies of one-parameter curves that cover a smooth surface, and their

tangent vectors v;w at an arbitrary point x on a surface are con-

jugate (see its formal definition in Section 3). These two families

of tangent directions v;w form a general cross field without the

requirement of orthogonality, which we call a conjugate direction

field (CDF) hereafter. The layout of a PQ mesh can be controlled

through the design of the CDF.



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