Connectivity Editing for Quadrilateral Meshes

晃晃

Connectivity Editing for Quadrilateral Meshes

Chi-Han Peng Eugene Zhang†  Yoshihiro Kobayashi   ‡ Peter Wonka§

Arizona State University Oregon State University Arizona State University Arizona State University /





Abstract

We propose new connectivity editing operations for quadrilateral

meshes with the unique ability to explicitly control the location,

orientation, type, and number of the irregular vertices (valence not

equal to four) in the mesh while preserving sharp edges. We provide

theoretical analysis on what editing operations are possible and im-

possible and introduce three fundamental operations to move and

re-orient a pair of irregular vertices. We argue that our editing op-

erations are fundamental, because they only change the quad mesh

in the smallest possible region and involve the fewest irregular ver-

tices (i.e., two). The irregular vertex movement operations are sup-

plemented by operations for the splitting, merging, canceling, and

aligning of irregular vertices. We explain how the proposed high-

level operations are realized through graph-level editing operations

such as quad collapses, edge flips, and edge splits. The utility of

these mesh editing operations are demonstrated by improving the

connectivity of quad meshes generated from state-of-art quadran-

gulation techniques.

Keywords: quadrilateral mesh editing, irregular vertex editing,

mesh optimization, mesh-based design, topology, geometry pro-

cessing



1 Introduction

Quadrilateral and hexahedral meshes are popular choices in simula-

tion and shape modeling due to the natural tensor product property

that they possess. Quadrilateral meshes can also facilitate architec-

tural modeling as well as texture and geometry synthesis. Impor-

tant aspects of a quadrilateral mesh include the location, orienta-

tion, type, and number of irregular vertices. While there has been

some work in quad mesh connectivity editing [Daniels et al. 2008;

Bommes et al. 2011], achieving irregular vertex control is challeng-

ing and many questions about what editing operations are possible

and impossible still need to be answered.

In this paper, we propose three operations that move an irregular

vertex pair (two valence 3, two valence 5, or one valence 3 and

one valence 5) over the mesh. To show that these are fundamental

operations for quad mesh editing, we will establish the following

properties:

 These editing operations impact the smallest possible region

on the mesh and are therefore as local as possible (in a convex

region).

 A region containing only one irregular vertex cannot be

edited.

 A region containing two irregular vertices can be edited by

changing the location of the irregular vertices within the re-

gion. However, they cannot be canceled. Some irregular ver-

tex pairs can be merged while others cannot, depending on

their graph distance in the initial configuration.

 A region with three irregular vertices can be edited by cancel-

ing or merging the irregular vertices.

 Our three movement operations can perform all possible edits

within a (convex) region that contains two irregular vertices.









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