Boundary Aligned Smooth 3D Cross-Frame Field

晃晃

Boundary Aligned Smooth 3D Cross-Frame Field

Jin Huang   Yiying Tong  Hongyu Wei  Hujun Bao

State Key Lab of CAD&CG, Zhejiang University    Michigan State University





Abstract

In this paper, we present a method for cons***cting a 3D cross-

frame field, a 3D extension of the 2D cross-frame field as applied

to surfaces in applications such as quadrangulation and texture syn-

thesis. In contrast to the surface cross-frame field (equivalent to a

4-Way Rotational-Symmetry vector field), symmetry for 3D cross-

frame fields cannot be formulated by simple one-parameter 2D ro-

tations in the tangent planes. To address this critical issue, we rep-

resent the 3D frames by spherical harmonics, in a manner invariant

to combinations of rotations around any axis by multiples of =2.

With such a representation, we can formulate an efficient smooth-

ness measure of the cross-frame field. Through minimization of

this measure under certain boundary conditions, we can cons***ct

a smooth 3D cross-frame field that is aligned with the surface nor-

mal at the boundary. We visualize the resulting cross-frame field

through restrictions to the boundary surface, streamline tracing in

the volume, and singularities. We also demonstrate the applica-

tion of the 3D cross-frame field to producing hexahedron-dominant

meshes for given volumes, and discuss its potential in high-quality

hexahedralization, much as its 2D counterpart has shown in quad-

rangulation.

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

ometry and Object Modeling—Geometric algorithms, languages,

and systems;

Keywords: hexahedral, spherical harmonics, N-RoSy frame field



1 Introduction

Many recent quadrangulation methods start by cons***cting a

smooth field of orientations determined up to a rotation of  or

=2. Substantial progress has been made towards the generation

of quadrilateral meshes with controlled element sizes and edge di-

rections by optimizing such fields. However, many applications

require discretization of 3D volumes rather than just their bound-

ary surfaces. Applications such as simulated elasticity of 3D volu-

metric objects, computational electromagnetics, and computational

fluid dynamics require Finite Element, Finite Volume, or Finite Dif-

ference methods on a discretized domain. These methods benefit

from a high-quality hexahedral mesh, since hexhedral meshes offer

several numerical advantages over tetrahedral meshes due to their

tensor product nature. They are also desirable for applications such

as geometric design and B-spline fitting, and amenable to applica-

tions such as 3D texture atlases. In addition, hexahedral meshes

often capture the symmetries of 3D objects and domains better than

tetrahedral meshes, thus making the model more intuitive to design-

ers or animators. However, the automatic generation of a hexahe-

dral mesh for a given curved 2D boundary with feature alignment,

sizing, and regularity control remains far more challenging than au-

tomatic tetrahedralization.



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