Simple Quad Domains for Field Aligned Mesh Parametrization

晃晃

Simple Quad Domains for Field Aligned Mesh Parametrization

Marco Tarini 1;2 Enrico Puppo3  Daniele Panozzo3   Nico Pietroni1  Paolo Cignoni1

1ISTI-CNR, Italy 2Universit` a dell’Insubria, Italy 3Universit` a di Genova, Italy





Abstract

We present a method for the global parametrization of meshes that

preserves alignment to a cross field in input while obtaining a para-

metric domain made of few coarse axis-aligned rectangular patches,

which form an abstract base complex without T-junctions. The

method is based on the topological simplification of the cross field

in input, followed by global smoothing.

CR Categories: I.3.5 [Computer graphics]: Computational geom-

etry and object modeling—Curve, surface, solid and object repres.

Keywords: quad mesh, mesh parametrization

becoming increasingly common.

A recent trend is to first define a cross field C over M, and then

to find f such that its gradient vectors match C as much as pos-

sible. Interestingly, each of the criteria above can be redefined in

terms of desired properties of C. Thus the task is shifted from the

definition of a good parameterization to the definition of a good

cross field C for a given M. High quality parametrization can be

obtained following this approach [Ray et al. 2006; Bommes et al.

2009]. It is now appearent that the definition of a good cross field

C implies, among other things, the good placing of a few irregular

points (a.k.a. cone singularities). Irregular points tend to be needed,

for example, in places where M exhibits high Gaussian curvature.

In this paper, we focus on an important additional criterion for the

quality of f, which we refer to as the simplicity of domain D (see

below for an informal definition). Simplicity determines how much

a parametrization f will be effectively useful in most applications,

just as much as the other criteria listed above. As we will show, a

cross field C designed to satisfy all the above conventional criteria,

but not simplicity, will usually fail producing an acceptably sim-

ple domain. Still, it is often the case that a slightly modified cross

field C0

exists, which is able to generate a parametrization f with a

dramatically simpler domain, while preserving to a large extent the

other qualities of C. This work presents a way to obtain the cross

field C0

, given C.

1.1 Objective: Domain Simplicity

For topologies ofM other than the disk, the domain D must neces-

sarily include discontinuities (a.k.a. cuts, or seams): two infinitesi-

mally close points m0 and m1 of M lying of different sides of the

cut may be mapped by f 1

to arbitrarily distant positions d0 and d1

of D. The values d0 and d1 are often constrained to be reciprocally

associated with a “transition function” associated to that cut.



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