Integrable cross-eld generation based on imposed singularity conguration the 2D manifold case

2025-05-05 0 0 9.33MB 25 页 10玖币
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Integrable cross-field generation based on imposed
singularity configuration
– the 2D manifold case –
Jovana Jezdimirovi´c, Alexandre Chemin, Jean-Fran¸cois Remacle
Universit´e catholique de Louvain, Louvain la Neuve, Belgium
jean-francois.remacle@uclouvain.be
Abstract
This work presents the mathematical foundations for the generation of inte-
grable cross-field on 2D manifolds based on user-imposed singularity configura-
tion. In this paper, we either use singularities that appear naturally, e.g., by
solving a non-linear problem, or use as an input user-defined singularity pattern,
possibly with high valence singularities that typically do not appear in cross-
field computations. This singularity set is under the constraint of Abel-Jacobi’s
conditions for valid singularity configurations. The main contribution of the
paper is the development of a formulation that allows computing an integrable
isotropic 2D cross-field from a given set of singularities through the resolution of
only two linear PDEs. To address the issue of possible suboptimal singularities’
distribution, we also present the mathematical setting for the generation of an
integrable anisotropic 2D cross-field based on a user-imposed singularity pat-
tern. The developed formulations support both an isotropic and an anisotropic
block-structured quad mesh generation.
Keywords: integrable 2D cross-field, valid singularity configuration, quad
layout, quad meshing
1. Introduction and related work
Numerous methods for surface parametrization/representation have been
developed for a large number of applications [1, 2, 3]. In cases when a shape ex-
hibits complex topological or geometrical characteristics, it is necessary to split
it into simple partitions to obtain a quad mesh. The special case of partitioning
it into a simply connected network of conformal quadrilateral partitions is called
the quad layout [4]. The latter manner of surface representation is a subject of
great interest in meshing and computer graphics communities, due to providing
a wide range of benefits [1, 2, 5]. Nevertheless, these advantages come with the
high price of dealing with complex and time-consuming algorithms [5].
Among the developed methods for the quad layout generation, a general
distinction can be made among the ones which are: computing a seamless global
Preprint submitted to arXiv.org October 7, 2022
arXiv:2210.02563v1 [math.NA] 23 Sep 2022
parametrization of the domain where integer iso-values of the parameter fields
form the sides [6, 7, 8], using Riemann geometry [9, 10, 11], or like in our case,
constructing a cross-field structure that will guide the integral lines emanating
from singularities [12, 13, 14, 15, 16, 17].
Although leaning on heterogeneous approaches, all the above-mentioned
methods share the common challenge: dealing with the inevitable singularity
configuration. A singularity appears where a cross-field vanishes and it repre-
sents an irregular vertex of a quad layout/quad mesh [18], i.e., a vertex which
doesn’t have exactly four adjacent quadrilaterals. The singular configuration is
constrained by the Euler characteristic χ, which is a topological invariant of a
surface. Moreover, a suboptimal number or location of singularities can have
severe consequences: causing undesirable thin partitions, large distortion, not
an adequate number and/or tangential crossings of separatrices as well as limit
cycles [5, 7, 17].
Cross-field guided methods can be very useful and flexible but they typically
lack direct control over the positions of the singularities and the structures of the
quad layout [19]. Our cross-field formulation, with mathematical foundations
detailed in Section 2, offers a contribution to this issue through the concept of
user-imposed singularity configuration in order to gain direct control over their
number, location, and valence (number of adjacent quadrilaterals). The user
is entitled to use either naturally appearing singularities, obtained by solving
a non-linear problem [14, 17, 20, 21], using globally optimal direction fields
[22], or to impose its own singularity configuration, possibly with high valences,
as illustrated in Fig. 1. It is important to note that the choice of singularity
pattern is not arbitrary, though. Moreover, it is under the direct constraint
of Abel-Jacobi theory [9, 10, 11] for valid singularity configurations. Here, the
singularity configuration is taken as an input and an integrable isotropic cross-
field is computed by solving only two linear systems, Section 3. Computation
of the scalar field Hused for this cross-field generation bears some resemblance
to the one developed for unstructured mesh generation on planar and curved
surface domains in [23]. Finally, the preliminary results of the developed cross-
field formulation for an isotropic block-structured quad mesh generation are
outlined using the 3-step pipeline [15] in Section 4.
Computing only one scalar field H(a metric that is flat except at singular-
ities) imposes a strict constraint on singularities’ placement, i.e., fulfilling all
Abel-Jacobi conditions. In practice, imposing suboptimal distribution of sin-
gularities may lead to not obtaining boundary-aligned cross-field, disabling an
isotropic quad mesh generation, Section 4.1 and 4.2. To bypass this issue,
we develop a new cross-field formulation on the imposed singularity configura-
tion, which considers the integrability, while relaxing the condition on isotropic
scaling of crosses’ branches. Here, two independent metrics H1and H2are
computed instead of only one as in the Abel-Jacobi framework, enabling an
integrable 2D cross-field generation with anisotropic scaling, Section 5.
Lastly, final remarks and some of the potential applications are discussed in
Section 6.
2
Figure 1: Three quad layouts of a simple domain. Singularities of valence 3 are colored in
blue, valence 5 in red, valence 6 in orange, and valence 8 in yellow.
2. Cross-field computation on prescribed singularity configuration
We define a 2D cross cas a set of 2 unit coplanar orthogonal vectors and
their opposite, i.e.,
c={u,v,−u,−v}
with {u.v= 0, |u|=|v|= 1}and u,vare coplanar. These vectors are called
cross’ branches.
A 2D cross-field CMon a 2D manifold M, now, is a map CM:X∈ M →
c(X), and the standard approach to compute a smooth boundary-aligned cross-
field is to minimize the Dirichlet energy:
min
CMZM
k∇CMk2(1)
subject to the boundary condition c(X) = g(X) on ∂M, where gis a given
function.
The classical boundary condition for cross-field computation is that ∀P∈
∂M, with T(P) a unit tangent vector to Mat P, one branch of c(P) has to
be colinear to T(P). In the general case, there exists no smooth cross-field
matching this boundary condition. The cross-field will present a finite number
of singularities Sj, located at Xjand of index kj, related to the concept of
valence as kj= 4 −valence(Sj).
We define a singularity configuration as the set
S={Sj, j ∈[|1, N|], N ∈Z}.
In the upcoming section, a method to compute a cross-field CMmatching a
given singularity configuration Sis developed. In other words, we are looking
for CMsuch as:
·if Xbelongs to ∂M, at least one branch of CM(X) is tangent to ∂M,
·singularities of CMare matching the given S
(the same number, location, and indices).
(2)
Before developing the method to compute such a cross-field, a few operators
on the 2D manifold have to be defined.
3
2.1. Curvature and Levi-Civita connection on the 2D manifold
Let E3be the Euclidean space equipped with a Cartesian coordinates system
{xi, i = 1,2,3}, and Mbe an oriented two-dimensional manifold embedded in
E3. We note n(X) the unit normal to Mat X∈ M. It is assumed that the
normal field nis smooth and that the Gaussian curvature Kis defined and
smoothed on M.
If γ(s) is a curve on Mparametrized by arc length, the Darboux frame is
the orthonormal frame defined by
T(s) = γ0(s) (3)
n(s) = n(γ(s)) (4)
t(s) = n(s)×T(s).(5)
One then has the differential structure
d
T
t
n
=
0κgκn
−κg0τr
−κn−τr0
T
t
n
ds(6)
where κgis the geodesic curvature of the curve, κnthe normal curvature of the
curve, and τrthe relative torsion of the curve. Tis the unit tangent, tthe
tangent normal and nthe unit normal.
Arbitrary vector fields Vand W∈E3can be expressed as
V=ViEi,W=WiEi
in the natural basis vectors {Ei, i = 1,2,3}of this coordinate system, and we
shall note
<V,W>=ViWjδij ,||V|| =p<V,V>
the Euclidean metric and the associated norm for vectors. The Levi-Civita
connection on E3in Cartesian coordinates is trivial (all Christoffel symbols
vanish), and one has
∇E
VW= (∇VWi)Ei.
The Levi-Civita connection on the Riemannian submanifold M, now, is not
a trivial one. It is the orthogonal projection of ∇E
Vin the tangent bundle TM,
so that one has
∇VW=PTM[∇E
VW] = (∇VWi)PTM[Ei] (7)
where PTM:E37→ TMis the orthogonal projection operator on TM.
An arbitrary orthonormal local basis (uX,vX,n) for every X∈ M, can be
represented through the Euler angles (ψ, γ, φ) which are C1on M, and with the
4
shorthands sφ≡sin φand cφ≡cos φ, as:
uX=
−sφsψcγ+cφcψ
sφcψcγ+sψcφ
sφsγ
,
vX=
−sφcψ−sψcφcγ
−sφsψ+cφcψcγ
sγcφ
,
n=
sψsγ
−sγcψ
cγ
(8)
in the vector basis of E3.
2.2. Conformal mapping
We are looking for a conformal mapping
F:P → M ⊂ E3
P= (ξ, η)7→ X= (x1, x2, x3)(9)
where Pis a parametric space. As finding Fright away is a difficult problem,
one focuses instead on finding the 3 ×2 jacobian matrix of F
J(P) = (∂ξF(P), ∂ηF(P)) ≡(˜
u(P),˜
v(P)),(10)
where ˜
u,˜
v∈TMare the columns vectors of J. The mapping Fbeing confor-
mal, the columns of J(P) have the same norm L(P)≡ ||˜
u(P)|| =||˜
v(P)|| and
are orthogonal to each other, ˜
u(P)·˜
v(P) = 0. We can also write:
J=L(u,v),n=u∧v
where
u=˜
u
||˜
u||
v=˜
v
||˜
v|| .(11)
Recalling that finding a conformal transformation Fis challenging, we will from
now on be looking for the jacobian J,i.e., the triplet (u,v, L).
The triplet (u,v,n) forms a set of 3 orthonormal basis vectors and can
be seen as a rotation of (uX,vX,n) among the direction n. Therefore, a 2D
cross c(X), X∈ M can be defined with the help of a scalar field θ, where
u=Rθ,n(uX) and v=Rθ,n(vX), and the local manifold basis (uX,vX,n) as:
u=cθuX+sθvX,v=−sθuX+cθvX.(12)
By using the Euler angles (ψ, γ, φ) and θ, the triplet (u,v,n) can also be
5
摘要：

Integrablecross- eldgenerationbasedonimposedsingularitycon guration{the2Dmanifoldcase{JovanaJezdimirovic,AlexandreChemin,Jean-FrancoisRemacleUniversitecatholiquedeLouvain,LouvainlaNeuve,Belgiumjean-francois.remacle@uclouvain.beAbstractThisworkpresentsthemathematicalfoundationsforthegenerationofin...

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