Weighted Minimal Surfaces and Discrete
Weighted Minimal Surfaces ?
Qin Zhang a,b , Guoliang Xu a,∗
a LSEC,
Institute of Computational Mathematics,
Academy of Mathematics and System Science, Chinese Academy of Sciences,
Beijing 100080, P. R. China.
b Department
of Basic Courses, Beijing Information Science and Technology
University, Beijing 100085, P. R. China.
Abstract
We introduce the concept of weighted minimal surface and define weighted mean
curvature with explicit formulas for functional, parametric and implicit surfaces. We
prove that the weighted minimal functional surface is stationary. The concept of the
discretized weighted minimal surface is proposed. Examples of weighted minimal
surfaces are presented.
Key words: Weighted minimal surface; Weighted mean curvature; Discretization.
1
Introduction
As is well known, a regular surface M ⊂ IR3 is minimal if the mean curvature of M vanishes everywhere. Equivalently, a minimal surface M is a critical
point of the area functional, i. e.,
Z
M = arg{min
Σ
dA},
Σ
where dA is the area element of surface Σ in IR3 . In this paper, we study
the weighted minimal surface M defined by zero weighted mean curvature
everywhere, which is also a critical point of the weighted area functional, i. e.,
? Partially supported by National Natural Science Foundation of China under
Grant No. 10371130 and National Key Basic Research Project of China under Grant
No. 2004CB318000.
∗ Corresponding author.
Email address: xuguo@lsec.cc.ac.cn (Guoliang Xu).
Z
φ(p, q)dA},
M = arg{min
Σ
(1)
Σ
where φ(p, q) is a weight function and is also called anisotropic function. This
weighted area functional appears in image analysis fields such as shape segmentation (see [5]), boundary detection (see [2]), tracking (see [4]).
In this paper, definitions of weighted mean curvature for various surface
representations and stationary weighted minimal surface are discussed. We
introduce the concept of discrete weighted minimal surface and give a discretization method. Brakke’s software Evolver [1] is used to conduct experiments. Comparisons of the weighted minimal surfaces with classical minimal
surface are carried out.
2
Weighted mean curvature
2.1 Anisotropic function
The definition of the anisotropic function and its important properties are
as follows.
Definition 1 Let φ : IR3 × IR3 \{0} → (0, +∞) be a C 2 (IR3 × IR3 \{0}) function. If φ is positively homogeneous of degree 1 for the second variable,
φ(p, λq) = λφ(p, q), p ∈ IR3 , q ∈ IR3 \{0}, λ > 0
(2)
and convex in the sense that there exists a constant c0 such that
∇2qq φ(p, q)z · z ≥ c0 |z|2
f or all p, q, z ∈ IR3
with
q · z = 0, |q| = 1,(3)
then φ is named as an anisotropic function. If φ = φ(q), then we say φ is
independent of the position.
Proposition 1 Let φ(p, q) be an anisotropic function. Then for any p ∈
IR3 , q ∈ IR3 \{0}, we have
∇q φ(p, q) · q = φ(p, q),
∇2qq φ(p, q)q = 0,
(4)
(5)
∇2qp φ(p, q)q = ∇p φ(p, q),
(6)
∇q φ(p, λq) = ∇q φ(p, q),

∇2qq φ(p, q)z
∇2qq φ(p, λq) =
Ã
c0  2
q
·z≥
|z| − z ·
|q|
|q|
!2 
,
1 2
∇ φ(p, q),
λ qq
∀z ∈ IR3 .
∀λ > 0
(7)
(8)
The proof of Proposition 1 is quite straightforward and notations are only
explained. Suppose p = (p1 , p2 , p3 )T and q = (q1 , q2 , q3 )T are two vectors in
IR3 . We use p · q and p × q to denote the usual inner product and outer product, respectively. ∇p φ(p, q) represents the gradient of φ(p, q) about variable p.
∇2qp φ(p, q) stands for the gradient of ∇q φ(p, q) about p variable whose ij entry
is φqj pi (p, q). ∇2qq φ(p, q) is the hessian of φ(p, q) about variable q with φqi qj as
its ij entry.
2.2
Weighted mean curvature
The Euler equation of the weighted area functional
Z
Eφ (M) =
φ(p, ~n)dA,
M
will be given and the definition of weighted mean curvature is formulated for
three types of surface M, where p and ~n are surface point and the unit normal
vector, respectively.
Functional Surface. Suppose surface M = {(x1 , x2 , u(x1 , x2 )); (x1 , x2 ) ∈
Ω ⊂ IR2 }. The weighted area of M is calculated by
Z
Z
q
φ((x1 , x2 , u), ~n(u)) 1 + |∇u|2 =
Eφ (u) =
Ω
φ((x1 , x2 , u), (∇u, −1)).
Ω
The Euler equation of this weighted area functional is
φp3 −
2
X
(φqj pj + φqj p3 uxj ) −
j=1
2
X
φqi qj uxi xj = 0.
(9)
i,j=1
For the first two terms of (9) depend on the position of the surface, we use
the last term of (9) to define the weighted mean curvature for a graph as
Hφ =
2
X
φqi qj ((x1 , x2 , u(x1 , x2 )), (∇u, −1))uxi xj .
i,j=1
Parametric Surface. For a regular parametric surface M parameterized as
r(u, v) ∈ IR3 on a domain Ω ⊂ IR2 , the weighted area is expressed as
ZZ
Eφ (M) =
ZZ
φ(r, ~n)|ru × rv | dudv =
Ω
φ(r, ru × rv )dudv.
Ω
The Euler equation of area functional is
h
i
∇p φ(r, ~n) · ~n − (~n × rv ) · (∇2qp φ(r, ~n)ru ) + (ru × ~n) · (∇2qp φ(r, ~n)rv )
h
−
i
(~n × rv ) · ∇2qq φ(r, ~n)(ru × rv )u + (ru × ~n) · (∇2qq φ(r, ~n)(ru × rv )v )
|ru × rv |2
= 0,
and the weighted mean curvature is defined as
h
Hφ =
i
(~n × rv ) · ∇2qq φ(r, ~n)(ru × rv )u + (ru × ~n) · (∇2qq φ(r, ~n)(ru × rv )v )
|ru × rv |2
.
Implicit Surface. If surface M is expressed by a level set of a function ϕ, i.
e., M = {x = (x1 , x2 , x3 ) ∈ IR3 ; ϕ(x1 , x2 , x3 ) = c}, where c is a constant, then
the weighted area of the surface M can be written as ([3], page 15)
Z
Eφ (M) =
Z
φ(x, ~n)δ(ϕ(x))|∇ϕ|dx =
IR3
φ(x, ∇ϕ)δ(ϕ(x))dx,
IR3
where δ is one-dimensional Dirac delta function. After calculating, the Euler
equation of weighted area functional is
3
X
φqi pi (x, ∇ϕ)δ(ϕ) + tr(∇2qq φ(x, ∇ϕ)∇2 ϕ(x))δ(ϕ) = 0,
i=1
where tr(·) stands for the trace of a square matrix. The weighted mean curvature of M is defined as
Hφ = tr(∇2qq φ(x, ∇ϕ)∇2 ϕ(x)),
(10)
because δ(ϕ) 6= 0 only on M.
3
Stationary weighted minimal surface
From now on, we restrict the anisotropic function φ(p, q) to be independent
of the first variable p.
Definition 2 (weighted minimal surface) A surface M is called a weighted
minimal surface if its weighted mean curvature Hφ is zero everywhere.
Definition 3 (stationary weighted minimal surface) For any variation
Mt of a weighted minimal surface M with fixed boundary, if the second variation of the weighted area of the surface is positive, i. e.,
¯
d2 Eφ (Mt ) ¯¯
¯
> 0,
¯
dt2
t=0
then M is named as a stationary weighted minimal surface.
Now suppose the weighted minimal surface M is described by a graph u(x)
as in section 2. We can compute the second variation of the weighted area for
η ∈ C0∞ (Ω) as follows:

¯

Z X
2
¯
d2
1
(∇u, −1) 
¯
q
E
(u
+
tη)
¯
=
φij  q
η xi ηxj
φ
2
¯
2
2
dt
1
+
|∇u|
1
+
|∇u|
i,j=1
t=0
Ω

Z
≥
c0
q
Ω
1 + |∇u|2

∇u

2
|∇η| − ∇η · q
1 + |∇u|2
2 
 
 > 0,
where ≥ is valid because of the property (8) of anisotropic function φ. Then
we obtain the following:
Theorem 1 The weighted minimal graph is a stationary weighed minimal
surface.
This stationary property of the weighed minimal graph is quite similar to the
classical minimal graph.
4
Discrete weighted minimal surface
For a discrete triangular surface mesh M ,
~ φ (vi ) ≈ − ∇Aφ (vi ) .
H
A(vi )
is a discrete approximation to the weighted mean curvature, where A(vi ) and
Aφ (vi ) are classical area and weighted area around a vertex vi ∈ M , respectively. Let [vi vj vj+1 ] be a neighbor triangle of vertex vi . The usual area of the
triangle [vi vj vj+1 ] is A(j) (vi ) = 12 |(vj − vi ) × (vj+1 − vi )| and the weighted area
(j)
can be computed by Aφ (vi ) = 12 φ((vj − vi ) × (vj+1 − vi )). So the area and
weighted area of the one ring neighborhood of vertex vi are
A(vi ) =
X
A(j) (vi ) and Aφ (vi ) =
j∈N (i)
X
(j)
Aφ (vi ),
j∈N (i)
respectively, where N (i) is the index set of the one ring neighbor vertices of
vertex vi . We can compute the gradient of Aφ (vi ) to vertex vi as
∇vi Aφ (vi ) =
1 X
(vj − vj+1 ) × ∇φ((vj − vi ) × (vj+1 − vi )).
2 j∈N (i)
(11)
Hence
~ φ (vi ) ≈ −
H
X
1
(vj − vj+1 ) × ∇φ((vj − vi ) × (vj+1 − vi )).
2A(vi ) j∈N (i)
(12)
Definition 4 Let φ be a given anisotropic function, and M be a triangular
surface mesh with interior vertices {vi }. If
X
(vj − vj+1 ) × ∇φ((vj − vi ) × (vj+1 − vi )) = 0,
(13)
j∈N (i)
for all interior vertices vi , then we say M is a weighted discrete minimal
surface.
5
Examples of weighted minimal surfaces
We present the classical Scherk’s minimal surface and weighted Scherk’s
minimal surfaces generated by the following anisotropic functions
φ1 (~n) = (n21 + 0.25n22 + 16n23 )1/2
and φ2 (~n) = (n41 + 4n42 + 4n43 )1/4 .
by Brakke’s software Evolver. The results for these three minimal surfaces
are shown in Table 1 and Fig. 1 to Fig. 3. Let us explain the various entries
that appear in Table 1. The left column, refine, is the number of refining.
facets, is the number of facet of the mesh. The third column, iters, is
approximate number of iterations. The next three columns with each one
including two columns, area, which is the classical discrete area of the whole
mesh, and energy, the whole weighted discrete area. From this table, we can
see that weighted minimal surface is not minimal surface, because when energy
is decreasing, the area may be increasing. These three weighted or classical
minimal surfaces are stationary because these three surfaces are graphs defined
on a domain. For example, Scherk’s minimal surface can be written in function
form as
cos y
, −π/2 < x, y < π/2.
z = ln
cos x
This verified that the weighted minimal graph is stationary which we have
proved in section 3.
Table 1: Discrete minimal Scherk’s surface and weighted discrete
minimal Scherk’s surfaces
refine
facets
iters
20
1
DM Scherk’s
WDM Scherk’s I
WDM Scherk’s II
area
energy
area
energy
area
energy
49.348
49.348
49.348
69.087
49.348
61.612
80
O(10)
44.740
44.740
45.273
56.856
45.786
47.822
2
320
O(102 )
44.060
44.060
44.876
53.036
44.851
46.563
3
1280
O(102 )
43.885
43.885
44.620
52.003
44.674
45.934
References
[1] K. A. Brakke. Surface Evolver Manual, version 2.24 for windows edition,
Ocotober 2004. Available on http://www.susqu.edu/brakke.
(a)
(b)
(c)
(d)
Fig. 1. Discrete classical Scherk’s minimal surface corresponding to the DM Scherk’s
in Table 1: (a) is the original model. (b) is obtained after refining the initial mesh
once and iterating O(10) times. (c) is obtained after further refining and iterating.
Another refinement and more iterations result in (d).
(a)
(b)
(c)
(d)
Fig. 2. Discrete minimal Scherk’s surface I weighted by anisotropic function φ1 (~n)
corresponding to WDM Scherk’s I in Table 1: (a) is initial model. (b) is obtained
after refining once and iterating. (c) and (d) are showed after further refinement
and iteration.
(a)
(b)
(c)
(d)
Fig. 3. Discrete minimal Scherk’s surface II weighted by function φ2 (~n) corresponding to WDM Scherk’s II in Table 1. Other explanations are similar to Fig. 2.
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