This article appeared in a journal published by Elsevier. The attached
copy is furnished to the author for internal non-commercial research
and education use, including for instruction at the authors institution
and sharing with colleagues.
Other uses, including reproduction and distribution, or selling or
licensing copies, or posting to personal, institutional or third party
websites are prohibited.
In most cases authors are permitted to post their version of the
article (e.g. in Word or Tex form) to their personal website or
institutional repository. Authors requiring further information
regarding Elsevier’s archiving and manuscript policies are
encouraged to visit:
http://www.elsevier.com/copyright
Author's personal copy
Computers and Mathematics with Applications 65 (2013) 673–681
Contents lists available at SciVerse ScienceDirect
Computers and Mathematics with Applications
journal homepage: www.elsevier.com/locate/camwa
Constructing PDE-based surfaces bounded by geodesics or lines
of curvature
Wei-Xian Huang, Hua-Jing-Ling Wu, Guo-Jin Wang ∗
Department of Mathematics, Zhejiang University, Hangzhou 310027, China
State Key Laboratory of CAD&CG, Zhejiang University, Hangzhou 310027, China
article
info
Article history:
Received 1 July 2012
Received in revised form 14 October 2012
Accepted 20 November 2012
Keywords:
Partial differential equations
Geodesic
Line of curvature
Surface modeling
Biharmonic surfaces
abstract
In order to explore a new approach to construct surfaces bounded by geodesics or lines of
curvature, a method of surface modeling based on fourth-order partial differential equations (PDEs) is presented. Compared with the free-form surface modeling based on finding control points, PDE-based surface modeling has the following three advantages. First,
the corresponding biharmonic surface can naturally be derived under some degenerative
conditions; second, the parameters in the PDE implicate some physical meaning, such as
elasticity or rigidity; third, there are only a few parameters that need to be evaluated, and
hence the computation is simple. In addition, this paper constructs two adjacent surfaces
with C1 continuity whose common boundary is the same given curve as well as respective
geodesic (or line of curvature). Examples show that this method to construct PDE-based
surfaces bounded by geodesics or lines of curvature is easy and effective.
© 2012 Elsevier Ltd. All rights reserved.
1. Introduction
Surface modeling is a central issue in computer-aided design, and surfaces are usually represented by implicit or
parametric forms. For example, [1] transformed the construction of a surface into Poisson problems and then obtained
implicit surfaces. Compared with implicit surfaces, parametric surfaces are used more widely, and the most usual modeling
technique is to construct free-form parametric surfaces based on determining control points. [2] constructed parametric
spline surfaces by using variable-degree polynomial splines. [3] constructed harmonic and homogenous biharmonic Bézier
surfaces bounded by the given curves. [4] constructed nonhomogenous biharmonic and tetraharmonic Bézier surfaces
according to given boundary curves and tangent conditions along them. In addition, for free-form parametric surfaces, there
is another modeling technique that needs to be paid attention to, which is the parametric surface construction based on
PDEs. [5] constructed parametric surfaces using a fourth-order PDE with three shape control parameters, and presented
exact analytic solutions of the PDE in some cases. [6] constructed surfaces using a sixth-order PDE, which interpolate the
given boundary curves, boundary tangents, and boundary curvatures. In recent years, people have become interested in two
additional problems in surface modeling. One is to construct surfaces bounded by given geodesic curves, and the other is to
construct surfaces bounded by lines of curvature.
A geodesic curve is intrinsic to the geometric characterization of surfaces. Geodesics are used in many fields. For example,
they are used in object segmentation [7,8] and multi-scale image analysis [9], in computer vision and image processing. In
surface modeling, [10] constructed parametric surfaces bounded by the given geodesic curves; [11] constructed polynomial
ruled surfaces with the given boundary as geodesic curves; similarly, [12] constructed cubic polynomial ruled patches with
the given geodesic boundary. Recently, [13] constructed Bézier surfaces bounded by the four given geodesic curves, and [14]
∗
Corresponding author at: Department of Mathematics, Zhejiang University, Hangzhou 310027, China.
E-mail address: wanggj@zju.edu.cn (G.-J. Wang).
0898-1221/$ – see front matter © 2012 Elsevier Ltd. All rights reserved.
doi:10.1016/j.camwa.2012.11.018
Author's personal copy
674
W.-X. Huang et al. / Computers and Mathematics with Applications 65 (2013) 673–681
constructed triangular Bézier surfaces using the methods in [13]. As we know, the existed method of constructing parametric
surfaces bounded by geodesic curves is usually based on finding corresponding control points, but a method based on PDEs
has not yet been published in the literature.
On the other hand, lines of curvature have important applications in computer graphics and product manufacturing. [15]
studied an approach to compute lines of curvature near the umbilical points on the surfaces, and applied it to extracting
the generic features of free-form parametric surfaces for shape interrogation. [16] presented a method to determine lines
of curvature in point cloud models, and applied it to reconstruct meshes that interpolate the given lines of curvature. [17]
showed an application of lines of curvature in plate-metal-based manufacturing. In the surface modeling literature, there
are only a few articles concerning lines of curvature. Recently, [18] constructed Bézier parametric surfaces bounded by the
four lines of curvature, but the construction of a PDE-based surface bounded by lines of curvature has not been studied in
the literature.
Hence, this paper aims to address the issue of constructing a PDE-based surface bounded by geodesics or lines of
curvature. The work of this paper can be listed as follows. First, the method to generate a surface bounded by geodesics or
lines of curvature based on a vector-valued fourth-order PDE is presented, and the exact solutions of the PDE for some given
conditions are obtained by using the method in [5]. Second, the numerical solution of the PDE by the least-square method
is given. Third, the process shows that a fourth-order PDE-based surface can be degenerated into a biharmonic surface by
choosing appropriate coefficients. Fourth, two adjacent surfaces are constructed so that they have a common boundary
which is the same given curve as well as the respective geodesic, and the continuity between them is also discussed.
The rest of the paper is arranged as follows. Section 2 presents the construction of surfaces bounded by geodesic curves.
Section 3 presents the construction of surfaces bounded by lines of curvature, and the conclusion is presented in the last
section.
2. Construction of surfaces bounded by geodesics
2.1. Description of surfaces with geodesic boundary curves
Suppose that the curves p0 (u), p1 (u), u ∈ [a, b] are given, where a, b are both arbitrarily real numbers. Then the surface
p(u, v) = (x(u, v), y(u, v), z (u, v))T
with the given curves as boundary curves should satisfy
p(u, 0) = p0 (u),
(1)
p(u, 1) = p1 (u).
(2)
In order to make the given curves p0 (u), p1 (u) be geodesics on the surface p(u, v), the necessary condition is that the
vectors b0 (u) = p′0 (u) × p′′0 (u) and b1 (u) = p′1 (u) × p′′1 (u) are all tangent to the surface. The vectors b0 (u) and b1 (u) are
parallel to the binormals of the curves p0 (u) and p1 (u), respectively. Specifically, the partial derivative with respect to v of
the surface should satisfy the following equations:
pv (u, 0) = α0 (u)p′0 (u) + β0 (u)b0 (u),
(3)
pv (u, 1) = α1 (u)p′1 (u) + β1 (u)b1 (u),
(4)
where αi (u), βi (u) ̸= 0, i = 0, 1 are any functions defined on [a, b]. In addition, suppose that the surface satisfies the
following fourth-order PDE:
∂4
∂4
∂4
a 4 +b 2 2 +c 4
∂u
∂ u ∂v
∂v


p(u, v) = 0,
(5)
where a = (ax , ay , az )T , b = (bx , by , bz )T , c = (cx , cy , cz )T are vectors with positive components.
The PDE in Eq. (5) includes all forms of the existing fourth-order PDEs used for surface generation [5], and another
advantage of this PDE is that the surface p(u, v) becomes a biharmonic surface if we choose the coefficients as a =
(1, 1, 1)T , b = (2, 2, 2)T , c = (1, 1, 1)T . So the surface
 Eq. (5) is interesting. A biharmonic surface is a surface
 satisfying
that satisfies the equation ∆2 p(u, v) = 0, where ∆ =
∂
∂ u2
+
∂
∂v 2
. Besides, the parameters in the PDE imply some physical
meaning. For example, a fourth-order PDE can be derived from the theory of bending thin elastic plates; thus the coefficients
of such a PDE are closely related to the physical properties of the surface that it represents [19]. The parametric surface
determined by Eqs. (1)–(5) is the one we need, i.e., the surface with the common boundary as respective geodesic.
Remark 1. The operations of the vectors defined in this paper are ab = (ax bx , ay by , ay by )T and a/b = (ax /bx , ay /by , ay /by )T ,
called vector multiplication or division, respectively, where a = (ax , ay , az )T , b = (bx , by , bz )T . In addition, a < b means
ax < bx , ay < by , az < bz , 1 − a = (1 − ax , 1 − ay , 1 − az ), ξ a = (ξ ax , ξ ay , ξ az ),

b
a
=

bx
ax
,

by
ay
,
 
bz
az
.
Author's personal copy
W.-X. Huang et al. / Computers and Mathematics with Applications 65 (2013) 673–681
675
2.2. Solution of the fourth-order PDE
For the fourth-order PDE, shown as in Eq. (5), with boundary conditions (1)–(4), explicit solutions do not always exist. [5]
presented three cases in which explicit solutions do exist. Hence, we divide the process of solving the PDE into two parts.
For cases where explicit solutions exist, we will simply present the explicit solutions [5]. For other cases, we will use the
least-square method to obtain the numerical solution.
We represent the boundary conditions written as Eqs. (1)–(4) as follows:
p(u, 0) =
J

aj,1 gj (ξ u),
pv (u, 0) =
j =0
p(u, 1) =
J

J

aj,2 gj (ξ u),
(6)
aj,4 gj (ξ u),
(7)
j =0
aj,3 gj (ξ u),
pv (u, 1) =
j =0
J

j =0
where {gj (ξ u) = (gj,x (ξ u), gj,y (ξ u), gj,z (ξ u))T , j = 0, . . . , J } are independent functions with respect to the parameter u,
the parameter ξ is a constant coefficient, and their corresponding coefficients are chosen as aj,i = (aj,i,x , aj,i,y , aj,i,z )T , i =
1, 2, 3, 4. The reason that we use Eqs. (6) and (7) to represent the boundary conditions is that the solution of the PDE with
the above boundary conditions can be represented by
p(u, v) =
J

Gj (v)gj (ξ u),
j =0
where Gj (v) = (Gj,x (v), Gj,y (v), Gj,z (v))T ; that is, the solution separates the parameters u and v . Therefore, our goal has
been transformed to deriving the functions Gj (v). If the basis functions satisfy one of the following three cases, then explicit
solutions exist.
Case 1. ∂∂u2 gj (ξ u) = ∂∂u4 gj (ξ u) = 0.
The unknown functions Gj (v) can be expressed as
4
2
Gj (v) =
4

cj,k v k−1 ,
j = 0, 2, . . . , J ,
k=1
where cj,k are unknown coefficients, which can be determined by Eqs. (6) and (7) uniquely, and J is the number of
independent functions in Eqs. (6) and (7).
2
Case 2. ∂∂u2 gj (ξ u) = −ξ 2 gj (ξ u).
If the coefficients in Eq. (5) satisfy 4ac < b2 , then the unknown functions Gj (v) can be expressed as
Gj (v) = cj,1 erj,1 v + cj,2 erj,2 v + cj,3 erj,3 v + cj,4 erj,4 v ,
where
 



b
4ac

rj,i = ±ξ
1± 1− 2 ,
erj,i v = (erj,i,x v , erj,i,z v , erj,i,z v )T ,
b
2a
i = 1, 2, 3, 4.
If the coefficients in Eq. (5) satisfy 4ac = b2 , then the unknown functions Gj (v) can be expressed as
Gj (v) = cj,1 + cj,2 v erj,1 v + cj,3 + cj,4 v erj,2 v ,




where

rj,i = ±ξ
b
2a
,
i = 1, 2.
It should be noticed that the computations of ac , b2 , ac /b2 , 1 − 4ac /b2 ,
rj,i , a, b, c are all vectors.
√
b/(2a), etc. are all defined in Remark 1, and
2
Case 3. ∂∂u2 gj (ξ u) = ξ 2 gj (ξ u).
If the coefficients in Eq. (5) satisfy 4ac < b2 , then the unknown functions Gj (v) can be expressed as
Gj (v) = cj,1 cos rj,1 v + cj,2 sin rj,1 v + cj,3 cos rj,2 v + cj,4 sin rj,2 v ,








Author's personal copy
676
W.-X. Huang et al. / Computers and Mathematics with Applications 65 (2013) 673–681
where
 



b
4ac
rj,i = ξ 
1± 1− 2 ,
cos rj,i v = cos rj,i,x v , cos rj,i,y v , cos rj,i,z v

b
2a

sin rj,i v = sin rj,i,x v , sin rj,i,y v , sin rj,i,z v









T
,





T
,
i = 1, 2.
If the coefficients in Eq. (5) satisfy 4ac = b2 , then the unknown functions Gj (v) can be expressed as
Gj (v) = cj,1 + cj,2 v cos rj,1 v + cj,3 + cj,4 v sin rj,1 v ,








where

rj,1 = ξ
b
2a
.
The unknown coefficients cj,k in the expressions of the above function Gj (v) can be determined by the boundary conditions
written as Eqs. (6) and (7) uniquely.
If the boundary conditions do not satisfy the above three cases, then we can use the following least-square method
to obtain the numerical solution. In order to satisfy the boundary conditions expressed as (6) and (7), we represent the
numerical solution as follows:
p(u, v) =

J
m


j =0

pj,i Bm
i (v) gj (ξ u),
(8)
i=0
where {gj (ξ u) = (gj,x (ξ u), gj,y (ξ u), gj,z (ξ u))T , j = 0, . . . , J } are independent functions with respect to the parameter u,
the parameter ξ is a constant coefficient, Bm
i (v) are Bernstein polynomials, and pj,i are unknown coefficients. It should be
noticed that the higher the degree m of the parameter v is, the smaller the error of the numerical solution is, which is similar
to numerical solution of a sixth-order PDE in [6]. In this paper, we take m = 6. Because Eq. (8) satisfies Eqs. (6) and (7), the
unknown coefficients pj,0 , pj,1 , pj,m−1 , pj,m can be derived as follows:
pj,0 = aj,1 ,
pj,1 = aj,1 + aj,2 /m,
pj,m−1 = aj,3 − aj,4 /m,
pj,m = aj,3 .
Substituting the above equations into Eq. (8), we can obtain that
p(u, v) =
J



m−2
f (v, aj,i ) +
j=0

pj,i Bm
i
(v) gj (ξ u),
(9)
i=2
where

f (v, aj,i ) = aj,1 (1 − v)m + m aj,1 +
aj,2 
m

aj,4 
(1 − v)m−1 v + m aj,3 −
(1 − v)v m−1 + aj,3 v m .
m
Substituting Eq. (9) into Eq. (5) and uniformly sampling n2 ≥ (J + 1)(m − 3) points in the parametric domain, denoted by
{(uk , vk ), k = 1, . . . , n2 }, then we can obtain the following n2 equations:
∆(uk , vk ) =
J m
−2


A(uk , vk )j,i pj,i + B(uk , vk ),
k = 1, . . . n2 ,
j=0 i=2
where
∂2 m
∂2
∂4 m
∂4
g
(ξ
u
)
+
b
B
(v
)
g
(ξ
u
)
+
cg
(ξ
u
)
B (vk ),
j
k
k
j
k
k
j
∂ u4
∂v 2 i
∂ u2
∂v 4 i

J 

∂4
∂2
∂2
∂4
B(uk , vk ) =
af (vk , aj,i ) 4 gj (ξ uk ) + b 2 f (vk , aj,i ) 2 gj (ξ uk ) + cg j (ξ uk ) 4 f (vk , aj,i ) .
∂u
∂v
∂u
∂v
j=0
A(uk , vk )j,i = aBm
i (vk )
For simplicity, Eq. (10) can be represented in matrix form as
∆ = AC − B,
where the vectors are
∆ = (∆(u1 , v1 ), . . . , ∆(un2 , vn2 ))T ,
B = (−B(u1 , v1 ), . . . , −B(un2 , vn2 )),
C = (p0,2 , p0,3 , . . . , p0,m−2 , p1,2 , . . . , pJ ,m−2 )T ,
(10)
Author's personal copy
W.-X. Huang et al. / Computers and Mathematics with Applications 65 (2013) 673–681
677
Fig. 1. The biharmonic surface with given geodesic boundaries; the boundaries are drawn in blue. (For interpretation of the references to colour in this
figure legend, the reader is referred to the web version of this article.)
and the elements of matrix A are
Ak,j(m−3)+i = A(uk , vk )j,i+1 ,
k = 1, . . . , n2 , j = 0, . . . , J , i = 1, . . . , m − 3.
In order to minimize the error ∆T ∆, according to the least-square method, it can be derived that
 −1
C = AT A

AT B.
(11)
Hence, the unknown coefficients in Eq. (9) can be obtained by Eq. (11).
2.3. Numerical examples
Two examples are presented in this section. Example 1 constructs a biharmonic surface bounded by the given geodesics,
and Example 2 constructs two adjacent surfaces that have a common geodesic boundary curve. Suppose that the two surfaces
are s1 (u, v) and s2 (u, v), and that the common boundary is p(u); that is, s1 (u, 1) = p(u) = s2 (u, 0). Also, suppose that the
surface s1 (u, v) chooses the functions α1 (u), β1 (u) in Eq. (4), and the surface s2 (u, v) chooses the functions α0 (u), β0 (u)
in Eq. (3). If the two surfaces choose the same functions α0 (u) = α1 (u), β0 (u) = β1 (u) on the common boundary which
is the given curve, then we can prove that they are C1 continuous along the common boundary, as follows. Since s1 (u, 1) =
∂ s (u,1)
= ∂ p∂(uu) =
p(u) = s2 (u, 0), the two surfaces are C 0 continuous. Denoting b(u) = p′ (u) × p′′ (u), it can be seen that 1∂ u
∂ s2 (u,0)
∂u
∂ s (u,1)
∂ s (u,0)
and 1∂v
== α1 (u)p′ (u) + β1 (u)b(u) = α0 (u)p′ (u) + β0 (u)b(u) = 2∂v , so the two surfaces are also C1
continuous. The proof is valid for lines of curvature too. If we need to make the two adjacent surfaces be C 2 continuous
along the common boundary, then a sixth-order PDE is needed to construct the corresponding surfaces; see [20].
Example 1 (To Construct Biharmonic Surface Bounded by Geodesics). Two curves p0 (u) = (sin u, cos u, 1)T , p1 (u) = (2 sin u,
3 cos u, 0)T , u ∈ [0, 2π ] are given, the functions in Eqs. (3) and (4) are chosen as α0 (u) = α1 (u) ≡ 0, β0 (u) ≡ 0.5, β1 (u) ≡
1, and the coefficients in Eq. (5) are chosen as a = (1, 1, 1)T , b = (2, 2, 2)T , c = (1, 1, 1)T (see Fig. 1). In this example, it
can be seen that J = 3, and the three independent functions are {sin u, cos u, 1}.
Example 2 (To Construct Two Adjacent Biharmonic Surfaces with Prescribed Geodesic Boundaries, One of them Common to Both
Surfaces). For Surface 1, the given geodesic boundary curves are
p0 (u) = (3 sin u, cos u, 4)T ,
p1 (u) = (2 sin u, 3 cos u, 0)T ,
u ∈ [0, 2π ],
and the functions in Eqs. (3) and (4) are chosen as α0 (u) = α1 (u) ≡ 0, β0 (u) ≡ 0.1, β1 (u) ≡ 1.
For Surface 2, the given geodesic boundary curves are
p0 (u) = (2 sin u, 3 cos u, 0)T ,
p1 (u) = (3 sin u, 2 cos u, −1)T ,
u ∈ [0, 2π ],
and the functions in Eqs. (3) and (4) are chosen as α0 (u) = α1 (u) ≡ 0, β0 (u) = β1 (u) ≡ 1. In addition, the coefficients in
Eq. (5) are chosen as a = (1, 1, 1)T , b = (2, 2, 2)T , c = (1, 1, 1)T . The two biharmonic surfaces are C1 continuous along the
common boundary (see Fig. 2).
In this example, it can also be seen that J = 3, and the three independent functions are {sin u, cos u, 1}.
Author's personal copy
678
W.-X. Huang et al. / Computers and Mathematics with Applications 65 (2013) 673–681
Fig. 2. Two biharmonic surfaces with the same given geodesic boundary. Surface 1 is drawn in red, and the other is drawn in green. (For interpretation of
the references to colour in this figure legend, the reader is referred to the web version of this article.)
3. Construction of surfaces bounded by lines of curvature
3.1. Rational rotation-minimizing frames for spatial Pythagorean-hodograph (PH) curves
[21] presented the sufficient and necessary condition that quintic PH curves have rational rotation-minimizing frames.
We simply describe it as follows.
If the derivative of the r (t ) satisfies
x′2 (t ) + y′2 (t ) + z ′2 (t ) = σ 2 (t ),
where r ′ (t ) = (x′ (t ), y′ (t ), z ′ (t )) and σ (t ) is a polynomial, then the curve r (t ) is called a spatial Pythagorean-hodograph
(PH) curve. What is more, there are polynomials u(t ), v(t ), p(t ), q(t ) such that they satisfy
x′ (t ) = u2 (t ) + v 2 (t ) − p2 (t ) − q2 (t ),
z (t ) = 2 [v(t )q(t ) − u(t )p(t )] ,
′
y′ (t ) = 2 [u(t )q(t ) + v(t )p(t )] ,
σ (t ) = u2 (t ) + v 2 (t ) + p2 (t ) + q2 (t ).
Meanwhile, the derivative of the PH curve r (t ) can also be represented by a quaternion, as follows:
r ′ (t ) = A(t )iA∗ (t ),
(12)
where
A(t ) = u(t ) + v(t )i + p(t )j + q(t )k ,
ii = jj = kk = −1,
ij = k ,
A∗ (t ) = u(t ) − v(t )i − p(t )j − q(t )k ,
ji = −k ,
jk = i,
kj = −i,
ki = j ,
ik = −j .
Suppose that the Frenet frame of the curve r (t ) is defined by
T =
r′
∥r ′ ∥
,
N =
r ′ × r ′′
∥r ′
×
r ′′ ∥
× T,
B=
r ′ × r ′′
∥r ′ × r ′′ ∥
;
then the curve’s angular velocity is given by
w = kB + τ T ,
where k is the curvature, and τ is the torsion. The Frenet frame is not a rotation-minimizing frame, because the angular
velocity contains the term τ T . A rotation of (N , B) through the angle
ϕ(ξ ) = ϕ0 −
ξ

τ (u)σ (u)du
0
can change the Frenet frame into a rotation-minimizing frame, such that the angular velocity only contains the term kB,
where ϕ0 is the free integration constant, σ (u) = ∥r ′ (u)∥. If ϕ(ξ ) is a rational function, then the Frenet frame is changed
into a rational rotation-minimizing frame.
[21] pointed out the sufficient and necessary condition that quintic spatial PH curves defined by Eq. (12) have rational
rotation-minimizing frames, namely, if and only if the coefficients of the quaternion polynomial
A(t ) = A0 (1 − t )2 + 2A1 (1 − t )t + A2 t 2
satisfy the expression A2 iA∗0 = A1 iA∗1 .
(13)
Author's personal copy
W.-X. Huang et al. / Computers and Mathematics with Applications 65 (2013) 673–681
679
If we take
A0 = 1,
A1 = u1 + v1 i + p1 j + q1 k,
A2 = u21 + v12 − p21 − q21 + 2 (u1 p1 − v1 q1 ) j + 2 (u1 q1 + v1 p1 ) k ,


where u1 , v1 , p1 , q1 are any constants, then A(t ) satisfies the requirement. The key polynomials u(t ), v(t ), p(t ), q(t ) can be
obtained as follows by simple computation:
u(t ) = (1 − t )2 + 2u1 (1 − t )t + u21 + v12 − p21 − q21 t 2 ,


v(t ) = 2v1 (1 − t )t ,
p(t ) = 2p1 (1 − t )t + 2(u1 p1 − v1 q1 )t 2 ,
q(t ) = 2q1 (1 − t )t + 2 (u1 q1 + v1 p1 ) t 2 .
In the following, all quintic spatial PH curves are defined in this way. We denote
a(t ) = (1 − t )2 + 2u1 (1 − t )t + u21 + v12 + p21 + q21 t 2 ,
(14)
b(t ) = 2v1 (1 − t )t .
(15)


Then Euler–Rodrigues [21] frame is defined as follows:
t =
A(t )iA∗ (t )
∥A(t )∥2
,
n=
A(t )jA∗ (t )
∥A(t )∥2
,
b=
A(t )kA∗ (t )
∥A(t )∥2
,
where
∥A(t )∥2 = u2 (t ) + v 2 (t ) + p2 (t ) + q2 (t ).
(16)
Further, we can obtain the rational rotation-minimizing frame of the curve r (t ) as
e1 = t ,
e2 =
a2 (t ) − b2 (t )
a2 (t ) + b2 (t )
n−
2a(t )b(t )
a2 ( t ) + b 2 ( t )
b,
e3 =
2a(t )b(t )
a2 ( t ) + b 2 ( t )
n+
a2 (t ) − b2 (t )
a2 (t ) + b2 (t )
b.
(17)
3.2. Description of surfaces bounded by lines of curvature
[18] presented a lemma stating that a curve on a smooth surface is a line of curvature of that surface if and only if the
Darboux frame is the same as the rational rotation-minimizing frame with respect to the curve’s tangent. Suppose that the
curve r (ξ ) = s(u(ξ ), v(ξ )) is a curve on the surface s(u, v); then the Darboux frame (t , n, h) along this curve is defined as
follows: t is the tangent to the curve r (ξ ), n is the surface normal along r (ξ ), and h = n×t.
Suppose that the curves p0 (u), p1 (u), u ∈ [0, 1] and their rotation-minimizing frames (ti , ni , hi ), i = 0, 1 are given,
where (ti , ni , hi ) is defined by (e1 , e2 , e3 ) as Eq. (17). And suppose that the surface bounded by the given curves as lines of
curvature is denoted by p(u, v) = (x(u, v), y(u, v), z (u, v))T . Then the surface should satisfy
p(u, 0) = p0 (u),
(18)
p(u, 1) = p1 (u).
(19)
According to the lemma in [18], mentioned above, in order to make the curves p0 (u), p1 (u) be lines of curvature on the
surface p(u, v), the partial derivative with respect to v of the surface should satisfy
pv (u, 0) = α0 (u)t0 (u) + β0 (u)h0 (u),
(20)
pv (u, 1) = α1 (u)t1 (u) + β1 (u)h1 (u),
(21)
where αi (u), βi (u) ̸= 0, i = 0, 1 are any functions defined on [0, 1]. In addition, suppose that the surface satisfies the fourthorder PDE defined by Eq. (5). Then the surface determined by Eqs. (18)–(21) and (5) is just the one bounded by the given
lines of curvature. Specially, the surface p(u, v) will be degenerated into a biharmonic surface if we choose the coefficients
as a = (1, 1, 1)T , b = (2, 2, 2)T , c = (1, 1, 1)T . The solution of the PDE has been given in Section 2.2.
3.3. Numerical examples
Example 3 (To Construct Biharmonic Surface Bounded by Lines of Curvature). Two quintic spatial PH curves, p0 (u) and p1 (u),
are given as follows.
The curve p0 (u) is defined as p0 (0) = (0, 0, 0), and the parameters in Eq. (13) are chosen as u1 = 1, v1 = 2, p1 = 3, q1 =
2. The functions in Eq. (20) are taken as α0 (u) ≡ 0, β0 (u) ≡ 0.0000017(a2 (u) + b2 (u))∥A(u)∥2 , where a(u), b(u), A(u) are
defined by Eqs. (14), (15) and (16), respectively.
The curve p1 (u) is defined as p1 (0) = (0, 0, −100), and the parameters in Eq. (13) are chosen as u1 = 1, v1 =
2, p1 = 3, q1 = 2. The functions in Eq. (21) are taken as α1 (u) ≡ 0, β1 (u) ≡ 0.0000017(a2 (u) + b2 (u))∥A(u)∥2 , where
a(u), b(u), A(u) are defined by Eqs. (14), (15) and (16), respectively.
Author's personal copy
680
W.-X. Huang et al. / Computers and Mathematics with Applications 65 (2013) 673–681
Fig. 3. The biharmonic surface is bounded by lines of curvature; the boundaries are drawn in blue. (For interpretation of the references to colour in this
figure legend, the reader is referred to the web version of this article.)
Fig. 4. Two adjacent approximate biharmonic surfaces are bounded by lines of curvature. Surface 1 is drawn in red, and the other is drawn in green. (For
interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
The coefficients in Eq. (5) are taken as a = (1, 1, 1)T , b = (2, 2, 2)T , c = (1, 1, 1)T , and m = 6, n = 20 in Eq. (10). In this
example, we can see thatJ = 8, and the independent functions are {ui |i = 0, . . . , 7}. Hence, we need to solve the PDE by the
least-square method. We can obtain the approximate biharmonic surface with given curves as boundary lines of curvature
as shown in Fig. 3.
Example 4 (To Construct Two Adjacent Biharmonic Surfaces Bounded by Lines of Curvature). For Surface 1, two quintic spatial
PH curves, p0 (u) and p1 (u), are given as follows.
The curve p0 (u) is defined as p0 (0) = (0, 0, 0), and the parameters in Eq. (13) are chosen as u1 = 1, v1 = 2, p1 = 3, q1 =
2. The functions in Eq. (20) are taken as α0 (u) ≡ 0, β0 (u) ≡ 0.000017(a2 (u) + b2 (u))∥A(u)∥2 , where a(u), b(u), A(u) are
defined by Eqs. (14), (15) and (16), respectively.
The curve p1 (u) is defined as p1 (0) = (0, 0, −100), and the parameters in Eq. (13) are chosen as u1 = 1, v1 = 3, p1 =
3, q1 = 2. The functions in Eq. (21) are taken as α1 (u) ≡ 0, β1 (u) ≡ 0.000017(a2 (u)+ b2 (u))∥A(u)∥2 , where a(u), b(u), A(u)
are defined by Eqs. (14), (15) and (16), respectively.
For Surface 2, two quintic spatial PH curves, p0 (u) and p1 (u), are given as follows.
The curve p0 (u) is defined as p0 (0) = (0, 0, 100), and the parameters in Eq. (13) are chosen as u1 = 1, v1 = 2, p1 =
2, q1 = 3. The functions in Eq. (20) are taken as α0 (u) ≡ 0, β0 (u) ≡ 0.000017(a2 (u)+ b2 (u))∥A(u)∥2 , where a(u), b(u), A(u)
are defined by Eqs. (14), (15) and (16), respectively.
The curve p1 (u) is defined as p1 (0) = (0, 0, 0), and the parameters in Eq. (13) are chosen as u1 = 1, v1 = 2, p1 = 3, q1 =
2. The functions in Eq. (21) are taken as α1 (u) ≡ 0, β1 (u) ≡ 0.000017(a2 (u) + b2 (u))∥A(u)∥2 , where a(u), b(u), A(u) are
defined by Eqs. (14), (15) and (16), respectively.
The coefficients in Eq. (5) are taken as a = (1, 1, 1)T , b = (2, 2, 2)T , c = (1, 1, 1)T , and m = 6, n = 20 in Eq. (10). In
this example, we can see that J = 8, and the independent functions are {ui |i = 0, . . . , 7}. Hence, we need to solve the PDE
by the least-square method. We can obtain two approximate biharmonic surfaces with the given curves as boundary lines
of curvature, see Fig. 4.
4. Conclusion
This paper presents a method to construct PDE-based surfaces bounded by geodesics or lines of curvature. Examples
show that this method is effective and easy. The method for constructing a fourth-order PDE-based surface has some
advantages that the one for constructing a free-form surface does not have. It is simple to obtain a biharmonic surface,
Author's personal copy
W.-X. Huang et al. / Computers and Mathematics with Applications 65 (2013) 673–681
681
and the parameters in the PDE possess some physical interpretations, such as elasticity or rigidity. Furthermore, this paper
gives an example of the construction of two adjacent surfaces with the same given curve as the common boundary, and they
are C1 continuous along the common boundary. It should be noticed that the method proposed in Section 3 can only deal
with a very important case of quintic PH curves constructed according to [21], which will be extended in the future.
Acknowledgment
This work was supported by National Nature Science Foundations of China under Grant No. 60933007 and No. 61070065.
References
[1] M. Kazhdan, M. Botitho, H. Hoppe, Poisson surface reconstruction, in: Proceedings of the Fourth Eurographics Symposium on Geometry Processing,
2006, pp. 61–70.
[2] P. Costantini, Curve and surface construction using variable degree polynomial splines, Computer Aided Geometric Design 17 (5) (2000) 419–446.
[3] J. Monterde, H. Ugail, On harmonic and biharmonic Bézier surface, Computer Aided Geometric Design 21 (6) (2004) 697–715.
[4] P. Centella, J. Monterde, E. Moreno, R. Oset, Two C1 -methods to generate Bézier surfaces from the boundary, Computer Aided Geometric Design 26
(2) (2009) 52–173.
[5] L.H. You, J.J. Zhang, P. Comninos, Blending surface generation suing a fast and accurate analytical solution of a fourth-order PDE with three shape
control parameters, The Visual Computer 20 (2) (2004) 199–214.
[6] J.J. Zhang, L.H. You, Fast surface modeling using a 6th order PDE, Computer Graphics Forum 23 (3) (2004) 311–320.
[7] V. Caselles, R. Kimmel, G. Sapiro, Geodesic active contours, International Journal of Computer Vision 22 (1) (1997) 61–79.
[8] L. Cohen, R. Kimmel, Global minimum for active contours models: a minimal path approach, International Journal of Computer Vision 24 (1) (1997)
57–78.
[9] R. Kimmel, intrinsic scale space for images on surfaces: the geodesic curvature flow, Graphical Models Image Process 59 (5) (1997) 365–372.
[10] G.J. Wang, K. Tang, C.L. Tai, Parametric representation of a surface pencil with a common spatial geodesic, Computer Aided Design 36 (5) (2004)
447–459.
[11] J. Sánchez-Reyes, R. Dorado, Constrained design of polynomial surfaces from geodesic curves, Computer Aided Design 40 (1) (2008) 49–55.
[12] M. Paluszny, Cubic polynomial patches through geodesics, Computer Aided Design 40 (1) (2008) 56–61.
[13] R.T. Farouki, N. Szafran, L. Biard, Construction of Bézier surface patches with Bézier curves as geodesic boundaries, Computer Aided Design 41 (10)
(2009) 772–781.
[14] R.T. Farouki, N. Szafran, L. Biard, Construction and smoothing of triangular Coons patches with geodesic boundary curves, Computer Aided Geometric
Design 27 (4) (2010) 301–312.
[15] T. Maekawa, F.E. Wolter, N.M. Patrikalakis, Umbilics and lines of curvature for shape interrogation, Computer Aided Geometric Design 13 (2) (1996)
133–161.
[16] E. Kalogerakis, D. Nowrouzezahrai, P. Simari, K. Singh, Extracting lines of curvature from noisy point clouds, Computer Aided Design 41 (4) (2009)
282–292.
[17] F.C. Munchmeyer, On surface imperfections, in: R. Martin (Ed.), Mathematics of Surfaces II, Oxford University Press, Oxford, 1987, pp. 459–474.
[18] L. Biard, R.T. Farouki, N. Szafran, Construction of rational surface patches bounded by lines of curvature, Computer Aided Geometric Design 27 (5)
(2010) 359–371.
[19] L.H. You, P. Comninos, J.J. Zhang, PDE blending surfaces with C2 continuity, Computers and Graphics 28 (5) (2004) 895–906.
[20] L.H. You, J.J. Zhang, Surface blending with curvature continuity, in: Ninth International Conference on Computer Aided Design and Computer Graphics,
Hong Kong, China, 2005, p. 8.
[21] R.T. Farouki, Quaternion and Hopf map characterizations for the existence of rational rotation-minimizing frames on quintic space curves, Advance
in Computational Mathematics 33 (3) (2010) 331–348.