Design of Rational Bézier Developable Surface Pencil Through A Common Isogeodesic
Yu Liu, Guojin Wang
Design of Rational Bézier Developable Surface Pencil Through A Common
Isogeodesic
1
Yu Liu, *2Guojin Wang
1,
State Key Laboratory of CAD&CG, Zhejiang University, Hangzhou 310027, P. R. China,
lhc_liuyu@163.com
*2,
State Key Laboratory of CAD&CG, Zhejiang University, Hangzhou 310027, P. R. China,
wanggj@zju.edu.cn
Abstract
This paper studies the problem of designing a rational Bézier developable surface pencil with a
common isogeodesic, and provides an algorithm for the representation of complicated geometric
models in industrial applications which need to satisfy that the shape surface can be developed
and a given curve is geodesic. By employing the local Frenet orthonormal frame, the explicit
expression of the rational Bézier developable surface pencil is derived. Furthermore, the order of
the rational Bézier developable surface pencil interpolating a planar or non-planar curve as its
geodesic is discussed. The formulae of the control net vertices for the derived surface are
presented. Finally, the effectiveness and correctness of the algorithm are verified by examples of
the rational Bézier developable surface pencil through a degree 2 or 3 Bézier curve as a common
geodesic.
Keywords: Geodesic; Rational Bézier Surface; Developable Surface Pencil; Interpolation.
1. Introduction
Developable surfaces, as one of the most important types of surface in differential geometry,
are a subset of ruled surfaces whose tangent planes are the same along every generatrix. They can
be unfolded onto a plane without stretching and tearing, which is well known as the
developability. Hence, developable surfaces are widely used in the manufacture of 3D objects,
such as fabrication of apparels [1,2] and automobile components [3]. The construction of
developable surfaces has always been an important research field in CAD, for which a mass of
results have been presented. Fernández-Jambrina [4] derived an algorithm for constructing
B-spline control nets for spline developable surfaces of arbitrary degree and number of pieces;
using the de Casteljaus algorithm, Aumann et al. [5-7] constructed Bézier developable patches
through a Bézier boundary; for the construction of developable surfaces interpolating two given
space curves defining a strip, Chu[8-12], Wang[1,13,14] and their cooperators carried out very
detailed studies and proposed some algorithms for designing low degree Bézier developable
patches.
A Geodesic, as one of the most important intrinsic geometric properties of a surface, is the local
shortest path connecting two points on a surface so that it is widely used in industrial applications
such as tent manufacturing [15-16], fabrication of apparels [17], recognition for deformed
surface[18], Detect Community in Social Networks[19]. Therefore, there obviously exist
practical needs to study an algorithm for constructing developable surfaces through potential
isogeodesics. In recent years, these kinds of study are concentrated in constructing developable
surfaces interpolating one geodesic. Zhao and Wang [20] designed a general parametric
International Journal of Digital Content Technology and its Applications(JDCTA)
Volume6,Number18,October 2012
doi:10.4156/jdcta.vol6.issue18.10
78
Design of Rational Bézier Developable Surface Pencil Through A Common Isogeodesic
Yu Liu, Guojin Wang
developable surface pencil through a given curve as its common geodesic; Li et al. [21] derived
an algorithm for creating the general parametric developable surfaces through a given low degree
Bézier curve as its geodesic. However, the algorithm for designing rational Bézier developable
surfaces through Bézier curves as geodesics has not been presented yet, and it is well known that
the kind of rational Bézier surfaces, which takes a dominant position in CAD system, is used
more widely than the general parametric surface. Since the recent algorithms cannot meet the
practical needs of expressing the modern large-scale industrial complex geometric models, it is
urgent to present an algorithm of designing a rational Bézier developable surface pencil through a
given Bézier curve as its common geodesic, for which this paper will go on in-depth study.
The reminder of the paper is organized as follows. In Section 2, some properties of Bézier curve
are reviewed. In Section 3, the algorithm of designing a rational Bézier developable surface
pencil through a given Bézier curve as its common geodesic is presented and the formulae of its
control points are derived. The algorithm is executed by programming and some examples are
illustrated in Section 4. Finally, this paper is summed up in Section 5.
2. Preliminaries
mi
Let L degree mi Bézier curves be Ri  r    B mj i  r  Q ij , 0  r  1, i  1, 2, , L . Denote the
j 0
dot and cross product between two vectors a and b by  1 and   2  , respectively. By
calculating the dot and cross products of the above Bézier curves, we can obtain
 


R1  r    m1 R2  r    mL1 RL  r  
m1  m2  mL

j 0
 m1  m2   mL  1

     Qk1   m1
k1  k2 k L  j  k1  k 2 
 kL 
 j ! m1  m2    mL  j  !
B mj 1  m2  mL  r  

  m1  m2    mL  !

Qk22   m2    mL1 QkmLL  , 0  r  1,

 
 


(1)
m
where  mi  1, 2, i  1, 2, , L  1,    0, i  m, Qki i  0 , ki  mi , .
i
The correctness of Eq. (1) is based on the mathematical induction and the following formulae,
which is obtained by calculating the dot and cross product between two degree mi1 and mi2 Bézier
curves, respectively,
79
Design of Rational Bézier Developable Surface Pencil Through A Common Isogeodesic
Yu Liu, Guojin Wang
Ri1  r      Ri2  r 



 B0 i1  r  Q0i1  B1 i1  r  Q1i1    Bmii1  r  Qmi1i     B0 i2  r  Q0i2  B1 i2  r  Q1i2    Bmii2  r  Qmi2i
m
m
1
B
mi1
0

m
 r  B  r  Q    Q
mi1  mi2

j 0
mi2
0
i1
0
m  mi2
B j i1
i2
0

m
B
mi1
1


i1

m
m

 mi2 !
mi2
0
i1
1
 mi1  mi2
 

ki1  ki2  j  k1  k 2

m
2
 r  B  r  Q    Q
 j ! mi  mi  j !
1
2
r  
1
i2
0
  B
mi1
mi1
2

 r  B  r  Q    Q
mi2
mi2
i1
mi1
i2
mi2
T

 i1
i
 Qki1     Qk2i2 , 0  r  1, 1  i1 , i2  L,   1, 2.



m
he s -th derivative vectors of a degree m Bézier curve R  r    B mj  r  Q j , 0  r  1 are [23]
j 0
R s   r  
m! m s ms
B j  r   s Q j , 0  r  1.

m
s
!


 j 0
(2)
 sˆ Q j    s 1  Q j 1  Q j  , sˆ  1, 2, , s; j  0,1, , m  sˆ,  0 Q j  Q j , j  0,1, , m.
ˆ
Let R  r  be a curve on the surface P  r , t  . According to the theorem about geodesics in
differential geometry [22], the expression of a developable surface pencil P  r , t  through a given
spatial curve R  r  as its common geodesic can be presented by [20]

P  r , t   R  r    t  t0    r   R  r   R  r    R  r    R  r   R  r 
2

 R  r   R  r    R  r   R  r   ,
2
0  r  1, 0  t , t0  1.
(3)
3. A rational Bézier developable surface pencil through a spatial curve as its
common geodesic
This section will design a rational Bézier developable surface pencil through a given curve as a
common geodesic, present the formulae to evaluate the control points and discuss the degree of the
derived rational Bézier developable surface.
m
Suppose R  r    Bim  r  Qi , 0  r  1 is the given degree m spatial Bézier curve. In order
i 0
to obtain the expression of the rational Bézier developable surface pencil, let n be a positive integer
to be fixed,  be a positive integer correlative with n, m and
ai i  0 ,  j  j  0
n

be two groups of
positive real number correlative with the shape of a developable surface in the pencil, then by setting
the control function   r  in Eq. 3 as follows:
80
Design of Rational Bézier Developable Surface Pencil Through A Common Isogeodesic
Yu Liu, Guojin Wang
 r  
a r 
 r 
, a r  

n
1
m6  m  1
2
 B  r  a ,   r    B  r   ,0  r  1,
i 0
n
i
i
j 0
j
j
(4)
defining similarly with Section 2,
m
2
   0, i  m;Tk  0 , k  m  1; Tk  0 , k  m  2;  Tk  0, k  m  3; ak  0, k  n.
i
 
and denoting
D1  r    R  r   R  r    R  r    R  r   R  r   a  r  ,
2
D 2  r   R  r   R  r    R  r   R  r    a  r  ,
2
we can obtain
D1  r  
6 m 9  n

i 0
Bi6 m 9  n  r  Di1 ,
(5)
Bi6 m 9  n  r  Di2
(6)
D2  r  
6 m 9  n

i 0
based on Eq.-s (1) and (2) finally, where
Di1 
 m  1 m  2   m  3   m  1  m  1  m  1  n 
m2




 



 6m  9  n  k1  k6  k7 i  k1  k2   k3   k4   k5   k6   k7 


i





(7)

 ak7  Tk1 , Tk2 ,  2Tk3  Tk4  Tk5   Tk6 ,


Di2 
m 1
 m  1 m  2   m  1  m  2   m  1 m  2  n 




 



6
m
9
n



 k1  k6  k7 i  k1  k2   k3   k4   k5  k6  k7 


i




 
  ak7  Tk1  Tk2  Tk3  Tk4   Tk5  Tk6


(8)
 ,
and k j  0, j  1, , 7. By substituting Eq.-s (4), (5) and (6) into Eq. (3), we have
81
Design of Rational Bézier Developable Surface Pencil Through A Common Isogeodesic
Yu Liu, Guojin Wang

P  r, t  
 B  r    B  r  Q   t  t   D  r   D  r  
j 0
m
j
j
i 0
m
i
1
i
2
0
 r 
, 0  r  1, 0  t , t0  1.
Furthermore, using Eq. (1), the above formula can be simplified to
P  r, t  
V  r    t  t0   D1  r   D 2  r  
 r 
,
0  r  1, 0  t , t0  1,
(9)
in which
V r  
 m
 B  r V , 0  r  1,
i 0
m
i
(10)
i
Vi 
min  i , 
   m 
1

 
 k Qi  k .
   m  k  max  0,i  m   k  i  k 


 i 
(11)
Suppose M  r , t   V  r    t  t0   D1  r   D 2  r   , 0  r  1, 0  t , t0  1 , and we discuss the
degree of the surface M  r , t  . Firstly, the following lemma is presented.
m
Lemma 1 Suppose R  r    Bim  r  Qi , 0  r  1 is a given degree m Bézier curve. The degree
i 0
2m  1 curve W  r   R  r   R  r  can be reduced by degree 1, which means the degree of
W  r  is 2m  2 .
m 1
Proof: According to Eq. (2), we obtain R  r   m Bim 1  r  Qi 1  Qi  . Denote Bim as a
i 0
short-hand for Bim  r  , and hence it is clear that for i, j  0, i  j , the coefficients of all of the vector
Qi  Q j comprised in the expansion of the vector function W  r  are
 Bim  B mj 1  Bim  B mj 11  B mj  Bim 1  B mj  Bim11  r i  j 1 1  r 
2 m  2   i  j 1
82
Design of Rational Bézier Developable Surface Pencil Through A Common Isogeodesic
Yu Liu, Guojin Wang
  m   m  1
 m  m  1  m  m  1
 m   m  1 
  r   
  1  r   
  r  
  1  r    

 i  j  1   j  i 
 j   i  1 
  i  j 
2 m  2   i  j 1 
   m  m   m  m    m  m  1  m  m  1 
 r i  j 1 1  r 
  r             
   

   i  j   j  i    i  j  1   j  i  1  
 Bi2mj 12 
 i  j  1! 2m  i  j !  m   m  1  m   m  1 
   
   
 .
 2m  2  !
 i   j  1   j   i  1  
Thus, for i, j  0, i  j , the degree of Qi  Q j that included in the vector function W  r  is 2m  2 ;
in addition, for the case of i  0 or j  0 , the degree of Qi  Q j is 2m  2 similarly. Therefore,
the degree of the curve W  r  is 2m  2 . This completes the proof.
Lemma 1 indicates that the Bézier curve obtained by the cross product between a given Bézier
curve and its first derivative curve can be reduced by degree 1. Thus for setting   5m  12  n , the
degree of the Bézier curve V  r  in Eq. 10 is less than or equal to 6m  12  n .
Now we can discuss the degree of the surface M  r , t  in two cases. For one case that
R  r  is a non planar Bézier curve, according to Lemma 1, the degree of D1  r  in Eq. (5) is
6m  10  n and the degree of D 2  r  in Eq. (6) is 6m  12  n , so the degree of the surface
M  r , t  is
 6m  10  n  1 ; for the other case that
R  r  is a planar Bézier curve, D1  r   0 and
the degree of D 2  r  is 6m  12  n , so the degree of M  r , t  is
 6m  12  n  1 .
In order to
obtain the uniform representation, no less the universality, we take M  r , t  as the surface with
degree of
 6m  9  n   1 .
Set t0  0,   5m  9  n . By elevating the degree   5m  9  n Bernstein polynomial   r 
by degree m , we obtain
 r  
6 m 9  n

j 0
B 6j m 9  n  r  mj ,
0 r  1,
(12)
where the Bézier vertical coordinates  mj  0 can be calculated by the following recursive formulae:
83
Design of Rational Bézier Developable Surface Pencil Through A Common Isogeodesic
Yu Liu, Guojin Wang




 0j   j , j  0, ,5m  9  n,
j
j
mˆ 1
 mjˆ  1 
 mjˆ11 , j  0, ,5m  9  n  mˆ ; mˆ  1, , m,
 j 
5m  9  n  mˆ
5m  9  n  mˆ
(13)
mˆ1  5mˆm  9  n  mˆ 1  0, mˆ  0,1, , m  1.
On the other hand, if D1  r  , D 2  r  and V  r  as shown in Eq.-s (5), (6) and (10) are rewritten by
D1  r  
6 m 9  n

i 0
V r  
Bi6 m 9  n  r  im Di1 , D 2  r  
6 m 9  n

6 m9 n
i
B
i 0
Di1 
Di1

m
i
,
 r 
Di2 
6 m 9  n

i 0
Bi6 m 9  n  r  im Di2 ,
Vi ,0  r  1,
m
i
Di2
im
Vi 
,
Vi
im
,
(14)
respectively, then Eq. (9) can be expressed as
6 m 9  n
P  r, t  
1  t  
Bi6 m 9  n  r  imVi  t
i 0
6m 9 n
1  t  
j 0
6 m 9  n
B 6j m 9  n  r   mj  t

Bi6 m 9  n  r  imU i

B 6j m 9  n  r   mj
i 0
6 m 9  n
j 0
,
0  r  1, 0  t  1.
(15)
Here the weights are calculated by Eq. (13) and the control points are
U i  Di1  Di2  Vi ,
(16)
where Di1 , Di2 and Vi are calculated by Eq.-s (7), (8), (11) and (14).
Eq. (15) is just the expression of the degree
 6m  9  n   1
rational Bézier developable
surface pencil P  r , t ; n, ai ,  j  through a given degree m Bézier curve R  r  as its common
geodesic, while Eq. (9) can be regarded as a simplified rational representation of Eq. (15), but not the
standard rational Bézier form.
4. Examples by program
According to the algorithm presented in Section 3, Section 4.1 will show examples of Bézier
developable surfaces by degree reduction or not for interpolating a degree 2 planar Bézier curve as its
common geodesic; Section 4.2 will show examples of rational Bézier surfaces interpolating a degree 3
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Design of Rational Bézier Developable Surface Pencil Through A Common Isogeodesic
Yu Liu, Guojin Wang
spatial Bézier curve as its common geodesic.
4.1. To interpolate degree 2 planar Bézier curve
2
Suppose R  r    Bi2  r  Qi , 0  r  1 is a degree m  2 planar Bézier curve, where the
i 0
control points are Q0 , Q1 , Q2    0, 0, 0  ,
 0, 2,1 , 1, 2, 0  .
Set n  2, t0  0,   3 . By reducing Eq. (9) by degree 3, the expression of the degree 5  1
rational Bézier developable surface pencil P  r , t ; ai ,  j  with the degree of freedom 7, through this
degree 2 planar Bézier curve as its common geodesic, can be expressed as
5
P  r , t ; ai ,  j  
2
 B  r V  t  B  r  a  T
i 0
5
i
i
i 0
2
i
i
0
3
 B3j  r   j
 T1 
, 0  r  1, 0  t  1,
j 0
where T0  Q1  Q0 , T1  Q2  Q1 and the control points Vi are calculated by Eq. (11). Fig.1
illustrates the impact of the selection of free variables ai and  j to the shape of this rational Bézier
developable surface pencil P  r , t ; ai ,  j  .
Next, set n  3, t0  0,   4 . By employing Eq. (15), the expression of the degree 6  1 rational
Bézier developable surface pencil P  r , t; ai ,  j  with the degree of freedom 9, through this degree 2
planar Bézier curve as its common geodesic, can be written as
6
P  r , t ; ai ,  j  
6
1  t   Bi6  r   2j Vi  t  Bi6  r   2j U i
i 0
i 0
6
6
1  t   B  r  
j 0
6
j
2
j
 t  B 6j  r   2j
, 0  r  1, 0  t  1,
j 0
where the control points Vi , U i ,  2j are calculated by Eq.-s (7), (8), (11), (14), (16) and (13). Fig.2
illustrates the impact of the selection of free variables ai and  j to the shape of this rational Bézier
developable surface pencil P  r , t ; ai ,  j  .
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Design of Rational Bézier Developable Surface Pencil Through A Common Isogeodesic
Yu Liu, Guojin Wang
4.2. To interpolate degree 3 spatial Bézier curve
3
Suppose R  r    Bi3  r  Qi , 0  r  1 is a degree m  3 spatial Bézier curve, where the
i 0
control points are Q0 , Q1 , Q2
Q3   1, 2,1 ,
 2,1.6,1.3 ,  2.5,1.3, 2.4  ,  3, 2, 0.5  .
Setting t0  0,   n  6 , by employing Eq. (15), the expression of the degree
 n  9  1
rational Bézier developable surface pencil P  r , t; ai ,  j  with the degree of freedom 2n  8 ,
through this degree 3 spatial Bézier curve as its common geodesic, can be expressed as
n 9
P  r , t; ai ,  j  
n 9
1  t   Bin 9  r   3jVi  t  Bin 9  r   3j U i
i 0
n 9
1  t   B  r  
j 0
n9
j
i 0
n9
3
j
 t  B nj  9  r   3j
, 0  r  1, 0  t  1,
j 0
where the control points Vi , U i ,  3j are calculated by Eq.-s (7), (8), (11), (14), (16) and (13). When set
n  0,3 , we can obtain the expression of the degree 9  1 or 12  1 rational Bézier developable
surface pencil P  r , t ; ai ,  j  through the degree 3 Bézier curve as its common geodesic, as shown in
Fig.3(a) or Fig.3(b), Fig.3(c), respectively.
(a) ai  0.1 ;
(b)  j  1 ;
(c) ai ,  j
Fig. 1 the degree 5  1 rational Bézier developable surface P  r , t ; n  2,  3, ai ,  j  by degree-3
reduction through a degree 2 planar Bézier curve as its common geodesic
(a) ai  0.1,  j  1 ;
(b) ai  0.2 ;
(c) ai ,  j
Fig. 2 the degree 6  1 rational Bézier developable surface P  r , t ; n  3;  4, ai ,  j  by non degree
reduction through a degree 2 planar Bézier curve as its common geodesic
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Design of Rational Bézier Developable Surface Pencil Through A Common Isogeodesic
Yu Liu, Guojin Wang
(a) n  0,  6, ai  0.1 ;
(b) n  3,   9, ai  0.1,  j  1 ;
(c) n  3,  9,  j  1
Fig. 3 the rational Bézier developable surface P  r , t ; n,  n  6, ai ,  j  through a degree 3 spatial Bézier
curve as its common geodesic
For above examples, when set  j  K , j  0, ,  , the rational Bézier developable surface pencil
P  r , t  is degenerated into a polynomial surface pencil, and at the same time, if the given Bézier
curve is a planar curve and set ai  L, i  0, , n , the pencil P  r , t  represented by Eq.-s (9) and (15)
is the type of cylinder.
5. Conclusion
In this paper, we presented the expression of a rational Bézier developable surface pencil with a
common geodesic and discussed the degree of the pencil for interpolating a planar or spatial Bézier
curve as a common geodesic. Besides, the formulae to calculate the control points of the pencil are
given. Our work proposes an effective algorithm for the representation of a complicated geometric
model in industrial applications which need to satisfy two geodesic and developable constrains.
6. Acknowledgment
This work was supported by the National Natural Science Foundations of China under Grant No.
61070065 and No. 60933007.
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