Communications in Information Science and Management Engineering (CISME)
A New Interpolation Method by NTP Curves and
Surfaces
Gang Liu a, b, Guo-Jin Wang a, b, *
a
b
Department of Mathematics, Zhejiang University, Hangzhou, 310027, China
State Key Laboratory of CAD&CG, Zhejiang University, Hangzhou, 310027, China
*wanggj@zju.edu.cn
Delgado and Peña [9] in 2003, called as DP basis by some
researchers later, gives a positive response. DP basis not only
is provided with NTP property, but also has linear time
complexity of evaluation algorithm. It makes up for the
shortage of both Wang-Ball basis and Said-Ball basis more or
less. Later, Jiang and Wang [10] researched some
transformation formulae between DP basis and Bernstein
basis.
Abstract-In this paper, we research on constructing an explicit
parametric curve to be taken as the limitation curve of
Progressive Iteration Approximation (PIA) which can
interpolate some scattered data points by using normalized
totally positive (NTP) basis. We prove that when the data points
and corresponding parameter list are given, the limitation curve
has the unique explicit expression. There is a similar conclusion
also for surface case. By specially choosing two kinds of NTP
bases, Said-Bézier type generalized Ball (SBGB) basis and DP
basis, based on the formula for the inverse matrix of
Vandermonde matrix, we deduce an explicit matrix solutions of
the limitation curve and surface corresponding to these NTP
bases. Our results avoid the tedious calculation of the inverse
matrix and hence will gain extensive application in reverse
engineering.
To sum up, SBGB curve and DP curve are all NTP curves
with excellent properties. In order to apply these curves in
reverse engineering, we often need to seek proper control
points to blend SBGB basis or DP basis and then generate
corresponding parametric curve to interpolate given data
points, so does parametric surface. It is to do this that is the
theme of our manuscript. There have been plentiful general
research works to focus on approximation and interpolation.
Recently,
the
progressive
iteration
approximation
(Abbreviated as PIA) method which utilizes NTP basis to
generate limitation curve and surface interpolating given
scattered data points attracts researchers’ great attention, the
corresponding early and uptodate researchers include Qi et al.
[13]
, de Boor [14], Lin et al. [15], [16], Lu [17], Lin [18], Chen et al.
[19]
, etc. It should be noted that Delgado and Peña [20]
compared the PIA convergence rates of different NTP bases
and proved that the convergence of normalized B-basis is
fastest.
Keywords-Normalized Totally Positive basis; Said-Bézier type
generalized Ball basis; DP basis; Progressive Iteration
Approximation; Interpolation
I.
INTRODUCTION
Since an effective rational cubic curve was developed by
Ball.A.A in the CONSURF system in 1974 [1], a mass of
treatises on the construction of its high degree generalized
forms has published. In 1987 and 1989, Wang-Ball curve and
Said-Ball curve [2], [3] emerged as the times require
respectively. In 1996, Hu et al. [4] gave some algorithms for
degree elevation, degree reduction and recursive evaluation of
both Wang-Ball and Said-Ball curves, and pointed out that all
recursive evaluations of Wang-Ball and Said-Ball curves are
faster than that of Bézier curve. In detail, the evaluation time
complexity of Wang-Ball curve declines to linear, however,
Said-Ball curve does not. In 2000, Wu [5] defined two new
families of generalized Ball curves with position parameters,
which are Said-Bézier generalized Ball curve and Wang-Said
generalized Ball curve, abbreviated as SBGB curve and
WSGB curve respectively. SBGB curve take Said-Ball and
Bézier curves as its two special cases and WSGB curve take
Wang-Ball and Said-Ball curves as its special cases. They
possess many common properties of both generalized Ball
curves and Bézier curve. In 2004, Wu [6] gave the dual bases
of these two basis families as well as the Marsden identities.
This paper deduces the PIA limitation form of NTP curve
and surface, and proves the uniqueness of the limitation curve
and surface. In the meantime, by deducing or summing up
many transformation matrices from SBGB basis to power
basis, DP basis to Bernstein basis, and Bernstein basis to
power basis, by utilizing the explicit inverse matrix algorithm
of Vandermonde matrix, we obtain matrix expressions of PIA
limitation curve and surface when interpolation basis is SBGB
basis or DP basis. The examples are used to demonstrate the
validity and rationality of our algorithm in the end.
II.
NTP BASIS AND ITS TRANSFORMATION WITH
POWER BASIS
A. SBGB Basis and Its Transformation
Generally speaking, in addition to pursuing quickness of
evaluation, shape designers also require that parametric curve
preserves the shape of the control polygon as far as possible.
Theoretically speaking, shape-preserving is closedly related to
normalized totally positive (Abbreviated as NTP) property of
the basis. Delgado and Peña [7] proved that Said-Ball basis is
NTP basis but Wang-Ball basis not, therefore WSGB curve is
Definition 1 [5] In general, SBGB basis is defined as
 
n 2
 + K +i i

n 2
 + K +1
,

 t (1 − t )
i




0≤i≤
n 2
 − K − 1;

 n 

α i (t ; n, K ) =   t i (1 − t ) n −i ,
 i 

n 2
−K ≤i≤
n 2
;


α n −i (1 − t ; n, K ),



n 2
 +1 ≤ i ≤ n

not NTP curve. Moreover, in 2005, Wang et al. [8] proved that
SBGB curve is NTP curve.
Then does the basis with both evaluation quickness and
shape preserving exist? A special new one, proposed by
C 2012 World Academic Publishing
CISME Vol.2 No.1 2012 PP.18-24 www.jcisme.org ○
- 18 -
Communications in Information Science and Management Engineering (CISME)
 n 2  and  n 2  are the maximum integer not
more than n 2 and the minimum integer not less than
n 2, respectively. SBGB basis blending the control points
Thus, this lemma can be proved by substituting the above
equation to the dual function.
in which
{ pi }i=0
n
generates a degree
B. DP Basis and Its Transformation
Definition 2 [9] In general, DP basis can be expressed as
n Said-Bézier type generalized
(1 − t )n ,
i=
0;

n −i
t 1 − t ) , 1 ≤ i ≤  n2  − 1;
Din ( t ) =  (
i =  n2  ;
K (t ) ,
 D n (1 − t ) ,  n  + 1 ≤ i ≤ n,
2
 n −i
Ball (Abbreviated as SBGB) curve:
P (t ; n, K )
∑
n
i =0
α i (t ; n, K ) pi
0 ≤ t ≤1 .
From the definition, the curves P (t ; n,0),
P ( t ; n,  n 2  )
are Said-Ball curve and Bézier curve respectively. When
in which
1 ≤ K ≤  n 2  − 1, P (t ; n, K ) is located between them.
=
K (t )
Theorem 1 [8] SBGB basis is an NTP basis.
Lemma 1 The transpose matrix of the matrix
( 12 ) 
 n2  −  n2 
1 − t  n2  +1 − (1 − t )  n2  +1 


+ (  n2  −  n2  ) t (1 − t )  2  .
 n +1
F(nn+,K1)×(n +1)
= (fi , j )(n +1)×(n +1) can transform SBGB basis to Power basis,
 n 2  and  n 2  have the same meaning as in Definition
n
1. DP basis blending control points { pi }i= 0 can generate a
degree n DP curve:
i.e.,
(1,t,...,t n ) = (α 0 (t;n,K ) ,α1 (t;n,K ) ,
D ( n, t )
...,α n (t;n,K ))( F(nn+,K1)×(n +1) )T=
.
Here
∑
n
i =0
Din (t ) Pi ,
Theorem 2 [9] DP basis
Definition 2 is an NTP basis.
0 ≤ t ≤ 1.
{Din (t )}in=0 mentioned in
Next, we introduce the following notations [10]:
 i −1   n 
(−1) j −i 
  ,
 j − 1  i 
 n −1− i   n 
j
1− ∑
(−1) j −i 
K 2 ( n, j ) =
  .
=i  n 2  +1
 j − i  i 
1 − ∑ i= j
K1 (n, j ) =
 n 2  −1
H (n +1)×(n +1)
Proof: Simplifying the dual functionals of SBGB basis
mentioned in Ref. [6], we can get
Bernstein basis, i.e.,
i

 i    n 2  + K + i  1 ( s )
0 ≤ i ≤  n 2  − K − 1;

 f ( 0) ,
∑
  
s
s =0  s  
 s!


i
 i   n  1 (s)

∑
    f ( 0) ,
 n 2 − K ≤ i ≤  n 2 ;

s =0  s   s  s !
λ fi = 
n −i n − i

 n
s 1
(s)

 n 2  + 1 ≤ i ≤  n 2  + K ;
∑

   ( −1) f (1) ,

s!
s =0  s   s 

 n −i  n − i    n 2  + K + n − i 
s 1
(s)
∑ 
 ( −1) f (1) ,
 
 n 2  + K + 1 ≤ i ≤ n.
s
!
s
−
n
i
 

 s =0 
Taking
i
f (t ) as t , we have
 0, i ≠ s;
f (s) ( 0) = 
i !, i = s,
f
(s)
 i!
,
(1) =  ( i − s )!
 0,

[10]
The transpose matrix of the matrix
= (hi , j )(n +1)×(n +1) can transform DP basis to
Lemma 2
s ≤ i;
s > i.
(B0n (t ) ,B1n (t ) ,...,Bnn (t ))
= (D0n (t ) ,D1n (t ) ,...,Dnn (t )) H (Tn +1)×(n +1) ,
here the non-zero elements in the matrix
written
as
1,

( −1)i − j  i − 1  n  ,

 

 j − 1 i 

n 
 1 A( n )
A(n)
n 2 − j   n 2  − 1 
( −1)(   )     n 2  ,
( 2 ) K 2 ( n, j ) +
−
n
2
j
2



       

hi , j = 
( 1 ) A( n ) + A ( n ) ( −1) j −n 2  n  ,


 2
2
  n 2  

 A( n )
n 
A(n)
j − n 2   n 2  − 1 
( 12 ) K3 ( n, j ) −
( −1)       n 2  ,
−
j
n
2
2

       
g
,
 n −i , n − j
C 2012 World Academic Publishing
CISME Vol.2 No.1 2012 PP.18-24 www.jcisme.org ○
- 19 -
H (n +1)×(n +1) can be
i= j= 0;
1 ≤ i ≤  n / 2  − 1,
1 ≤ j ≤ i;
i =  n 2  ,
1 ≤ j ≤  n 2  − 1;
i =  n 2  ,
 n 2  ≤ j ≤  n 2  ;
i =  n 2  ,
 n 2  + 1 ≤ j ≤ n − 1;
i ≥  n 2  + 1,
Communications in Information Science and Management Engineering (CISME)
Lemma
in which
=
A(n)
(  n
α = {ai }i =0 ,
n −1
2  −  n 2  ) .
( B0n (t ), B1n (t ), , Bnn (t )) = (1, t , , t n )Q(Tn +1)×( n +1) ,
(1, t , , t n ) = ( B0n (t ), B1n (t ), , Bnn (t )) S(Tn +1)×( n +1) .
Proof: If Denote
n
1
n
matrix generated when the
j -th row of the matrix Vnα−1 is
(1, x,..., x n −1 ) as V j , and let
= (1, x,..., x n −1 )V * V ,
then
V
=
p j ( x ) V=
j
z11
z22
Since
we have

k −1
i =1
V * , V as the adjoint matrix and
( p0 ( x), p1 ( x),..., pn −1 ( x)) = (1, x,..., x n −1 )(Vnα−1 ) −1
z00
∑
 a0n −1 

 a1n −1 
  

 ann−−11 n×n
Vnα−1 , respectively, express the
n −1
can be recursively computed by using the
following generalized Yanghui triangle:
z12
matrix
determinant of the matrix
replaced by the vector
n
n
z20
inverse
j= 0,1,..., n − 1, m= 0,1,...n − 2.
( x − a0 )( x − a1 ) ( x − an −1 ) =
F ( x) =
z x +z x
in which zk0 =
1, z1k =
the
zn0
znn −m
α
α
, rm, j − a j r=
,
r=
m +1, j
F′(aj )
F′(aj )
0,
j < i;


=
i+ j  n   j 
( − 1)  j   i  , j ≥ i,
  

0,
j < i;


=  n − i   n 
   , j ≥ i.

 j − i   j 
z10
in
α
n −1, j
D. The Inverse Matrix of the Vandermonde Matrix
Lemma 4 The coefficients of a degree n polynomial
written as
+ + z
elements
parameters
have the following recursive relation:
Here
0
n
different
1 a0

1 a1
α
Vn −1 = 
 

1 an −1
can transform power basis into Bernstein basis, and vice
versa:
si , j
n
Vandermonde matrix
(=
qi , j )( n +1)×( n +1) , S( n +1)×( n +1) ( si , j )( n +1)×( n +1)
qi , j
the
Given
Rnα−1 = (riα, j ) n×n of the following n × n non-degenerate
C. The Transformation between Bernstein Basis and Power
Basis
Lemma 3 [21] The transpose matrices of the matrices
Q( n +1)×( n +1)
[22]
5
(− ai ), zkk =∏ i =1 (− ai ), and
k −1
n −1
x − ai
∏ a=
−a
i =0
i≠ j
j
i
F ( x)
F ′ ( a j )( x − a j )
.
Rnα−1 is the inverse matrix of the matrix Vnα−1 ,
p j ( x) = (1, x,..., x n −1 )(r0,α j , r1,αj ,..., rnα−1, j )T .
the recursion formula is
i.e.,
zki =zki −1 + zki −−11 ( −ak −1 ) , k =3, 4,..., n, i =2..., k − 1.
Proof: The coefficients of the polynomial F ( x ) can be
zn0 x n + z1n x n −1 +  + znn
= rnα−1, j x n −1 +  + r1,αj x + r0,α j .
F ′(a j )( x − a j )
summarized as
 zn0 = 1,
 1
n −1
=
 zn ∑ i =0 (− ai ),

n−2
n −1
 zn2 =
(− ai )∑ j = i +1 (− a j ),
∑
i= 0


 ,
 m
n−m
n − m +1
n −1
(− ai1 )∑ i =
(− ai2 ) ∑ i =
(− aim ),
∑ i1 =
 zn =
0
2 i1 +1
m im−1 +1

 ,
n −1
 n
=

 zn ∏ i =0 (−ai ).
Therefore Lemma 4 can be proved by a direct computation.
Comparing the coefficients of=
x , i 0,1,..., n − 1, which
is in the both sides of the above expression, the proof of the
lemma can be completed.
i
III. THE PIA ALGORITHM OF NTP CURVE AND
SURFACE AND ITS INTERPOLATION
A. An Explicit Algorithm of PIA
Given an NTP basis
U = {ui (t )}in=0 and a sequence of
R 2 or R 3 , and given T = {ti }in=0 , a
group of increasing parameters, t0 < t1 <  < tn , in which
ti is assigned to the points pi , i = 0,1, , n. Firstly, we
the data points in
C 2012 World Academic Publishing
CISME Vol.2 No.1 2012 PP.18-24 www.jcisme.org ○
- 20 -
Communications in Information Science and Management Engineering (CISME)
p u (t ), p
∑=
n
generate an initial
curve C (t )
=
0
i =0
0
i i
0
i
P k +1 = P + ( I − B ) P k ,
pi ,
i = 0,1, , n. Then we iteratively compute a sequence of
C
the curve
k +1
in which
(t ) = ∑ i =0 p u (t ), where
k +1
i
i
n
j
j
j
P j = [ p0,0
, p0,1
, , p0,j n , p1,0
, p1,1j , , p1,j n , ,
j
T
pmj ,0 , pmj ,1 , , p=
0,1, , k + 1,
m,n ] , j
i 0,1, , n.
pik +1= pik + ∆ ik , ∆ ik= pi − C k (ti ), =
P = [ p0,0 , , p0,n , p1,0 , , p1,n , , pm ,0 , , pm ,n ]T ,
Thus the control points of the iteration sequence can be
written as the following matrix form:
P
k +1
=j 0,=
, m
B =⊗
B1 =
B2 , B1 =
(u j (ti ))i 0,=
(v j ( si ))ij 0,0,,,nn
, m , B2 =
Considering B is an invertible matrix [16], we have
− P = ( I − B) P ,
k
in which
P k +1 − B −1 P =−
( I ω B )( P k − B −1 P )
=
P [ p , p , , p=
] , j 0,1, , k + 1,
j
j
0
j
1
j T
n
=
( I − ω B ) k +1 ( P − B −1 P ), k =
0,1,
P = [ p0 , p1 , , pn ]
T
Since U , V are NTP bases [16], the iteration sequence is
convergent. Then lim k →∞ P k +1 = B −1 P .
and the collocation matrix is
 u0 (t0 ) u1 (t0 )

u (t ) u1 (t1 )
B= 0 1
 


 u0 (tn ) u1 (tn )
B. An Interpolation Algorithm of NTP Curve and Surface
Theorem 3 Given an ordered data points set { pi ∈ R 3 | i =
 un (t0 ) 

 un (t1 ) 
.

 

 un (tn ) 
=
τ {ti }in=0 , t0 < t1 <  < tn , a sequence
0,1, , n} and
of the real increasing parameters, then the NTP curve
interpolating the i-th point
pi at the parameter
ti (i = 0,1, , n) can be expressed as
(1)
Considering that
P ( t ) = ∑ i =0 N in ( t ) p i ,
n
B is an invertible matrix [16], we have
P k +1 − B −1 P =−
( I ω B )( P k − B −1 P )
in which
(p 0 , p 1 ,..., p n )T = Z (n +1)×(n +1) (p0 , p1 ,..., pn )T .
=
( I − ω B ) k +1 ( P − B −1 P ), k =
0,1,.
Since U is an NTP basis [16], the above iteration sequence is
convergent. Thus
Furthermore, if there is the following relation
lim k →∞ P k +1 = B −1 P .
(1,t , ,t n ) = (N 0n (t ),N1n (t ), ,N nn (t )) AT
The case of tensor product surface can be analyzed in a
m
similar way. Given two NTP bases U = {ui (t )}i = 0 ,
V = {v j ( s )}
n
j =0
m,n
i , j =i 0,=j 0
, given the data points { p }
between the NTP basis
in
R and two group of increasing parameters T = {t } ,
t0 < t1 <  <=
tm ; S {s j }nj =0 , s0 < s1 <  < sn .
every pair of the parameters
point
pi , j , i
=
initial
surface
generated, where
Then,
a
∑ ∑
n
m
=i 0=j 0
k +1
i, j
Z (n +1)×(n +1) = AT ( Rnτ )T .
Here
Specially, if the NTP interpolation curve is SBGB curve
or DP curve, then we have
(ti , s j ) are assigned to the
0,1,
=
 , m; j 0,1, , n. Above all, the
S ( t ) ==
∑i
m
0
∑
n
0=j 0
0
i, j
ui (t )v j ( s ) p
Z (n +1)×(n +1) = ( F(nn+,K1)×(n +1) )T ( Rnτ )T ,
is
Z (n +1)×(n +1) = H (Tn +1)×(n +1) S(Tn +1)×(n +1) ( Rnτ )T .
=
m; j 0,1, , n.
pi0, j p=
0,1, ,=
i, j , i
sequence
of
k +1
i, j
ui (t )v j ( s ) p
the
surfaces
k
i, j
k
i, j
(2)
Proof: Suppose
S k +1 ( t ) =
(N 0n (t ),N1n (t ), ,N nn (t )) = (1,t , ,t n ) C T ,
is generated iteratively, in which
it is obvious that
p = p + ∆ , ∆= pi , j − S (ti , s j ).
k
i, j
and the power basis
{t i }in=0 , then we have
m
i i =0
3
{N in (t )}in=0
k
C T = ( AT ) −1. Noting that the curve
P ( t ) = ∑ i =0 N in ( t ) p i interpolates the point pi at the
n
The control points of the iteration sequence can be
expressed as the following matrix form:
parameter
ti , i = 0,1, , n, we have
C 2012 World Academic Publishing
CISME Vol.2 No.1 2012 PP.18-24 www.jcisme.org ○
- 21 -
Communications in Information Science and Management Engineering (CISME)
 N 0n ( t0 )
 n
N (t )
B= 0 1
 
 n
 N 0 ( tn )
N1n ( t0 ) 
N1n ( t1 ) 


n
N1 ( tn ) 
1

1
=


1
N nn ( t0 ) 

N nn ( t1 ) 



N nn ( tn ) 
product NTP surface interpolating the data point
pi , j at the
parameter point
=
(ti , s j ), i 0,1,
=
 , m; j 0,1, , n can
be expressed as
P (t , s ) ==
∑i
m
in=
which M i (t ), i
m
t0  t0n 

t1  t1n  T
C .
  

t2  tnn 
∑
n
0=j 0
M im (t ) N nj ( s ) p i , j ,
0,1,...,
=
m; N nj ( s), j 0,1,..., n are two
NTP bases. If we write the control point set
the data point set
{ p i , j }im, ,j=n 0,0 and
{ pi , j }im, ,j=n 0,0 as the following vector forms:
T
P =  p 0,0 , , p 0,n , p1,0 , , p1,n , , p m,0 , , p m,n  ,
Thus
=
B −1 ((=
Vnt )T C T ) −1 AT ( Rnτ )T .
we
can
T
P =  p0,0 , , p0,n , p1,0 , , p1,n , , pm,0 , , pm,n  ,
complete the proof of Eq. (2) by Lemma 1, 2, 3.
Corollary 1 The PIA limitation curve interpolating the
data points pi , i = 0,1, , n at the parameter ti which is
generated by the basis of polynomial function space with the
degree not more than n is unique.
This paper deduces the PIA limitation form of NTP curve
and then there exist an ((m + 1)(n + 1)) × ((m + 1)(n + 1)) matrix
Z such that
Proof: Suppose U = {ui (t )}i = 0 and V = {vi (t )}i = 0
are two bases of the polynomial function space with degree
not more than n, then there exist two ( n + 1) × ( n + 1)
n
n
P = ZP ,
in which
=
Z Z (Mm +1)×( m +1) ⊗ Z (Nn +1)×( n +1) . If M im (t ),
M U and M V , such that
=
i 0,1,...,
=
m; N nj ( s ), j 0,1,..., n are SBGB basis or DP
n
U
(1, t , , t ) = (u0 (t ), un (t ), , un (t )) M ,
M
N
basis, the matrices Z ( m +1)×( m +1) , Z ( n +1)×( n +1) have the same
form as in (2).
(1, t , , t n ) = (v0 (t ), vn (t ), , vn (t )) M V .
invertible matrices
If
P
U
there
are
two
( t ) = ∑ i =0 ui (t ) p
n
interpolating
U
i
the
and
data
point
curves
written
as
V
n
V
i
P
pi
Corollary 2 Given a space data points set
) p
i
( t ) = ∑ i =0 vi (t=
at
the
{ pi , j |
0,1,
=
 , m; j 0,1, , n} and a real parameter
sequence
=
(ti , s j ), i 0,1,
=
 , m; j 0,1, , n, the PIA
parameter
ti , i = 0,1, , n, then we have
limitation interpolation tensor product surface is unique if we
choose the NTP bases as some basis of polynomial function
space with degree not more than m and n respectively.
( p 0U , p 1U ,..., p Un )T = M U ( Rnτ )T ( p0 , p1 ,..., pn )T ,
( p 0V , p 1V ,..., p Vn )T = M V ( Rnτ )T ( p0 , p1 ,..., pn )T .
Thus, the interpolation curves have the following relation:
P U (t ) = (u0 (t ), u1 (t ),..., un (t ))( p 0U , p 1U ,..., p Un )T
= (u0 (t ), u1 (t ),..., un (t )) M U ( Rnτ )T ( p0 , p1 ,..., pn )T
(1,
=
t ,..., t n )( Rnτ )T ( p0 , p1 ,..., pn )T P V (t ).
This completes the proof.
Theorem
4
{ pi , j | i = 0,1, ,
sequence
Given
a
space
data
points
set
m; j = 0,1, , n} and a real parameter
Fig. 1 The interpolation curve of example 1
(ti , s j ),=i 0,1,
=
 , m; j 0,1, , n, satisfying
t0 < t1 <  < tm ,
Dashed curve and Doted curve are SBGB curve with K=1 and DP curve,
respectively, taking the initial data points as control points; the solid curve is
the curve interpolation those data points.
s0 < s1 <  < sn , then the tensor
C 2012 World Academic Publishing
CISME Vol.2 No.1 2012 PP.18-24 www.jcisme.org ○
- 22 -
Communications in Information Science and Management Engineering (CISME)
The sequence of the control points
TABLE 1. THE CONTROL POINTS OF THE INTERPOLATION CURVES
K=0 (Said-Ball)
SBGB K=1
SBGB K=3
K=5 (Bézier)
DP curve
(0, 0)
(0, 0)
(0, 0)
(0, 0)
(0, 0)
(1.05, −1.45)
(0.90, −1.25)
(0.70, −0.97)
(0.63, −0.87)
(−1.07e+02, −2.00e+03)
(1.80, 0.78)
(1.57, 0.28)
(1.26, −0.22)
(1.26, −0.22)
(4.11e+02, 5.63e+03)
(1.60, −6.90)
(1.59, −4.50)
(1.54, −3.16)
(1.54, −3.16)
(−5.38e+02, −5.26e+03)
(0.85, 15.90)
(1.15, 7.74)
(1.15, 7.74)
(1.15, 7.74)
(2.41e+02, 1.63e+03)
(0, 0)
(0, 0)
(0, 0)
(0, 0)
(0, 0)
(−0.85, −15.90)
(−1.15, −7.74)
(−1.15, −7.74)
(−1.15, −7.74)
(−2.41e+02, −1.63e+03)
(−1.60, 6.90)
(−1.59, 4.50)
(−1.54, 3.16)
(−1.54, 3.16)
(5.38e+02, 5.26e+03)
(−1.80, −0.78)
(−1.57, −0.28)
(−1.26, 0.22)
(−1.26, 0.22)
(−4.11e+02, −5.63e+03)
(−1.05, 1.45)
(−0.90, 1.25)
(−0.70, 0.97)
(−0.63, 0.87)
(1.07e+02, 2.00e+03)
(0, 0)
(0, 0)
(0, 0)
(0, 0)
(0, 0)
 (0 0.00 0) (1 0.00

 (0 1.67 0) (1 1.67
 (0 2.50 0) (1 2.50

 (0 2.50 0) (1 2.50
 (0 3.33 0) (1 3.33

 (0 5.00 0) (1 5.00
IV. EXAMPLES
Example 1 In this example, we sample 11 points
from
the
Lemniscate
of
Gerono,
{ pi }10
i= 0
( x(t ), y (t )) = (cos(t ), sin(t ) cos(t )), t ∈ [0, 2π ] in
the following way: pi =
( x(ti ), y(ti )), ti =
0,1, ,10.
π 2 + π i 5, i =
0.00) (2 0.00 0.00) (3 0.00 0.00) (4 0.00 0.00) (5 0.00 0.00) 

4.91) (2 1.67 − 0.76) (3 1.67 2.96) (4 1.67 1.26) (5 1.67 1.66) 
− 3.21) (2 2.50 7.47) (3 2.50 − 0.21) (4 2.50 3.40) (5 2.50 2.53) 
.
2.49) (2 2.50 − 0.41) (3 2.50 3.50) (4 2.50 1.93) (5 2.50 2.43) 
0.60) (2 3.33 2.70) (3 3.33 2.11) (4 3.33 2.89) (5 3.33 2.95) 

1.00) (2 5.00 2.02) (3 5.00 2.70) (4 5.00 3.18) (5 5.00 3.54) 
For the sake of convenience, we choose uniform parameter
{ti = i 10}10
i = 0 . Thus by Theorem 3, the control points of
SBGB curves and DP curve interpolating the above sample
points can be computed directly (See Table 1).
Example 2 We uniformly choose 36 points from the
surface
=
z ( x, y ) ( x, y, xy x + y ) in the interval
[0, 5; 0, 5]. The corresponding sample points are
2
2
 (0 0 0) (1 0 0) (2 0 0) (3 0 0) (4 0 0) (5 0 0) 


 (0 1 0) (1 1 0.71) (2 1 0.89) (3 1 0.95) (4 1 0.97) (5 1 0.98) 
 (0 2 0) (1 2 0.89) (2 2 1.41) (3 2 1.66) (4 2 1.79) (5 2 1.86) 


 (0 3 0) (1 3 0.95) (2 3 1.66) (3 3 2.12) (4 3 2.40) (5 3 2.57) 
 (0 4 0) (1 4 0.97) (2 4 1.79) (3 4 2.40) (4 4 2.83) (5 4 3.12) 


 (0 5 0) (1 5 0.98) (2 5 1.86) (3 5 2.57) (4 5 3.12) (5 5 3.54) 
Then by Theorem 4, we can directly obtain the NTP
interpolation surface, to be seen as in Fig.2. If we choose
two Bernstein bases to generate a surface interpolating the
given sample points, then the control points of interpolation
surface are
 (0 0 0) (1 0 0.00) (2 0 0.00) (3 0 0.00) (4 0 0.00) (5 0 0.00) 


 (0 1 0) (1 1 2.94) (2 1 − 0.45) (3 1 1.77) (4 1 0.76) (5 1 1.00) 
 (0 2 0) (1 2 − 0.45) (2 2 4.25) (3 2 0.76) (4 2 2.42) (5 2 2.02) 


 (0 3 0) (1 3 1.77) (2 3 0.76) (3 3 3.00) (4 3 2.34) (5 3 2.70) 
 (0 4 0) (1 4 0.76) (2 4 2.42) (3 4 2.34) (4 4 3.01) (5 4 3.18) 


 (0 5 0) (1 5 1.00) (2 5 2.02) (3 5 2.70) (4 5 3.18) (5 5 3.54) 
If we choose one Bernstein basis and one Said-Ball basis to
generate a surface interpolating the given sample points,
then the control points are
Fig. 2. Up: the 6×6 sample data points.
C 2012 World Academic Publishing
CISME Vol.2 No.1 2012 PP.18-24 www.jcisme.org ○
- 23 -
Communications in Information Science and Management Engineering (CISME)
Middle: interpolation surfaces blended by two Bernstein bases
Down: interpolation surfaces blended by one Bernstein basis and one
Said-Ball basis. The circles and stars are the control points of interpolation
surfaces and sample data points, respectively.
[7]
[8]
[9]
V. CONCLUSIONS
This paper sufficiently investigates the matrices
transformation between some usual NTP bases and power
basis as well as the invertible matrix formula of the
Vandermonde matrix. By expressing the PIA limitation of
NTP curves or surfaces, we obtain some explicit matrix
forms of the curves and surfaces generated by these NTP
bases which can interpolate the given scattered data points,
to make the goal of inverse engineering accomplished
quickly and accurately.
[10]
[11]
[12]
[13]
Parameter choices of PIA make a big difference to the
final interpolation form of the curves and surfaces, thus the
influence effect and control will be our future research
work.
[14]
[15]
ACKNOWLEDGMENT
This work was supported by the National Nature
Science foundations of China under Grant No.60933007 and
No.61070065.
[16]
[17]
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[18]
Ball. A. A. (1974) CONSURF Part 1: Introduction of conic lofting
title. Computer Aided Design, 6(4): 243-249
Wang G. J. (1987) Ball curve of high degree and its application.
Applied Mathematics, A Journal of Chinese Universities, 1987, 2(1):
126−140 (in Chinese with English abstract).
Said H. B. (1989) Generalized ball curve and its recursive algorithm.
ACM Transaction on Graphics, 8(4): 360−371.
Hu S. M., Wang G. Z., Jin T. G. (1996) Properties of two types of
generalized ball curves. Computer Aided Design, 28(2): 125−133.
Wu H. Y. Unifying representation of Bézier curve and generalized
Ball curves. Appl. Math. Journal of Chinese University Ser. B, 2000,
15(1): 109−121.
Wu H. Y. Dual bases for a new family of generalized ball bases.
Journal of Computational Mathematics, 2004, 22(1): 79 −88.
[19]
[20]
[21]
[22]
Delgado J., Peña J. M. On the generalized ball bases. Advances in
Computational Mathematics, 2006, 24: 263−280.
Wang J. P., Yu H.J., Wu H.Y. Total Positivity of Said-Bézier Type
Generalized Ball Bases, 2005, 17(6): 1186-1190 (in Chinese).
Delgado J., Peña J. M. A shape preserving representation with an
evaluation algorithm of linear complexity. Computer Aided
Geometric Design, 20(1): 1-10.
Jiang S. R., Wang G. J. Conversion and evaluation for two types of
parametric surfaces constructed by NTP bases. Computers &
Mathematics with Applications, 49(2-3): 321-329.
Barsky B. A., Greenberg D. P. (1980) Determining a set of B-Spline
control vertices to generate an interpolating surface. Computer
Graph and Image Process, 14(3): 203-226.
de Boor C. (1976) Total positivity of the spline collocation matrix,
Indiana University Mathematics Journal, 25(6), 541-551.
Qi D., Tian Z., Zhang Y., and Zheng J.-B. (1975). The method of
numeric polish in curve fitting. ACTA MATHEMATICA SINICA,
18(3), 173-184.
de Boor C. (1979) How does Agee’s smoothing method work? In:
Proceedings of the 1979 Army Numerical Analysis and Computers
Conference, ARO Report 79-3, Army Research Office, 299-302.
Lin H. W, Wang G. J, and Dong C. S. (2004) Constructing Iterative
Non-Uniform B-spline Curve and Surface to Fit Data Points.
SCIENCE IN CHINA, Series F, 47(3), 315-331.
Lin H. W, Bao H. J, and Wang G. J. (2005) Totally Positive Bases
and Progressive Iteration Approximation. Computers and
Mathematics with Applications, 50(3-4), 575-586.
Lu L. Z. (2010) Weighted progressive iteration approximation and
convergence analysis. Computer Aided Geometric Design, 27(2):
129-137
Lin H. W. (2010) Local progressive-iterative approximation format
for blending curves and patches. Computer Aided Geometric Design
27(4), 322-339.
Chen J. and Wang G. J. (2011) Progressive-iterative approximation
for triangular Bézier surfaces. Computer Aided Design 43(8),
889-895.
Delgado J., Peña J. M. (2007) Progressive iterative approximation
and bases with the fastest convergence rates. Computer Aided
Geometric Design, 24(1), 10-18.
Wang G. J., Wang G. Z., Zheng J. M. Computer Aided Geometric
Design. Beijing: Higher Education Press; Heidelberg:
Springer-Verlag, 2001 (in Chinese).
Yan S. H., Yang A. M., (2009) Explicit Algorithm to the Inverse of
Vandermonde Matrix, IEEE International Conference on Test and
Measurement. Hong Kong, 176-179.
C 2012 World Academic Publishing
CISME Vol.2 No.1 2012 PP.18-24 www.jcisme.org ○
- 24 -