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Author's personal copy
Computer-Aided Design 43 (2011) 889–895
Contents lists available at ScienceDirect
Computer-Aided Design
journal homepage: www.elsevier.com/locate/cad
Progressive iterative approximation for triangular Bézier surfaces
Jie Chen, 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 23 August 2010
Accepted 11 March 2011
Keywords:
Triangular Bézier surface
Bernstein operator
Fitting scattered data points
Progressive iterative approximation
abstract
Recently, for the sake of fitting scattered data points, an important method based on the PIA (progressive
iterative approximation) property of the univariate NTP (normalized totally positive) bases has been
effectively adopted. We extend this property to the bivariate Bernstein basis over a triangle domain for
constructing triangular Bézier surfaces, and prove that this good property is satisfied with the triangular
Bernstein basis in the case of uniform parameters. Due to the particular advantages of triangular Bézier
surfaces or rational triangular Bézier surfaces in CAD (computer aided design), it has wide application
prospects in reverse engineering.
© 2011 Elsevier Ltd. All rights reserved.
1. Introduction
In a lot of applications of CAD (computer aided design) and
reverse engineering, the interpolation or approximation problems
of scattered data points for constructing a parametric curve or
a parametric surface is one of the significant research areas. In
the previous literature, many researchers studied the problems
in this area, among which an effective approach is the use of
the PIA (progressive iterative approximation) property of the
univariate NTP (normalized totally positive) bases. In [1], a type
of totally positive matrices and their properties were discussed
by Ando, this provided partial theoretic base of the research of
the PIA property. In [2,3], the PIA property for the uniform cubic
B-spline basis was considered by Qi et al. and de Boor, respectively.
Furthermore, Lin et al. [4] extended this result to the non-uniform
cubic B-spline basis and the non-uniform bicubic tensor product
B-spline surfaces, then they pointed out that the non-uniform
cubic B-spline curves generated by this method satisfied NURBS
standard and possessed some good properties such as explicit
representation, local support, convexity preserving, etc. In [5], it
was proved by Lin et al. that both curves and tensor product
surfaces generated by the NTP basis satisfied the PIA property.
In [6], the convergence rates of the different NTP bases were
compared by Delgado and Peña, and it was shown that the
normalized B-basis satisfied the PIA property with the fastest
convergence rate. In [7], Lin presented the idea of local PIA which
∗ Corresponding author at: Department of Mathematics, Zhejiang University,
310027 Hangzhou, China. Tel.: +86 571 87951609 8306.
E-mail address: wanggj@zju.edu.cn (G.-J. Wang).
0010-4485/$ – see front matter © 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.cad.2011.03.012
approximated only a chosen subset of the initial control points, not
all of them, so it was more flexible.
Compared with the tensor product parametric surfaces over
a rectangle domain, the triangular parametric surfaces over a
triangle domain not only possess all the basic advantages of the
former [8] (e.g. convexity preserving and convex hull), but also
have their own particular advantages [9] (e.g. splicing flexible,
inexistence of the degenerate control points). And their common
forms are triangular Bézier surfaces or rational triangular Bézier
surfaces. Since the triangular Bernstein basis is more complex
than both the univariate Bernstein basis and the B-spline basis,
to this day, the PIA property of the multivariate Bernstein basis
over a triangle domain is still an open question. But the research
in this area is very significant and the difficulties can actually be
overcome.
The main work of this article is that we extend the PIA property
of the univariate NTP basis to the bivariate Bernstein basis over
a triangle domain. The basic technique is to give a progressive
iterative scheme of triangular Bézier surfaces. By using the fact that
the range of the eigenvalues of the bivariate Bernstein operator
is the same as that of the univariate Bernstein operator [10], we
find the relation of the eigenvalues and the eigenvectors between
the bivariate Bernstein operator and the collocation matrix of
the triangular Bernstein basis with the uniform parameters,
respectively. On this basis, the explicit representation of the above
collocation matrix’s eigenvalues is derived, and it is proved that
the PIA property is satisfied with the triangular Bernstein basis
in the case of uniform parameters. This result provides a solid
theoretic base for applications of polynomial surfaces and rational
polynomial surfaces in reverse engineering.
This paper is organized as follows, in Section 2 we review the
PIA algorithms of the Bézier curves, in Section 3 we give the PIA
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J. Chen, G.-J. Wang / Computer-Aided Design 43 (2011) 889–895
algorithms of the triangular Bézier surfaces, next some examples
are illustrated in Section 4, the PIA algorithms of the rational
triangular Bézier surfaces of low degree with different weights are
discussed in Section 5, and we briefly discuss about application in
reverse engineering in Section 6, finally the conclusion is given in
Section 7.
2. Preliminaries: The PIA for Bézier curves
Given a sequence of points {Pi ∈ R 2 |i = 0, 1, . . . , n}, each point
Pi is assigned to a parameter value ti , i = 0, 1, . . . , n, satisfying
t0 < t1 < · · · < tn .
n

With the initial control points Pi0 = Pi i=0 and an NTP Bern-

stein basis Bni (t ) =

n
i

(1 − t )n−i t i ≥ 0|t ∈ R, i = 0, 1, . . . , n ,
we obtain an initial curve, that is
n

C 0 (t ) =
Pi0 Bi (t ).
(1)
i =0
By computing the adjusting vector
∆0i
= Pi − C 0 (ti ),
i = 0, 1, . . . , n,
(2)
for every control point Pi , and letting
Pi1
=
Pi0
∆0i
+
,
i = 0, 1, . . . , n,
(3)
we can get the second curve
C 1 (t ) =
n

Pi1 Bi (t ).
(4)
i =0
Similarly, we can get the (k + 1)st curve C k (t ) after the kth
iteration, then let
∆ki
= Pi − C (ti ),
i = 0, 1, . . . , n,
,
i = 0, 1, . . . , n,
Pik+1
k
=
Pik
+
∆ki
n

T (u, v, w) =
is generated. If limk→∞ C (ti ) =
, i = 0, 1, . . . , n, we call that
the initial curve C 0 (t ) has the PIAproperty, or equally we call the
n
univariate Bernstein basis Bni (t ) i=0 has the PIA property [5].
Next we analyze the reason for the convergence of the PIA
simply. It is obvious that [5]
k
∆ki Bi (tj ),
Pi0
j = 0, 1, . . . , n; k = 0, 1, . . .
(7)
i=0
 k+1 k+1
T

T
∆0 , ∆1 , . . . , ∆nk+1 = D ∆k0 , ∆k1 , . . . ∆kn ,
(8)
k = 0, 1, . . . .
where Iis the (n + 1) ×
 (n + 1) identity matrix, and
Bn0 , Bn1 , . . . , Bnn
B := B
t0 , t1 , . . . , tn
Bn9 (t0 )
n
B0 (t1 )

= 
 ...
Bn0 (tn )
(10)
where u + v + w = 1, and Bni,j,k (u, v, w) = i!nj!!k! ui v j w k are the
bivariate Bernstein basis functions, Ti,j,k are the control points.
Generally speaking, there are (n + 1)(n + 2)/2 control points of
this surface.
Definition 3.2 ([11]). Lexicographic order: Given two d-Dimensional (d ≥ 1) vectors α and β. The vector α is arranged before
the vector β, denoted as α ≻ β, if the first nonzero entry in the
difference α − β = (α1 − β1 , . . . , αd − βd ) is positive.
Definition 3.3. The Bernstein basis functions of degree n over a triangle domain T can be arranged according to the subscripts in lexicographic order
 and expressed as a (n + 1)(n + 2)/2-dimensional

vector Bn = Bnn,0,0 , Bnn−1,1,0 , Bnn−1,0,1 , . . . , Bn0,n,0 , . . . , Bn0,0,n .
Example 3.1. The Bernstein basis functions of degree 4 over a
triangle domain T can be arranged according to the subscripts in
lexicographic order and expressed as
B4 = B44,0,0 , B43,1,0 , B43,0,1 , B42,2,0 , B42,1,1 , B42,0,2 , B41,3,0 ,

B41,2,1 , B41,1,2 , B41,0,3 , B40,4,0 , B40,3,1 , B40,2,2 , B40,1,3 , B40,0,4 .

Lemma 3.1 ([12]). The univariate Bernstein operator B̃n : C [0, 1] →
C [0, 1] is given by
B̃n f (x) =
Bn1 (t0 )
Bn1 (t1 )
..
.
Bn1 (tn )
···
···
···
···
k=0

(9)
n
(t ) i=0 at
is the collocation matrix of the NTP Bernstein basis
the parametric values {t0 , t1 , . . . , tn }.
Given a matrix A, we denote the spectrum radius of A as ρ(A),
then
i = 0, 1, . . . , n.
 
k
n
.
(11)
Then all the eigenvalues of B̃n are
λni =
n!
1
(n − i)! ni
,
i = 0 , 1 , . . . , n.
(12)
Lemma 3.2 ([10]). The s + 1 variables (s ≥ 2) Bernstein operator is
given by
n
.. 
. 
n
Bn (tn )
Bni
so [5] lim C k (ti ) = Pi0 ,
k
xk (1 − x)n−k f
n   

n
α
B (fs+1 ) =
x 1 · · · xαs s (1 − x1 − · · · − xs )n−k
α 1
k=0 |α|=k


α1
αs n − α1 − · · · αs
×f
,..., ,
Bnn (t0 )
Bnn (t1 )
0 ≤ ρ(D) = ρ(I − B) < 1,
n  

n
n

k→∞
Bni,j,k (u, v, w)Ti,j,k ,
i+j+k=n
we can rewrite the above formulas in matrix form as
D = I − B;

(6)
i =0
∆kj +1 = ∆kj −
Definition 3.1. A triangular Bézier surface of degree n over a
triangle domain T := {(u, v, w) : u, v, w ≥ 0, u + v + w = 1} is
defined by
3.2. PIA for triangular Bézier surfaces
Pik+1 Bi (t )
n

3.1. Triangular Bézier surface and lexicographic order
(5)
so that the curve
C k+1 (t ) =
3. PIA for triangular Bézier surfaces
n
n
(13)
where f is a s + 1 variables function, and
α = (α1 , α2 , . . . , αs ) ,
|α| =
s

αi ,
i =0
 
n
α
=
n!
α1 !α2 ! · · · αs !(n − α1 − · · · − αs )!
(14)
.
Then the s+1 variables Bernstein operator Bn is diagonalizable, and
the range of all the eigenvalues is the same as that of the univariate
Author's personal copy
J. Chen, G.-J. Wang / Computer-Aided Design 43 (2011) 889–895
Bernstein operator. That is, suppose


n!
1
n
n

Ω = λi =
i = 0, 1, . . . , n ,
(n − i)! ni 

(15)
then for any eigenvalue λ of Bn , we have λ ∈ Ωn .
Next, we discuss the case of s = 2. The Bernstein basis
functions
of degree n over a triangle domain T can be expressed
 n
as Bn,0,0 , Bnn−1,1,0 , Bnn−1,0,1 , . . . , Bn0,n,0 , . . . , Bn0,0,n in lexicographic
order. Letting the uniform parameters be
tin,j,k =

i
j
k
, ,

n n n
,
i + j + k = n; i, j, k = 0, 1, . . . , n.
(16)
We suppose that
,
,
,...,
,...,
is a
corresponding parameter sequence in lexicographic order. Given
the control points {Pi,j,k }i+j+k=n in R 3 , then the original triangular
surface

tnn,0,0

G 0 (u, v, w) =
tnn−1,1,0
tnn−1,0,1
t0n,n,0
t0n,0,n

Pi0,j,k Bni,j,k (u, v, w)
collocation matrix of the triangular Bernstein
basis functions
is the

Bnn,0,0 , Bnn−1,1,0 , Bnn−1,0,1 , . . . , Bn0,n,0 , . . . , Bn0,0,n at the parametric


values tnn,0,0 , tnn−1,1,0 , tnn−1,0,1 , . . . , t0n,n,0 , . . . , t0n,0,n .
Theorem 3.1. Let Bn =
Bn0,0,n
be the triangular Bernstein basis of degree n in lexicographic
order over a triangle domain T , then a triangular Bézier surface
constructed by these basis functions
 j has
 the PIA property with the
uniform parameters tin,j,k = ni , n , nk , i + j + k = n; i, j, k =
0, 1, . . . , n.
Proof. Let {λi,j,k }i+j+k=n be all (n + 1)(n + 2)/2 eigenvalues of the
collocation matrix B. The trivariate Bernstein operator is given by
Bn (f3 ) =
i+j+k=n
(18)
i + j + k = n,
(19)
Pi1,j,k Bni,j,k (u, v, w)
(20)
i+j+k=n
over a triangle domain T . Similarly, we can get the (S + 1)st
triangular surface G S (u, v, w) after the Sth iteration. Next letting
PiS,j+,k1
=
PiS,j,k
+
∆Si,j,k
,
i + j + k = n,
(21)
i + j + k = n,
G S +1 (u, v, w) =

T
υn−1,0,1 , . . . , υ0,0,n be its associated eigenvector. Now we will
prove that λ is also an eigenvalue of the trivariate Bernstein
f = f (u, v, w) =
over a triangle domain T is generated. If limS →∞ G (
tin,j,k

υi,j,k Bni,j,k (tnn,0,0 )



+j+k=n
 i

f (tnn,0,0 )

n
υi,j,k Bni,j,k (tnn−1,1,0 )


f (t
)
i+j+k=n
=
 · n· −· 1· ,·1·,0 


  ······



f (t0n,0,n )
υi,j,k Bni,j,k (t0n,0,n )

λυn,0,0
λυ

=  n−1,1,0  .
······
λυ0,0,n
) =
Pi0,j,k , i + j + k = n, we call that the initial surface G 0 (u, v, w) has
the PIA property,
or equally we call the triangular Bernstein

 basis
functions Bnn,0,0 , Bnn−1,1,0 , Bnn−1,0,1 , . . . , Bn0,n,0 , . . . , Bn0,0,n over a
triangle domain T have the PIA property.
It is obvious that
∆Sp,q,r Bnp,q,r (tin,j,k ),

. . . , Bn0,0,n , we have


f (tin,j,k )Bni,j,k (u, v, w) =
λυi,j,k Bni,j,k (u, v, w)
i+j+k=n
(23)
we can rewrite the above formulas in matrix form as
 S +1

 S
T
1
S +1 T
S
S
∆n,0,0 , ∆Sn+
,
−1,1,0 , . . . , ∆0,0,n = D ∆n,0,0 , ∆n−1,1,0 , . . . , ∆0,0,n
D = I − B; S = 0, 1, . . . .
(24)
where I is the ((n + 1)(n + 2)/2) × ((n + 1)(n + 2)/2) identity
matrix, and
Bnn,0,0 , Bnn−1,1,0 , . . . , Bn0,0,n
tnn,0,0 , tnn−1,1,0 , . . . , t0n,0,n


···
···
.
Bnn−1,1,0 (t0n,0,n )
···
···
Bn0,0,n (tnn,0,0 )
Bn0,0,n (tnn−1,1,0 )

..
Bn0,0,n
(
.
t0n,0,n
 (25)

)
(29)
This shows that λ is also an eigenvalue of the trivariate
Bernstein operator Bn (f3 ), and f is its associated eigenvector.
Contrarily, let λ be an eigenvalue of the trivariate Bernstein
operator Bn (f3 ), and let f (u, v, w) be its associated eigenvector. Now we will prove that λ is also an eigenvalue of the
collocation
matrix B of the triangular Bernstein
basis Bn =

 n
Bn,0,0 , Bnn−1,1,0 , Bnn−1,0,1 , . . . , Bn0,n,0 , . . . , Bn0,0,n with the uniform
parameters tin,j,k , and the corresponding eigenvector is

T
υ = f (tnn,0,0 ), f (tnn−1,1,0 ), . . . , f (t0n,0,n ) .
Bnn−1,1,0 (tnn,0,0 )
Bnn−1,1,0 (tnn−1,1,0 )
..
i+j+k=n
= λf (u, v, w).
i + j + k = n; S = 0, 1, . . . .
(28)
Then, multiplying at the right hand side of both sides of the above
expression by a row matrix Bnn,0,0 , Bnn−1,1,0 , Bnn−1,0,1 , . . . , Bn0,n,0 ,
p+q+r =n
 Bn (t n )
n,0,0 n,0,0
n
n
Bn,0,0 (tn−1,1,0 )

= 
..
.
Bnn,0,0 (t0n,0,n )
(27)
In fact, since λ is an eigenvalue of the matrix B and υ its
associated eigenvector we have the expression Bυ = λυ , that is
(22)
S
B := B
υi,j,k Bni,j,k (u, v, w).

PiS,j+,k1 Bni,j,k (u, v, w)


i+j+k=n
i+j+k=n
1
S
∆Si,+
j,k = ∆i,j,k −
(26)
i+j+k=n
so that the triangular surface

n
Let λ be any one among the set λi,j,k , and let υ = υn,0,0 , υn−1,1,0 ,

∆Si,j,k = Pi,j,k − G S (tin,j,k ),
n
operator associated to the eigenvector
we can get the second triangular surface

α1 !α2 !(n − α1 − α2 )!


α1 α2 n − α1 − α2
× uα1 v α2 w n−α1 −α2 f
, ,
.
n
for every control point Pi,j,k , and letting
G 1 (u, v, w) =
n!

0≤α1 +α2 ≤n
(17)
over a triangle domain T can be generated, where Pi0,j,k = Pi,j,k .
By computing the adjusting vector
Pi1,j,k = Pi0,j,k + ∆0i,j,k ,
Bnn,0,0 , Bnn−1,1,0 , Bnn−1,0,1 , . . . , Bn0,n,0 , . . . ,


i+j+k=n
∆0i,j,k = Pi,j,k − G 0 (tin,j,k ),
891
(30)
In fact, since λ is an eigenvalue of the trivariate Bernstein
operator Bn (f3 ) and f is the corresponding eigenvector, so

i+j+k=n
f (tin,j,k )Bni,j,k (u, v, w) = λ f (u, v, w).
(31)
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892
J. Chen, G.-J. Wang / Computer-Aided Design 43 (2011) 889–895
Then, evaluating the above expression at the parameter values
tnn,0,0 , tnn−1,1,0 , . . . , t0n,0,n , it follows that

f (tin,j,k )Bni,j,k (tpn,q,r ) = λ f (tpn,q,r ) p + q + r = n;
i+j+k=n
p, q, r = 0, 1, . . . , n.
(32)
We can rewrite the above formulas in matrix form as
f (tnn,0,0 )
f ( t n

 n−1,1,0 )

B
f (tnn,0,0 )
f (t n

 n−1,1,0 )


 = λ
···
f (t0n,0,n )

···
f (t0n,0,n )
.
(33)
This means that λ is also an eigenvalue of the collocation matrix B,
and υ is the corresponding eigenvector.
According to Lemmas 3.1 and 3.2, it is easy to know that
the range of all eigenvalues (including the multiple roots) of the
collocation matrix B is the same as that of the univariate Bernstein
operator. That is, for any eigenvalue λi,j,k of the collocation matrix
B, we have λi,j,k ∈ Ωn , where
Ωn =

λni =


 i = 0, 1, . . . , n .
(n − i)! ni 
n!
1
Fig. 4.1. Original degree 2 triangular Bézier surface.
(34)
Summarizing above all, we know that all the eigenvalues λi,j,k of
the collocation matrix B satisfy 0 < λi,j,k ≤ 1, i + j + k = n.
This result implies that the spectral radius of the matrix D = I − B
satisfies 0 ≤ ρ(D) = ρ(I − B) < 1, thus the above collocation
matrix B satisfies the convergence conditions
PIA, which is like

of
n
that of the univariate Bernstein basis Bni (t ) i=0 at the parametric values {t0 , t1 , . . . , tn }. So the triangular Bernstein basis, or the
triangular Bézier surface G 0 (u, v, w) has PIA property. This completes the proof. 
Example 3.2. Let B3 =
B33,0,0 , B32,1,0 , B32,0,1 , B31,2,0 , B31,1,1 , B31,0,2 ,
Fig. 4.2. After the first iteration.
,
,
,
be the triangular Bernstein basis of
degree 3 over a triangle domain T , then all the eigenvalues λi,j,k (i +
j + k = 3) (including the multiple roots) of the corresponding
collocation matrix B with the uniform parameters ti3,j,k can be
B30,3,0
B30,2,1
B30,1,2
B30,0,3

computed by Eq. (34) in the case of n = 3, and we have λi,j,k ∈ Ω3 ,
hence 0 < λi,j,k ≤ 1, i + j + k = 3, because they are the same
as the eigenvalues of the univariate Bernstein operator B̃3 . That is,
they are

1
3!

· 0 = 1,
λ3,0,0 =


(3 − 0)! 3



3!
1


λ2,1,0 = λ2,0,1 =
·
= 1,
(3 − 1)! 31
3!
1
2


λ1,2,0 = λ1,1,1 = λ1,0,2 =
· 2 = ,



3
−
2
3
3
(
)!


1
3!
2

λ
·
= .
0,3,0 = λ0,2,1 = λ0,1,2 = λ0,0,3 =
9
(3 − 3)! 33
(35)
4. Examples and error analysis
In this section, we give two examples to show the iterative
approximation results of the triangular Bézier surface sequences
of degrees 2 and 3 respectively after the different iteration levels.
We will see that with the increase in the number of iterations,
the surface sequences G S (u, v, w) approximate the original control
points quickly with the uniform parameters.
  j 
Taking the fitting error as error = maxi+j+k=n G S ni , n , nk

−Pi,j,k  in the L∞ norm, for the case of degree 3 surface sequences,
we list the fitting errors of the surface sequences after specific
Fig. 4.3. After the fifth iteration.
iteration levels in Table 4.1 and plot the error curves to show them
in Figs. 4.7 and 4.8.
Example 4.1. Given a degree 2 triangular Bézier surface, its control
points in lexicographic order are
{(0, 3, 0.3) , (−0.5, 2, 3) , (0.5, 2, 4) , (−1, 1.5, 1) , (0, 1.4, 3) ,
(1, 1.5, 0.5)} .
And the original degree 2 triangular Bézier surface, the surface after
the first iteration, the surface after the fifth iteration are illustrated
in Figs. 4.1–4.3
 respectively. All the eigenvalues of the collocation
matrix B are 1, 1, 1, 12 , 12 , 12 .
Author's personal copy
J. Chen, G.-J. Wang / Computer-Aided Design 43 (2011) 889–895
893
Fig. 4.4. Original degree 3 triangular Bézier surface.
Fig. 4.6. After the fifth iteration.
1.8
1.6
1.4
Error
1.2
1.0
0.8
0.6
0.4
0.2
0
0
Fig. 4.5. After the first iteration.
Example 4.2. Given a degree 3 triangular Bézier surface, its control
points in lexicographic order are
5
10
Iteration Level
15
20
Fig. 4.7. Error curve of degree 2.
{(6, 5, 2) , (5.2, 3, 4) , (6.5, 3, 4) , (4.5, 1.5, 5) , (6.1, 1, 5.2) ,
(7, 1.5, 3.5) , (4, 0.5, 2) , (5.5, 0.1, 4) , (6.5, 0.2, 3.5) ,
(7.2, 0.4, 2.5)} .
1.2
1.0
5. PIA algorithms of rational triangular Bézier surfaces of
degrees 2 and 3
0.8
Error
And the original degree 3 triangular Bézier surface, the surface after
the first iteration, the surface after the fifth iteration are illustrated
in Figs. 4.4–4.6
of the collocation

 respectively. All the eigenvalues
matrix B are 1, 1, 1, 32 , 32 , 23 , 92 , 29 , 29 , 92 .
0.6
0.4
Obviously, a rational triangular Bézier surface
R (u, v, w) =

rin,j,k (u, v, w)Ri,j,k ,
0.2
i+j+k=n
rin,j,k (u, v, w) =
Bn (u, v, w)ωi,j,k
 i,j,k n
,
Bi,j,k (u, v, w)ωi,j,k
0
i+j+k=n
u, v, w ≥ 0,
u + v + w = 1,
(36)
has the PIA property which is similar to that of the triangular Bézier
surface. In this section we consider the case of rational triangular
Bézier surfaces of degrees 2 and 3.
5
10
Iteration Level
15
20
Fig. 4.8. Error curve of degree 3.
First, let us start with a rational triangular Bézier surface of
degree 2. Supposed that the rational triangular Bernstein basis
Author's personal copy
894
J. Chen, G.-J. Wang / Computer-Aided Design 43 (2011) 889–895
r33,0,0 , r23,1,0 , r23,0,1 , r13,2,0 , r13,1,1 , r13,0,2 , r03,3,0 , r03,2,1 , r03,1,2 , r03,0,3


=


B33,0,0 , ωB32,1,0 , ωB32,0,1 , ωB31,2,0 , ωB31,1,1 , ωB31,0,2 , B30,3,0 , ωB30,2,1 , ωB30,1,2 , B30,0,3
B33,0,0 + ωB32,1,0 + ωB32,0,1 + ωB31,2,0 + ωB31,1,1 + ωB31,0,2 + B30,3,0 + ωB30,2,1 + ωB30,1,2 + B30,0,3
,
ω > 0.
Box I.
Table 4.1
Fitting errors of the surface sequences in Examples 4.1 and 4.2.
Iteration level
0th
1th
5th
10th
20th
Degree 2
Degree 3
1.8043e000
1.2475e000
9.0220e−001
5.8350e−001
5.6400e−002
1.1420e−001
1.8001e−003
3.1502e−002
1.7207e−006
2.6025e−003
functions of degree 2 in lexicographic order are given by
r22,0,0 , r12,1,0 , r12,0,1 , r02,2,0 , r02,1,1 , r02,0,2


=



B22,0,0 , ωB21,1,0 , ωB21,0,1 , B20,2,0 , ωB20,1,1 , B20,0,2
,
B22,0,0 + ωB21,1,0 + ωB21,0,1 + B20,2,0 + ωB20,1,1 + B20,0,2
ω > 0.
(37)
Then,
 taking
the uniform parameter sequence
j
tin,j,k = ni , n , nk , i + j + k = 2, we get the collocation matrix

M
r22,0,0 , r12,1,0 , . . . , r02,0,2
t22,0,0 , t12,1,0 , . . . , t02,0,2

1
1

 2 (1 + ω)

1


=  2 (1 + ω)

0



0
0
0
0
0
1
0
ω
1+ω
0
0
2 (1 + ω)
0
0
0
0
0
0
0
0
0
1
1
0
2 (1 + ω)
0
ω
1+ω
0



1


2 (1 + ω)  .

0


1

2 (1 + ω)
M
r33,0,0 , r23,1,0 , . . . , r03,0,3
t33,0,0 , t23,1,0 , . . . , t03,0,3


=
A
0
B
C
0
0
0
0
0






9 (1 + 2ω)  ,


0



1

3 (1 + 8ω)
1
ω
0
1 + 8ω
0
1
8
0
4ω
0
2ω
3 (1 + 2ω)
2ω
3 (1 + 2ω)
4ω
3 (1 + 2ω)
0
3 (1 + 2ω)
0
(41)
8
0
1


9 (1 + 2ω) 
.

8
9 (1 + 2ω)
1
(42)

4ω
3(1+2ω)
1

, nj , nk , i + j + k = 3, we get the collocation matrix

0
0
It is easy to compute the eigenvalues of the above collocation
matrix M are 1(with multiplicity 3), 3(12+ω2ω) (with multiplicity 3),

It is easy to compute the eigenvalues of the above collocation
ω
matrix M are 1(with multiplicity 3) and 1+ω
(with multiplicity 3)
respectively, the latter is strictly increasing for all ω > 0. That
is to say, in order to get the fastest convergence rate, we should
take ω as bigger as possible, i.e., ω = 2128 in single precision and
ω = 21024 in double precision. With these choices of the weight
ω
the eigenvalue 1+ω
is almost 1.
Analogously, we can assume that the rational triangular
Bernstein basis functions of degree 3 in lexicographic order are
given in Box I. Then, taking the uniform parameter sequence tin,j,k =
n
0
ω

 9 (1 + 2ω)
C =

1

9 (1 + 2ω)
(38)
i
0
0
1 + 8ω
0
0
0
0


ω
1+ω
0
1

 9 (1 + 2ω)



0

B=
8


 9 (1 + 2ω)

1

3 (1 + 8ω)
(with multiplicity 2) and 3(16+ω2ω) (with multiplicity 2), the
three last eigenvalues are strictly increasing for all ω > 0. That
is to say, in order to get the fastest convergence rate, we should
take ω as bigger as possible, i.e., ω = 2128 in single precision and
ω = 21024 in double precision. With these choices of the weight
the three last eigenvalues are almost 13 , 23 , 1 respectively.
Example 5.1. Considering the sphere given by x2 + y2 + z 2 = 1.
A sequence of 6 points is sampled from this surface and expressed
in lexicographic order as
{(−1, 0, 0) , (−0.25, −0.75, 0.61) , (−0.25, 0.75, 0.61) ,
(0.87, −0.5, 0) , (−0.79, 0, 0.61) , (0.87, 0.5, 0)} .
And the degree 2 rational triangular Bézier surfaces after the
second iteration with the different weights ω = 2 and ω = 3 are
illustrated in Figs. 5.1 and 5.2 respectively.
2

(39)
y2
Example 5.2. Considering the Ellipsoid given by 3x 2 + 22 + z 2 = 1.
A sequence of 6 points is sampled from this surface and expressed
in lexicographic order as
{(−3, 0, 0) , (−1.5, −1, 0.71) , (−1.5, 1, 0.71) , (0, −2, 0) ,
(0, 0, 1) , (0, 2, 0)} .
where

1
0
0
0
0
0
8
4ω
2ω


0
0
0
 9 (1 + 2ω) 3 (1 + 2ω)

3 (1 + 2ω)


4
ω
2
ω
8


0
0
0


 9 (1 + 2ω)
3 (1 + 2ω)
3 (1 + 2ω) 


1
2ω
4ω
A=
,
0
0
0


9
1
+
2
ω)
3
1
+
2
ω)
3
1
+
2
ω)
(
(
(




1
ω
ω
ω
2ω
ω


 3 (1 + 8ω)
1 + 8ω
1 + 8ω
1 + 8ω
1 + 8ω
1 + 8ω 


1
2ω
4ω
0
0
0
9 (1 + 2ω)
3 (1 + 2ω)
3 (1 + 2ω)

(40)
And the degree 2 rational triangular Bézier surfaces after the fifth
iteration with the different weights ω = 2 and ω = 3 are
illustrated in Figs. 5.3 and 5.4 respectively.
6. Briefly discussing about application in reverse engineering
In reverse engineering for scattered data points, the literature [13] introduced a method of surface reconstruction based on
hierarchical space decomposition, and in the literature [14] surface
reconstruction using octree representation was presented. Thus, in
Author's personal copy
J. Chen, G.-J. Wang / Computer-Aided Design 43 (2011) 889–895
895
Fig. 5.3. After the fifth iteration, ω = 2.
Fig. 5.1. After the second iteration, ω = 2.
Fig. 5.4. After the fifth iteration, ω = 3.
References
Fig. 5.2. After the second iteration, ω = 3.
order to apply our PIA algorithm for triangular Bézier surface to reverse engineering, if the un-organized cloud points are given, first
we can construct a triangular mesh with proper topology by the
similar method in [13] or [14]. Then it is easy to classify the data
n(n+1)
points so that the number of the data points is 2 in each group.
Finally, for the data points in each group, the fitting surface can be
derived by our PIA algorithm. The concrete method, example and
the smooth joining problem of multiple pieces of surfaces could be
discussed in future work in detail.
7. Conclusion
This paper derives the PIA property of triangular Bézier
surfaces with the uniform parameters, that is, given some
scattered data points to form an initial control mesh, the limit
surface can interpolate these original points by constructing an
iterative sequence of triangular Bézier surfaces. Moreover, the
PIA algorithms using rational triangular Bernstein basis of low
degree are also discussed. The PIA property of triangular Bézier
surfaces with the general parameters could be a future research
project.
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