Appl. Math. J. Chinese Univ.
2011, 26(1): 47-56
A new type of the generalized Bézier curves
CHEN Jie
WANG Guo-jin
Abstract. In this paper, we improve the generalized Bernstein basis functions introduced by
Han, et al. The new basis functions not only inherit the most properties of the classical Bernstein
basis functions, but also reserve the shape parameters that are similar to the shape parameters
of the generalized Bernstein basis functions. The degree elevation algorithm and the conversion formulae between the new basis functions and the classical Bernstein basis functions are
obtained. Also the new Q-Bézier curve and surface constructed by the new basis functions are
given and their properties are discussed.
§1
Introduction
In CAGD/CAD system, parametric representation of curves and surfaces and its shape
control are under wide consideration and research, and it is very important to choose the appropriate basis functions. Based on their good properties, the classical Bernstein basis functions
play a significant role in CAGD/CAD system. In recent years, the parametric representation
of curves and surfaces with shape parameters has received attention in [3,4].
Han et al. [4] introduced a new type of the generalized Bernstein basis functions associated
with the shape parameters and Bézier curves and surfaces which seems to be entirely new,
and these curves and surfaces have simpler form and clearer geometric meaning than rational
Bézier curves and surfaces. Note that the mathematical preciseness about the statement in [4]
should be doubted. It is well-known that the number of basis functions is n + 1 in the space of
polynomials of degree n, that is, the degree of the polynomials is lower than the dimension of the
corresponding polynomial space by one. But according to the definition of the basis functions
there, the degree and the number of so-called basis functions are the same. In fact, though
all functions forming this group are linear independent to each other, they don’t construct a
maximal linearly independent set. So it seems to be inappropriate to call them basis functions.
Received: 2009-12-29.
MR Subject Classification: 65D18.
Keywords: Shape parameter, shape control, degree elevation.
Digital Object Identifier(DOI): 10.1007/s11766-011-2453-8.
This work was supported by the National Natural Science Foundations of China (61070065, 60933007).
Corresponding author: wanggj@zju.edu.cn (G.-J.Wang).
48
Appl. Math. J. Chinese Univ.
Vol. 26, No. 1
Further more, according to that construction methods, the conversion between the classical
Bernstein basis functions and this group of functions can’t be executed. It brings a lot of
inconvenience in the design of geometric system.
In order to amend the defects in [4], in the present paper, we do some appropriate modification in the mathematical construction about the generalized Bernstein basis functions in [4], so
as to make the total number of the functions, i.e., the dimension of the corresponding polynomial space, is just higher than the degree of themselves by one. The curve constructed by the
new basis functions also has the good property of shape control, and the geometric meaning of
the shape parameters is clear. Moreover, it can be executed to convert the new basis functions
into the classical Bernstein basis functions and vice versa, so it can be used widely in geometric
shape design.
The present paper is organized as follows. First, we introduce the main idea and the
construction process of the new basis functions in Section 2, so that give a base to propose their
strict definition naturally. Some elementary algebraic properties of the basis functions and the
constructed curve are discussed in Section 3 and 4. Also the degree elevation algorithms and the
conversion formulae between the new basis functions and the classical Bernstein basis functions
are derived. In section 7, we give some examples about the shape control of the parametric
curve by the different shape parameters in the case of low degree.
§2
Issues raised and the idea of generalized Bernstein basis function
Han et al [4] introduced a new kind of the so-called generalized Bernstein basis functions
associated with shape parameters. The motivation is to introduce shape parameters which can
adjust the shape of the curve. It can be explained in detail by the following formulae:
n
n
r(t) =
i=0 Pi bi (t)
n
[ n2 ]
n
n
n+1−i i
=
t di − i=[ n ]+1 λi (1 − t)n+1−i ti di ,
i=1 λi (1 − t)
i=0 Pi Bi (t) +
for t ∈ [0, 1], where
2
di = Pi − Pi−1 .
For the sake of the similar effect of shape control, by taking into account the degree of the
basis functions, we do some appropriate modification as follows:
n
n
r(t) =
i=0 Pi bi (t)
n
[ n2 ]
n
n
n−i i
=
t di − i=[ n ]+1 λi (1 − t)n−i ti di ,
i=1 λi (1 − t)
i=0 Pi Bi (t) +
2
for t ∈ [0, 1], where the meaning of di is the same as above.
By merging the coefficients of the same control point, we can obtain a new kind of generalized
Bernstein basis functions which will be introduced in this paper.
Now we give an example in the case of low degree for n = 4 as
CHEN Jie, WANG Guo-Jin.
4
r(t) =
A new type of the generalized Bézier curves
49
Pi b4i (t)
i=0
4
=
Pi Bi4 (t) +
i=0
2
λi (1 − t)4−i ti di −
i=1
4
λi (1 − t)4−i ti di
i=3
P0 (B04 (t) − λ1 (1 − t3 )t) + P1 (B14 (t) + λ1 (1 − t)3 t − λ2 (1 − t)2 t2 )
=
+P2 (B24 (t) + λ2 (1 − t)2 t2 + λ3 (1 − t)t3 )
+P3 (B34 (t) − λ3 (1 − t)t3 + λ4 t4 ) + P4 (B44 (t) − λ4 t4 ).
By simple calculation, we obtain the generalized Bernstein basis functions of degree 4
⎧
4
3
⎪
⎪ b0 (t) = (1 − t) ((1 − t) − λ1 t),
⎪
⎪
⎪
⎪ b41 (t) = t(1 − t)2 (4(1 − t) + λ1 (1 − t) − λ2 t),
⎨
b42 (t) = t2 (1 − t)(6(1 − t) + λ2 (1 − t) + λ3 t),
⎪
⎪
⎪
b43 (t) = t3 (4(1 − t) − λ3 (1 − t) + λ4 t),
⎪
⎪
⎪
⎩ 4
b4 (t) = t4 (1 − λ4 ).
In next section, we will give the definition of the generalized Bernstein basis functions of
arbitrary degree, and some good geometric properties are also derived.
§3
Generalized Bernstein basis functions
Definition 3.1. For every nature number n (n ≥ 2) and n arbitrarily selected real values of
λi , i = 1, 2, . . . , n, we call polynomial functions
⎧
bn0 (t)
= (1 − t)n−1 ((1 − t) − λ1 t),
⎪
⎪
⎪
⎪
n
⎪ b (t)
= ti (1 − t)n−1−i ( ni (1 − t) + λi (1 − t) − λi+1 t),
i = 1, . . . , [ n2 ] − 1,
⎪
⎨ i
n
n
bn[ n ] (t) = t[ 2 ] (1 − t)n−1−[ 2 ] ( [ nn ] (1 − t) + λ[ n2 ] (1 − t) + λ[ n2 ]+1 t),
(1)
2
⎪
n 2
⎪
n
n
i
n−1−i
⎪
⎪
bi (t)
= t (1 − t)
( i (1 − t) − λi (1 − t) + λi+1 t),
i = [ 2 ] + 1, . . . , n − 1,
⎪
⎪
⎩ n
n
= t (1 − λn )
bn (t)
are the new generalized Bernstein basis functions of degree n associated with the shape parameters {λi }ni=1 , where
λi ∈
(− ni , 0] for i = 1, 2, . . . , [ n2 ] and λi ∈ [0, ni ) for i = [ n2 ] + 1, . . . , n
with [ n2 ] =
n
2,
n+1
2 ,
n is even,
n is odd.
It is obvious that, for λi = 0, i = 1, 2, . . . , n, the basis functions are degenerated as classical
Bernstein polynomials of degree n.
Theorem 3.1. The new generalized Bernstein basis functions of degree n, shown as in (1),
associated with the shape parameters {λi }ni=1 , have the following properties:
(a) nonnegativity: bni (t) ≥ 0 for i = 0, 1, . . . , n;
n n
(b) partition of unity:
i=0 bi (t) = 1;
50
Appl. Math. J. Chinese Univ.
(c) linear independently: if
Proof.
n
i=0
Vol. 26, No. 1
αi bni (t) = 0, then αi = 0 for i = 0, 1, . . . , n.
(a) By the range of λi for i = 1, 2, . . . , n in Definition 3.1, we have
(1 − t) − λ1 t ≥ 0,
n
(1 − t) + λi (1 − t) − λi+1 t ≥ 0,
i
n
(1 − t) + λ[ n2 ] (1 − t) + λ[ n2 ]+1 t ≥ 0,
[ n2] n
(1 − t) − λi (1 − t) + λi+1 t ≥ 0,
i
and
Thus it is obvious that
(b)
n
(c)
n
n
i=0 bi (t)
=
n
i=0
bni (t)
1 − λn ≥ 0.
≥ 0 for i = 0, 1, . . . , n.
n i
n−i
≡ 1.
i t (1 − t)
n
αi bni (t) = i=0 βi bni (t), where Bin (t) (i = 0, 1, . . . , n) are classical Bernstein basis
functions. It is easy to know that
⎧
⎪
⎪ β0 = α0 ,
⎪
⎪
⎪
αi ((n
⎪ β
i )+λi )−αi−1 λi
⎨
,
i = 1, 2, . . . , [ n2 ],
=
i
(ni)
n
αi (( i )−λi )+αi−1 λi
⎪
⎪ βi =
,
i = [ n2 ] + 1, . . . , n − 1,
⎪
⎪
(ni)
⎪
⎪
⎩ β
= α
λ + α (1 − λ ).
i=0
n
n−1 n
n
n
Because of the linear independence of Bernstein basis functions, it follows that βi = 0
(i = 0, 1, . . . , n) and αi = 0 (i = 0, 1, . . . , n).
§4
Construction of Q-Bézier curves
Definition 4.1. Given n + 1 control points Pi (i = 0, 1, . . . , n), we call the curve
n
Pi bni (t),
t ∈ [0, 1],
r(t) =
i=0
Q-Bézier curve with the shape parameters λi ∈ [−
for i = [ n2 ] + 1, . . . , n.
n
n
n
i , 0] for i = 1, 2, . . . , [ 2 ] and λi ∈ [0, i ]
From the definition, we obtain the following theorem.
Theorem 4.1. Q-Bézier curves have the following properties.
(a) Terminal properties:
r(0) = P0 ,
r(1) = Pn if λn = 0.
(2)
CHEN Jie, WANG Guo-Jin.
A new type of the generalized Bézier curves
51
(b) Geometric invariance:
r(t; {λi }ni=1 ; P0 + q, P1 + q, . . . , Pn + q) = r(t; {λi }ni=1 ; P0 , P1 , . . . , Pn ) + q,
r(t; {λi }ni=1 ; P0 ∗ T , P1 ∗ T , . . . , Pn ∗ T ) = r(t; {λi }ni=1 ; P0 , P1 , . . . , Pn ) ∗ T ,
where q is an arbitrary vector in R2 or R3 , and T is an arbitrary 2 × 2 or 3 × 3 matrix.
(c) Convex hull properties:
The entire Q-Bézier curve segment must lie inside its control polygon spanned by P0 , P1 ,
. . ., Pn .
§5
Degree elevation of Q-Bézier curves
When n ≥ 4, if the shape parameters {λi }ni=1 of bni (t) (i = 1, 2, . . . , n) satisfy the following
conditions
⎧
n
)+λi−1 )λi−1
((i−1
⎪
λ
=
,
i = 2, 3, . . . , [ n2 ] + 1,
⎪
i
n
⎪
(
)+λi−2
⎪
i−2
⎪
⎪
⎪
([ nn ])+λ[ n ] λ[ n ]
⎪
⎪
⎨ λ[ n2 ]+1 = − ( n2n )+λ2 n 2 ,
[ ]−1
2
[ 2 ]−1
n
+λ
n
(
)
⎪
[ n ]+1 λ[ n ]+1
[ ]+1
⎪
2
2
2
⎪
,
λ[ n2 ]+2 = −
⎪
⎪
([ nn2 ])+λ[ n2 ]
⎪
⎪
n
⎪
⎪
((i−1)−λi−1 )λi−1
⎩ λi
=
,
i = [ n2 ] + 3, . . . , n
n
(i−2
)−λi−2
where λ0 = 0, then the degree n Q-Bézier curve r(t) can be expressed as a Q-Bézier curve of
degree n + 1, that is,
n
n+1
r(t) =
bni (t)Pi =
bn+1
(t)Qi (n ≥ 4),
i
i=0
i=0
where λi (i = 1, 2, . . . , n) are the shape parameters of bni , and μi (i = 1, 2, . . . , n + 1) are the
shape parameters of bn+1
.
i
Then the first and the last control points of the curve after degree elevation coincide with
the first and the last control points of the original curve respectively, and the rest each control
point can be expressed as a convex linear combination by two neighboring control points of the
original curve. That is, the control polygon of the curve after degree elevation can be produced
by the control polygon of the original curve using a corner cutting algorithm.
When n is odd, we
⎧
⎪
Q0
⎪
⎪
⎪
⎪
⎪
Qi
⎪
⎪
⎪
⎪
⎨
Q[ n+1 ]+1
2
⎪
⎪
⎪
⎪
⎪
⎪
Qi
⎪
⎪
⎪
⎪
⎩
Qn+1
have
= P0 ,
n
)+λi−1
(i−1
(ni)+λi
=
Pi−1 + n+1
Pi ,
i = 1, . . . , [ n+1
2 ],
+μ
(n+1
)
(
i
i
i )+μi
n
n
([ n2 ])+λ[ n2 ]
([ n2 ]+1)−λ[ n2 ]+1
=
P[ n2 ] + n+1
Pn ,
n+1
−μ
([ n+1
)
(
)−μ[ n+1 ]+1 [ 2 ]+1
n+1
n+1
]+1
]+1
]+1
[
[
2
2
2
2
n
(i−1
(ni)−λi
)−λi−1
=
Pi−1 + n+1
Pi ,
i = [ n+1
2 ] + 1, . . . , n,
−μ
(n+1
)
(
i
i
i )−μi
= Pn .
When n is even, we have
52
Appl. Math. J. Chinese Univ.
Vol. 26, No. 1
⎧
⎪
Q0
= P0 ,
⎪
⎪
n
⎪
)+λi−1
(i−1
(ni)+λi
⎪
⎪
Qi
=
Pi−1 + n+1
Pi ,
i = 1, . . . , [ n+1
⎪
n+1
⎪
2 ] − 1,
+μ
(
)
(
i
⎪
i
i )+μi
⎪
n
n
⎨
([ n2 ])+λ[ n2 ]
([ n2 ]+1)−λ[ n2 ]+1
P[ n2 ] + n+1
P[ n2 ]+1 ,
Q[ n+1 ]+1 =
n+1
2
+μ
(
)
(
n+1
n+1
n+1 )+μ n+1
⎪
[
[
]
]
]
]
[
[
⎪
2
2
2
2
⎪
n
⎪
(i−1
(ni)−λi
)−λi−1
⎪
⎪
=
P
+
P
,
i = [ n+1
Q
⎪
i
i−1
i
n+1
n+1
2 ] + 1, . . . , n,
⎪
( i )−μi
( i )−μi
⎪
⎪
⎩
= Pn .
Qn+1
The relation between the shape parameters {λi }ni of the original curve and the shape parameters {μi }n+1
i=1 of the curve after degree elevation can be shown as follows.
When n is odd,
⎧
⎪
μ1
⎪
⎪
⎪
⎪
⎪
μ
⎪
⎨ i
μ[ n+1 ]+1
2
⎪
⎪
⎪
μ
i
⎪
⎪
⎪
⎪
⎩ μn+1
When n is even,
⎧
⎪
μ1
⎪
⎪
⎪
⎪
⎪
μ
⎪
⎨ i
μ[ n+1 ]
2
⎪
⎪
⎪
μi
⎪
⎪
⎪
⎪
⎩ μ
n+1
§6
6.1
=
=
=
=
=
=
=
=
=
=
λ1 ,
λi−1 + λi ,
i = 2, . . . , [ n+1
2 ],
n
n
−λ[ 2 ] + λ[ 2 ]+1 ,
λi−1 + λi ,
i = [ n2 ] + 2, . . . , n,
λn (1−λn )
λn + n −λ .
(n−1) n−1
λ1 ,
λi−1 + λi ,
i = 2, . . . , [ n+1
2 ] − 1,
λ[ n2 ] − λ[ n2 ]+1 ,
λi−1 + λi ,
i = [ n2 ] + 2, . . . , n,
n)
λn + λnn (1−λ
.
(n−1)−λn−1
Conversion formulae between generalized and
classical Bernstein basis
Conversion formulae from classical Bernstein basis to generalized
Bernstein basis
⎧
⎪
bn0
= B0n − λn1 B1n ,
⎪
⎪
⎪
⎪
(ni)+λi n
⎪
⎪
bni
Bi − λi+1
Bn ,
=
i = 1, . . . , [ n2 ] − 1,
n
n
⎪
⎪
(
)
(i+1
) i+1
i
⎪
⎨
n
([ n2 ])+λ[ n2 ] n
λ[ n ]+1
B n + 2n B[nn ]+1 ,
bn[ n ] =
n
([ n2 ]) ([ n2 ]+1) 2
([ n2 ])
2
⎪
⎪
⎪
n
⎪
−λi n
)
(
⎪
i
n
n
⎪
Bi + λi+1
Bi+1
bi
=
,
i = [ n2 ] + 1, . . . , n − 1,
⎪
n
n
⎪
(
(
i)
i+1)
⎪
⎪
⎩ bn
= (1 − λn )Bnn .
n
Let b = (b0 , b1 , . . . , bn )T , B = (B0 , B1 , . . . , Bn )T , and Mn+1 be the conversion matrix. The
above formulae can be expressed as
b = Mn+1 B.
CHEN Jie, WANG Guo-Jin.
53
A new type of the generalized Bézier curves
For instance, we give the conversion matrixes when n = 2 and n = 3.
For n = 2,
⎛
For n = 3,
⎛
⎜
⎜
M3 = ⎜
⎝
6.2
⎧
⎪
B0n
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
Bin
⎪
⎪
⎪
⎪
⎪
⎪
⎨
λ1 +2
2
0
− λ31
1
0
0
0
⎞
0
⎟
λ2 ⎠ .
1 − λ2
− λ21
1
⎜
M2 = ⎝ 0
0
λ1 +3
3
0
− λ32
0
0
0
λ2 +3
3
0
0
λ3
1 − λ3
⎞
⎟
⎟
⎟.
⎠
Conversion formulae from generalized Bernstein basis to classical
Bernstein basis
=
=
⎪
⎪
⎪
⎪
B[nn ] =
⎪
⎪
2
⎪
⎪
⎪
⎪
⎪
⎪
⎪
Bin =
⎪
⎪
⎪
⎩ n
Bi =
bn0 +
[ n2 ]
j
k=1 λk
n
k +λk
k=1
j=1 j
(( )
)
bnj +
n
j
n
j=[ n
2 ]+1
(−1)j−[ 2 ] [ n2 ]
n
k
k=1 (( )+λk )
k=1 λk
j
k=[ n ]+1
2
((nk)−λk )
bnj ,
(ni) jk=i+1 λk n
(ni) n [ n2 ]
b
b
+
j
n
j=i+1
+λk ) j
( )+λi i
k=i (( k )
n j
n
λk
( )
bnj ,
+ j=[ n ]+1 (−1)j−i+1 [ n2 ] n i k=i+1
i = 1, . . . , [ n2 ] − 1,
j
n
2
+λ
−λ
((
)
)
((
)
)
k
k
n
k
k
k=i
k=[ ]+1
2
([ nn2 ]) jk=[ n2 ]+1 λk
([ nn2 ])
n
j−[ n
]
n
2 bn ,
b
+
−1
n
n
j=[ 2 ]+1
([ nn2 ])+λ[ n2 ] [ 2 ]
([ nn2 ])+λ[ n2 ] jk=[ n2 ]+1 ((nk)−λk ) j
(ni) jk=i+1 λk n
(ni) n n
j−i bj ,
b
+
(−1)
i = [ n2 ] + 1, . . . , n − 1,
j
n
n
i
j=i+1
( i )n−λi
k=i (( k )−λk )
bn
1−λn .
n
i
Let b = (b0 , b1 , . . . , bn )T , B = (B0 , B1 , . . . , Bn )T , and Fn+1 be the conversion matrix. The
above formulae can be expressed as
B = Fn+1 b.
For instance, we give the conversion matrixes when n = 2 and n = 3.
For n = 2,
⎛
1
⎜
F2 = ⎝ 0
0
For n = 3,
⎛
1
⎜
⎜ 0
F3 = ⎜
⎜ 0
⎝
0
λ1
3+λ1
3
3+λ1
0
0
λ1
2+λ1
2
2+λ1
λ2
− (2+λλ11)(1−λ
2)
⎟
2
.
− (2+λ12λ
)(1−λ2 ) ⎠
0
λ1 λ2
(3+λ1 )(3+λ2 )
3λ2
(3+λ1 )(3+λ2 )
3
3+λ2
0
⎞
1
1−λ2
λ1 λ2 λ3
− (3+λ1 )(3+λ
2 )(1−λ3 )
⎞
⎟
3λ2 λ3
⎟
− (3+λ1 )(3+λ
2 )(1−λ3 ) ⎟
⎟.
3
− (3+λ23λ
⎠
)(1−λ3 )
1
1−λ3
54
Appl. Math. J. Chinese Univ.
§7
Vol. 26, No. 1
Shape control of the Q-Bézier curves
We rewrite (2) as
r(t) =
n
Pi bni (t)
i=0
where Bin (t) =
=
n
i=0
n
Pi Bin (t)
+
[2]
λi (1 − t)n−i ti di −
i=1
n
λi (1 − t)n−i ti di ,
t ∈ [0, 1],
i=[ n
2 ]+1
n
n−i i
t are classical Bernstein functions of degree n and di = Pi − Pi−1 .
i (1 − t)
It should be noted that, the shape parameters λi only affect the curve on the corresponding
control edges di . Specifically, as λi (i = 1, . . . , [ n2 ]) increases, the curve moves in the direction
of di ; as λi (i = 1, . . . , [ n2 ]) decreases, the curve moves in the opposite direction of di . The
same effects on the edge di (i = [ n2 ] + 1, . . . , n) can play by the corresponding shape parameters
λi (i = [ n2 ] + 1, . . . , n).
The following figures show the demonstration of shape control of the curve by different shape
parameters.
Figure 1: The effect of the shape of cubic QBézier curves by λ1 .
Figure 2: The effect of the shape of cubic QBézier curves by λ2 .
Figure 3: The effect of the shape of cubic Q-Bézier curves by λ1 , λ2 and λ3 .
In Fig.1, λ2 = λ3 = 0 (solid lines), λ1 = −1 (dashed lines), and λ1 = 2 (dashdotted lines).
In Fig.2, λ1 = λ3 = 0 (solid lines), λ2 = 1 (dashed lines), and λ2 = 2 (dashdotted lines).
In Fig.3, λ1 = −1, λ2 = −2, λ3 = 0 (solid lines), λ1 = 1, λ2 = −1, λ3 = 0 (dashed lines),
and λ1 = 2, λ2 = 0, λ3 = 0 (dashdotted lines).
In Fig.4, λ2 = λ3 = λ4 = 0 (solid lines), λ1 = −1 (dashed lines), and λ1 = 2 (dashdotted
lines).
CHEN Jie, WANG Guo-Jin.
A new type of the generalized Bézier curves
55
Figure 4: The effect of the shape of quartic
Q-Bézier curves by λ1 .
Figure 5: The effect of the shape of quartic
Q-Bézier curves by λ2 .
Figure 6: The effect of the shape of quartic
Q-Bézier curves by λ3 .
Figure 7: The effect of the shape of quartic
Q-Bézier curves by λ1 , λ2 , λ3 , λ4 .
In Fig.5, λ1 = λ3 = λ4 = 0 (solid lines), λ2 = −1 (dashed lines), and λ2 = 2 (dashdotted
lines).
In Fig.6, λ1 = λ2 = λ4 = 0 (solid lines), λ3 = −1 (dashed lines), and λ3 = 2 (dashdotted
lines).
In Fig.7, λ1 = λ2 = λ3 = −1, λ4 = 0 (solid lines), λ1 = 0, λ2 = 1, λ3 = 2, λ4 = 0 (dashed
lines), and λ1 = 1, λ2 = 2, λ3 = 2, λ4 = 0 (dashdotted lines).
§8
Q-Bézier surface
n
Given the generalized Bernstein basis functions bm
i (t), bj (t) of degree m and n respectively,
using tensor product, we can construct generalized Bézier surface.
m n
n
Pi,j bm
0 ≤ u, v ≤ 1.
S(u, v) =
i (u)bj (v),
i=0 j=0
Pi,j is the control points in 3D space. Tensor product of Q-Bézier curves has properties similar
to those of tensor product of Bézier curves.
§9
Conclusion
This paper presents a new type of the generalized Bernstein basis functions by modifying
and improving a group of linear independent functions (not the basis) in [4]. It inherits the
properties of the original functions, for example, the shape controlled by the shape parameters.
56
Appl. Math. J. Chinese Univ.
Vol. 26, No. 1
And Q-Bézier curve constructed by the new basis functions possesses the most properties of
Bézier curve. Moreover, it can be executed to convert the generalized Bernstein basis functions
into the classical Bernstein basis functions of same degree, or vice versa. This provides more
convenience in the application of CAD/CAM system. In order to play a more important role in
application of Q-Bézier curve, to research some fast evaluation algorithms and degree reduction
approximation approach is needed in the future work.
References
[1] Q Duan, L Wang, E H Twizell. A new weighted rational cubic interpolation and its approximation,
Appl Math Comput, 2005, 168(2): 990-1003.
[2] G Farin. Class a Bézier curves, Comput Aided Geomet, 2006, 23(7): 573-581.
[3] X Han. Cubic trigonometric polynomial curves with a shape parameter, Comput Aided Geom
Design, 2004, 21(6): 535-548.
[4] X A Han, Y C Ma, X L Huang. A novel generalization of Bézier curve and surface, J Comput
Appl Math, 2008, 217(1): 180-193.
[5] H Oru, G M Phillips. q-Bernstein polynomials and Bézier curves, J Comput Appl Math, 2003,
151(1): 1-12.
[6] J Sánchez-Reyes. p-Bézier curves, spirals, and sectrix curves, Comput Aided Geomet Design,
2002, 19(6): 445-464.
[7] R Winkel. Generalized Bernstein polynomials and Bézier curves: an application of umbral calculus to computer aided geometric design, Adv Appl Math, 2004, 27(1): 51-81.
Department of Mathematics, Zhejiang University, Hangzhou 310027, China
Email: wanggj@zju.edu.cn