Appl. Math. J. Chinese Univ. Ser. B
2006, 21(3): 350-356
A CLASS OF INEQUALITIES ABOUT BERNSTEIN
POLYNOMIAL AND THEIR APPLICATIONS
Deng Chongyang Yang Xunnian
Abstract. In this paper a class of new inequalities about Bernstein polynomial is established.
With these inequalities, the estimation of heights, the derivative bounds of Bézier curves and
rational Bézier curves can be improved greatly.
§1 Introduction
Bernstein polynomial plays very important roles in the fields of function approximation as
well as computer aided geometric design (CAGD). For function approximation the well-known
Weierstrass theorem states that any continuous function in a close interval can be uniformly
approximated by polynomials. While in the field of CAGD, the well-known Bézier curves and
surfaces are also established based on Bernstein basis functions. In order to comprise the
free-form curves and analytical curves like conics simultaneously, rational Bézier curves are
introduced in CAGD.
Without loss of generality, we can assume that a Bézier curve is defined by
P (t) =
n
X
Bin (t)Pi , 0 ≤ t ≤ 1,
(1)
i=0
a rational Bézier curve is defined as
Pn
n
P (t)
i=0 Bi (t)ωi Ri
= P
R(t) =
, 0 ≤ t ≤ 1,
n
n
ω(t)
i=0 Bi (t)ωi
(2)
and a Bernstein polynomial can be defined as
B n (t) =
Ã
n
X
Bin (t)di , 0 ≤ t ≤ 1,
(3)
i=0
!
n
where Bin (t) =
(1−t)n−i ti (i = 0, 1, . . . , n), and di (i = 0, 1, . . . , n) is a set of non-negative
i
real numbers, i.e. di ≥ 0.
Received:2005-01-08.
MR Subject Classification:65D17,26D05.
Keywords: Bernstein polynomial, inequality, Bézier curve, rational Bézier curve, height, derivative bound.
Supported by the National Natural Science Foundation of China (60303015, 60333010).
Deng Chongyang, et al.
A CLASS OF INEQUALITIES ABOUT BERNSTEIN POLYNOMIAL... 351
When processing some problems of Bézier curves and rational Bézier curves, we are often
required to estimate the derivative bounds, the heights of Bézier curves or rational Bézier
n
P
Bin (t)di (di ≥ 0) . In former
curves, which can be formulated as computing the bounds for
i=0
literatures [1-4], authors estimated the bounds by the weighting property of Bernstein basis,
i.e.
n
X
"
Bin (t)di
≤
i=0
n
X
"
Bin (t)di
i=0
≥
n
X
#
Bin (t)
i=0
n
X
max di = max di ,
(4)
min di = min di .
(5)
0≤i≤n
0≤i≤n
#
Bin (t)
i=0
0≤i≤n
0≤i≤n
In this paper we show that the heights, the derivative bounds of Bézier curves and rational
Bézier curves can be improved by establishing a set of new inequalities of Bernstein polynomial.
The paper is organized as follows. In §2 we present the inequalities. As applications of these
inequalities, in §3 we improve the estimation of heights, the derivative bounds of Bézier curves
and rational Bézier curves. Finally we give the conclusion in §4.
§2 A class of inequalities about Bernstein polynomial
In this paper, we assume n, i, f, s ∈ N, 0 ≤ i, f, s ≤ n. By classifying the distribution of di
into three cases, we discuss the bound computation for each case independently.
Pn
(1)When max di = d0 (or dn ), by inequality (4) we have i=0 Bin (t)di ≤ d0 (or dn ), and
0≤i≤n
the function reaches the height at t = 0(or t = 1). In this case, we use the bound by inequality
(4) as height computation too.
(2)If df = max di 6= max{d0 , dn }, and df > max di , we have
0≤i≤n
0≤i≤n,i6=f
Theorem 2.1. If n ≥ 2, df = max di , ds =
0≤i≤n
n
X
max
0≤i≤n,i6=f
di , then
Bin (t)di ≤ ds + Bfn (f /n)(df − ds ).
(6)
i=0
Proof. It is clear that Bfn (t) ≤ Bfn (f /n). Therefore
n
X
Bin (t)di
≤ Bfn (t)df +
i=0
n
X
Bin (t)ds
i=0,i6=f
≤ ds + Bfn (f /n)(df − ds )


n
 X

£ n
¤
£
¤
+ Bf (t) − Bfn (f /n) df +
Bin (t) − 1 − Bfn (f /n) ds


i=0,i6=f
≤ ds +
Bfn (f /n)(df
− ds ).
(3)If there are at least two numbers of di (i = 1, 2, . . . , n − 1) equal to the maximum one,
the heights defined by inequalities (4) and (6) will be the same. In this case we can improve
inequality (4) as
352
Appl. Math. J. Chinese Univ. Ser. B
Vol. 21, No. 3
Theorem 2.2. If n ≥ 3, k ∈ N, k ≥ 2, and there is a set F = {f1 , f2 , . . . , fk } satisfying
df = max di for every f ∈ F, ds = max di , df > ds , then
i≤i≤n−1
0≤i≤n,i∈F
/
n
X
³
´
Bin (t)di ≤ ds + 1 − 2−(n−1) (df − ds ).
(7)
i=0
Proof. It is clear that (1 − t)n + tn ≥ 2−(n−1) . So it is easy to see that
X
Bin (t) ≤ 1 − [B0n (t) + Bnn (t)] ≤ 1 − 2−(n−1) .
i∈F
Therefore
n
X
Bin (t)di
≤
i=0
X
X
Bin (t)df +
i∈F
0≤i≤n,i∈F
/
³
ds
−(n−1)

"
´
+ 1−2

+
Bin (t)ds ≤
(df − ds ) +

X
Bin (t)
³
−(n−1)
´
− 1−2
#
df
i∈F
³
´
Bin (t) − 2−(n−1)  ds ≤ ds + 1 − 2−(n−1) (df − ds ).
X
0≤i≤n,i∈F
/
That is to say, if there exists max di > max(d0 , dn ), we can get a result better than in0≤i≤n
equality (4). This is a similarity for Theorem 2.3 and between the applications of Theorem 2.1
and Theorem 2.3, so we don’t explain it any more.
Similarly, in order to obtain tighter lower bound than inequality (5), we have
Theorem 2.3. If n ≥ 2, df = min di , ds = min di , then
0≤i≤n
n
X
0≤i≤n,i6=f
Bin (t)di ≥ ds + Bfn (f /n)(df − ds ).
(8)
i=0
When estimating the heights of Béizer curves, the coefficients of B0n (t), Bnn (t) vanish, in this
case we have
Theorem 2.4. If n ≥ 3, and df = max di , ds =
max
di , then
1≤i≤n−1
n−1
X
Bin (t)di
³
−(n−1)
≤ 1−2
i=1
1≤i≤n−1,i6=f
´
ds +
Bfn
µ ¶
f
(df − ds ).
n
Proof. From the above obvious inequality
(1 − t)n + tn ≥ 2−(n−1) ; Bfn (t) ≤ Bfn (f /n),
(9)
A CLASS OF INEQUALITIES ABOUT BERNSTEIN POLYNOMIAL... 353
Deng Chongyang, et al.
we have
n−1
X
Bin (t)di
i=1
≤ Bfn (t)df +
X
Bin (t)ds
1≤i≤n−1,i6=f
µ ¶
·
µ ¶¸
´
f
f
1 − 2−(n−1) ds + Bfn
(df − ds ) + Bfn (t) − Bfn
df
n
n


·
µ ¶¸

X
f
n
−(n−1)
n
+
Bi (t) − 1 − 2
− Bf
ds

n 
1≤i≤n−1,i6=f
"n−1
#
µ ¶
³
´
³
´
X
f
−(n−1)
n
n
−(n−1)
≤
1−2
ds + Bf
(df − ds ) +
Bi (t) − 1 − 2
ds
n
i=1
µ ¶
³
´
f
−(n−1)
n
≤
1−2
ds + Bf
(df − ds ).
n
³
≤
§3 Applications
¯ n
¯
¯P n
¯
¯
Bi (t)di ¯¯, the inequalities we
In the cases of estimating the range of expressions like ¯
i=0
established are always available. We show here two examples.
3.1 Improve the estimation of derivative bounds of Bézier curves and rational Bézier
curves
Theorem 3.1.
Let P (t) be a Bézier curve defined by (1), df = max kPi Pi+1 k , ds =
0≤i≤n−1
max
0≤i≤n−1,i6=f
kPi Pi+1 k , then
£
¤
kP 0 (t)k ≤ n ds + Bfn (f /n)(df − ds ) .
(10)
Proof. By the expression[5]
P 0 (t) = n
n−1
X
Bin−1 (t)(Pi+1 − Pi )
i=0
and Theorem 2.1 we obtain inequality (10) immediately.
Remark. Because
P (k) (t) =
n−k
X
n!
B n−k (t)∆k Pi , k = 1, 2, . . . , n,
(n − k)! i=0 i
(11)
where ∆ is a forward difference operator, i.e. ∆Pi = Pi+1 −Pi , for the higher derivative bounds,
we have estimation like (10). Especially, we can improve priori formula of subdivision depths
for Bézier curves in [4].
For derivative bounds of rational Bézier curves Floater gave estimation in [1]

2
°
°
max ωi
° dR(t) °
°
°
 0≤i≤n  max kRi Ri+1 k .
° dt ° ≤ n
0≤i≤n−1
min ωi
0≤i≤n
(12)
354
Appl. Math. J. Chinese Univ. Ser. B
Vol. 21, No. 3
We improve (12) as follows.
Theorem 3.2. Let R(t) be a rational Bézier curve defined by (2),ωi > 0, ωf = min ωi , ωs =
0≤i≤n
min
0≤i≤n,i6=f
ωi , then
!2
Ã
°
°
max ωi
° dR(t) °
0≤i≤n
°
°
max kRi Ri+1 k .
° dt ° ≤ n ωs + B n (f /n)(ωf − ωs )
0≤i≤n−1
f
(13)
Proof. From [2] we know that
à max ω !2
°
°
i
° dR(t) °
° ≤ n 0≤i≤n
°
max kRi Ri+1 k .
° dt °
0≤i≤n−1
ω(t)
(14)
.
By Theorem 2.3 we have
ω(t) ≥ ωs + Bfn (f /n)(ωf − ωs ).
Thus we complete the proof.
3.2 Estimate the heights of Béizer curves and rational Béizer curves
Theorem 3.3. Let P (t) be a Bézier curve defined by (1), write di := d(Pi , l(P0 , Pn )), df =
max di , ds =
max
di , and hn (t) is the height of P (t), then
1≤i≤n−1
1≤i≤n−1,i6=f
³
hn (t) ≤ 1 − 2
−(n−1)
´
ds +
Bfn
µ ¶
f
(df − ds ).
n
(15)
Proof. It is easily to see that
hn (t) ≤
n−1
X
Bin (t)di .
i=1
By Theorem 2.4 we have (15) immediately.
In the cases of rational Bézier curves, we have
Theorem 3.4. Let R(t) be a rational Bézier curve defined by (2), write di := d(Ri , l(R0 , Rn )),
and hn (t) is the height of R(t), then
Ã
!−1
1
max di ,
(16)
hn (t) ≤ 1 + n−1 n
1≤i≤n−1
2
Bf (f /n)(ωf∗ − ωs∗ ) + (2n−1 − 1)ωs∗
r³ ´ ³ ´
where ωi > 0, ωi∗ =
n
1
ωn
i
Proof. Because[6]
hn (t) ≤
n−1
X
1
ω0
,
Bin (t)ωi∗ di
i=1
n−i
n
X
ωi , i = 0, 1, . . . , n.ωf∗ =
Bin (t)ωi∗
≤
i=0
" n−1
X
i=1
max ωi∗ , ωs∗ =
1≤i≤n−1
,
Bin (t)ωi∗
n
X
i=0
i=0
1≤i≤n−1
ωi∗ .
#
Bin (t)ωi∗
by the inequality (1 − t)n + tn ≤ 21−n and Theorem 2.4 we have
" n−1
, n
#
X
X
n
∗
n
∗
Bi (t)ωi
Bi (t)ωi
max di =
i=1
max
1≤i≤n−1,i6=f
max di ,
1≤i≤n−1
A CLASS OF INEQUALITIES ABOUT BERNSTEIN POLYNOMIAL... 355
Deng Chongyang, et al.
(
n
n
1 + [(1 − t) + t ]/
)−1
n−1
X
Bin (t)ωi∗
max di ≤
1≤i≤n−1
i=1
Ã
1
1 + n−1 n
∗
2
Bf (f /n)(ωf − ωs∗ ) + (2n−1 − 1)ωs∗
!−1
max di .
1≤i≤n−1
Remark 1. It is obvious that inequality (15) is better than[3]
³
´
hn (t) ≤ 1 − 2−(n−1)
max di .
(17)
1≤i≤n−1
Examples. We present two examples for subdivision of planar Bézier curves. The control
points for these two Bézier curves are generated randomly, and they are listed as follows.
C1 (t) : (1.01, 3.99)(1.95, 1.21)(7.0, 5.58)(7.29, 2.59)
C2 (t) : (2.49, 3.4)(2.84, 1.06)(6.4, 1.84)(1.89, 2.87)(5.37, 3.81)
We tessellate the curves into piecewise linear segments with tolerance ε = 0.01, 0.001, 0.0001.
The numbers of segments computed by different formulae are presented in Table 1 and Table
2.
Table 1 Subdivision segments of C1 (t) computed by different height estimations
ε = 0.01
ε = 0.001
ε = 0.0001
Convex hull property
26
90
228
inequality(17)
25
80
208
inequality(15)
22
79
207
Table 2 Subdivision segments of C2 (t) computed by different height estimations
ε = 0.01
ε = 0.001
ε = 0.0001
Convex hull property
28
84
243
inequality(17)
26
78
233
inequality(15)
23
68
229
Remark 2. Since
ωi∗
r³
=
n
1
ωn
´i ³
1
ω0
´n−i
ωi ≤
ωi
min(ω0 ,ωn ) ,
we have
2n−1 Bfn (f /n)(ωf∗ − ωs∗ ) + (2n−1 − 1)ωs∗
≤
2n−1 Bfn (f /n)ωf + (2n−1 − 1 − 2n−1 Bfn (f /n))ωs
min(ω0 , ωn )
max(ω0 , ωn ) max ωi
1≤i≤n−1
≤
(2n−1 − 1).
ω0 ωn
356
Appl. Math. J. Chinese Univ. Ser. B
Vol. 21, No. 3
Therefore we can see that
Ã
!−1
1
1 + n−1 n
2
Bf (f /n)(ωf∗ − ωs∗ ) + (2n−1 − 1)ωs∗

−1
max(ω0 , ωn ) max ωi
1≤i≤n−1
≤ 1 − 1 +
(2n−1 − 1) .
ω0 ωn
So inequality (16) is better than[3]


−1 
max(ω0 , ωn ) max ωi
1≤i≤n−1


hn (t) ≤ 1 − 1 +
(2n−1 − 1)  max di .
1≤i≤n−1
ω0 ωn
(18)
§4 Conclusion
In this paper we have established a class of new inequalities about Bernstein polynomial,
and the bound estimation of Bézier curves and rational Bézier curves has been improved by
these inequalities. Under the condition that max {di } is larger than d0 and dn , we can get
0≤i≤n
better results than some previous results of derivative bounds. As far as the heights of Bézier
curves or rational Bézier curves are concerned, we can achieve better estimation, too.
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161-174.
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Dept. of math., Zhejiang Univ., Hangzhou 310027, China
Institute of Computer Images and Graphics, Zhejiang Univ., Hangzhou 310027, China.