PERGAMON
Applied Mathematics Letters 0 (2004) 1–0
www.elsevier.com/locate/aml
The Proof of Hermann’s Conjecture
Renjiang Zhang
Department of Mathematics, Zhejiang University
Hangzhou, 310027, P.R. China
and
State Key Lab of CAD and CG, Zhejiang University
Hangzhou, 310027, P.R. China
and
China Institute of Metrology, Zhejiang
Hangzhou, 310034, P.R. China
renjiang@mail.hz.zj.cn
Guojin Wang
Department of Mathematics, Zhejiang University
Hangzhou, 310027, P.R. China
and
State Key Lab of CAD and CG, Zhejiang University
Hangzhou, 310027, P.R. China
amawgj@mail.hz.zj.cn
(Received revised and accepted September 2003)
Communicated by R. P. Agarwal
Abstract—This note proves Thomas Hermann’s conjecture on the comparison between two
boundaries of the derivatives of rational cubic Bézier curves. The result is valuable for computer
c 2004 Elsevier Science Ltd. All rights reserved.
aided geometric design. °
Keywords—Computer aided geometric design, Rational Bézier curve, Derivatives, Boundary.
In the geometric design, estimating the boundary of the derivatives of rational Bézier curves is
an important subject [1,2]. In [3], Hermann has investigated the boundaries of the derivatives of
rational Bézier curves
n
P
Ci ωi bn,i (t)
Q(t) = i=0
,
(1)
n
P
ωi bn,i (t)
i=0
This work was supported by the National Natural Science Foundation of China (No. 60173034), the National
Natural Science Foundation for Innovative Research Groups (No. 60021201) and the Foundation of State Key
Basic Research 973 Item (No. 2002CB312101).
c 2004 Elsevier Science Ltd. All rights reserved.
0893-9659/04/$ - see front matter °
PII:00
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2
R. Zhang and G. Wang
in the case of n = 2 and n = 3 [3] and proved that when n = 3,
½
1
kQ (t)k ≤ 3 max kCi+1 − Ci k max a,
0≤i≤2
a
0
where
½
a=
ω3
ω0
m = min {ω̂1 , ω̂2 } ,
ω̂1 =
¾
A(m, M ),
(2)
¾1/3
,
(3)
M = max {ω̂1 , ω̂2 } ,
ω1
2/3 1/3
ω0 ω3
ω̂2 =
(4)
ω2
,
1/3 2/3
ω0 ω3
(5)

1 + 2M
(2 − m)M − 1


if
≤ m < 1;
 M (5m − 2m2 − 1) M − 1 ,
3M
¾
½
A(m, M ) =

3 + 10M + 3mM

 max M,
, otherwise.
(1 + 3m)2
(6)
Hermann pointed out that if ω0 = ω3 then (2) is less or equal than Floater’s formula [4]
kQ0 k ≤ n
W2
ω2
max kCi+1 − Ci k ,
(7)
0≤i≤n−1
where
ω = min ωi ,
W = max ωi .
0≤i≤n
(8)
0≤i≤n
In the general case, he has conjectured that the above result is true as well, but he has been unable
to prove it. Whereas, the important effect of rational cubic Bézier curves, we have considered
this problem and affirmed his conjecture is right. Now we will give the proof in this note. This
conjecture can be written in essence as follows.
¾
½
W2
1
A(m, M ) ≤ 2 .
max a,
a
ω
Conjecture.
(9)
To prove the conjecture, we need the following two lemmas.
Lemma 1.
a
M
W2
≤
mi
ω2
(i = 0, 1).
(10)
Proof. Since the proof of case i = 0 is similar with that of case i = 1, we only prove the
case i = 1. We distinguish two cases.
Case (1). If ω̂1 ≥ ω̂2 , then by (4) and (8) have
a
M
=
m
½
ω3
ω0
¾1/3
1/3
2/3
2/3
ω 1 ω3
ω1 ω0 ω3
=
≤
2/3
1/3
ω2 ω ω
ω2 ω 2/3
0
3
0
µ
W
ω
¶5/3
≤
W2
.
ω2
(11)
Case (2). If ω̂1 < ω̂2 , then by (4) and (8) have
M
=
a
m
µ
ω3
ω0
¶1/3
Combining (11) and (12), (10) holds.
2/3
1/3
W2
ω2
W
ω2 ω0 ω3
≤
=
≤
.
ω1 ω 1/3 ω 2/3
ω1
ω
ω2
0
3
(12)
Proof of Hermann’s Conjecture
3
Lemma 2. If α1 ± α2 = α3 + α4 ≤ 2 (0 ≤ αi ≤ 2, i = 1, 2, 3, 4), then
ωiα1 ωj±α2
W2
≤
ωkα3 ωlα4
ω2
(i, j, k, l ∈ {0, 1, 2, 3}).
(13)
Proof. Noticing (4) and (8), this lemma is simple. We omit it’s proof.
The conjecture’s proof. Without loss of generality we can suppose ω3 ≥ ω0 , then
½
1
max a,
a
¾
= a.
(14)
Now we distinguish two cases.
Case (1). (1 + 2M )/3M ≤ m ≤ 1. Noticing (9) and (6), we need to prove
M
M
(2 − m)M − 1
≤
.
(5m − 2m2 − 1) M − 1
m
(15)
Equation (15) is equivalent to
¡
¢
m[(2 − m)M − 1] ≤ 5m − 2m2 − 1 M − 1,
or
3M m − m2 M − M + m − 1 ≥ 0.
Noticing 3M m ≥ 2M + 1 and m < 1, we have
3M m − m2 M − M + m − 1 ≥ M (1 − m2 ) + m > 0.
So in this case (15) holds.
Case (2). m ≥ 1 or 0 < m ≤ (1 + 2M )/3M . Since
aM ≤
W2
ω2
(16)
is (10) when i = 0, we only need to prove
a
W2
3 + 10M + 3mM
≤ 2.
2
(1 + 3m)
ω
(17)
To prove (17), we need to distinguish two cases again.
2/3
1/3
1/3
2/3
Case A. (ω̂1 ≥ ω̂2 ). That is M = ω1 /(ω0 ω3 ), m = ω2 /(ω0 ω3 ), then we have
´
³
¾1/3 3 + 10 ω /ω 2/3 ω 1/3 + 3 (ω ω /ω ω )
1
1 2
0 3
0
3
3 + 10M + 3mM
ω3
=
a
n
³
´o2
2
(1 + 3m)
ω0
1/3 2/3
1 + 3 ω2 /ω0 ω3
½ ¾2/3
1/3
2/3
ω3
3ω0 ω3 + 10ω0 ω1 ω3 + 3ω1 ω2
=
.
n
o2
ω0
1/3 2/3
ω0 ω3 + 3ω2
½
(18)
Applying average inequality, we have
1/3
2/3
ω0 ω3
1/3
2/3
+ 3ω2 = ω0 ω3
´1/4
³
1/3 2/3
1/12 3/4 1/6
+ ω2 + ω2 + ω2 ≥ 4 ω0 ω3 ω2 ω2 ω2
= 4ω0 ω2 ω3 . (19)
4
R. Zhang and G. Wang
Substituting (19) into (18) and using (13), we have
a
3 + 10M + 3mM
≤
(1 + 3m)2
½
ω3
ω0
¾2/3
1/3
2/3
3ω0 ω3 + 10ω0 ω1 ω3
+ 3ω1 ω2
1/6 3/2 1/3
16ω0 ω2 ω3
1/6
4/3
3 ω0 ω3
=
16 ω 3/2
2
1/3
1/3
10 ω1 ω3
3 ω1 ω3
W2
+
+
≤ 2.
1/2
3/2
5/6
1/2
16 ω ω
16 ω ω
ω
0
2
0
2
2/3
2/3
(20)
1/3
Case B. (ω̂1 < ω̂2 ). That is M = ω2 /(ω0 ω3 ), m = ω1 /(ω0 ω3 ), then we have
a
3 + 10M + 3mM
=
(1 + 3m)2
½
³
´
¾1/3 3 + 10 ω /ω 1/3 ω 2/3 + 3 (ω ω /ω ω )
2
1 2
0 3
0
3
ω3
n
³
´o2
ω0
2/3 1/3
1 + 3 ω1 /ω0 ω3
2/3
1/3
(21)
3ω0 ω3 + 10ω0 ω1 ω3 + 3ω1 ω2
=
.
n
o2
2/3 1/3
ω0 ω3 + 3ω1
Applying average inequality, we have
2/3
1/3
ω0 ω3
2/3
1/3
+ 3ω1 = ω0 ω3
´1/4
³
2/3 1/3
1/6 3/4 1/12
+ ω1 + ω1 + ω1 ≥ 4 ω0 ω3 ω1 ω1 ω1
= 4ω0 ω1 ω3 . (22)
Substituting (22) into (21) and using (13), we have
2/3
a
1/3
3ω0 ω3 + 10ω0 ω1 ω3 + 3ω1 ω2
3 + 10M + 3mM
≤
1/3 3/2 1/6
(1 + 3m)2
16ω ω ω
3
=
16
0
1
2/3 5/6
ω0 ω3
10
+
3/2
16
ω1
3
1/3 1/6
ω0 ω3
1/2
ω1
ω2
3
W2
+
≤
.
16 ω 1/3 ω 1/2 ω 1/6
ω2
0
1
3
(23)
By (20) and (23), we know (17) holds. Combining (15)–(17), we know the conjecture has been
proved.
REFERENCES
1. T. Hermann, On a tolerance problem of parametric curves and surfaces, Computer Aided Geometric Design
9, 109–117, (1992).
2. G.J. Wang, T.W. Sederberg and T. Saito, Partial derivatives of rational Bézier surfaces, Computer Aided
Geometric Design 14, 377–381, (1997).
3. T. Hermann, On the derivatives of second and third degree rational Bézier curves, Computer Aided Geometric
Design 16, 157–163, (1999).
4. M.S. Floater, Derivatives of rational Bézier curves, Computer Aided Geometric Design 9, 161–174, (1992).