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The proof of hermann's conjecture
Applied , Mathematics Letters
Available online at www.sciencedirect.com
:SCIENCE ~DIFtECT ELSEVIER
e
Applied Mathematics Letters 17 (2004) 1387-1390
www.elsevier.com/locat e/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 CO, Zhejiang University Hangzhou, 310027, P.R. China and China Jiliang University, Zhejiang Hangzhou, 310018, P.R. China renj iang©mail, hz. z j . 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. z j . cn
(Received and accepted July 2003) Communicated by R. P. Agarwal A b s t r a c t - - T h i s note proves Thomas Hermann's conjecture on the comparison between two boundaries of the derivatives of rational cubic B~zier curves. The result is valuable for computer aided geometric design. (~) 2004 Elsevier Ltd. All rights reserved.
Keywords--Computer
aided geometric design, Rational B~zier curve, Derivatives, Boundary.
In the geometric design, estimating the boundary of the derivatives of rational B~zier curves is an important subject [1,2]. In [3], Hermann has investigated the boundaries of the derivatives of rational B~zier curves
Ciwibn,i(t) Q(t) = i=o
,
(1)
i=O This work was supported by the National Natural Science Foundation of China (No. 60373033 and No. 60333010), the National Natural Science Foundation for Innovative Research Groups (No. 60021201) and the Foundation of State Key Basic Research 973 Item (No. 2002CB312101). 0893-9659/04/$ - see front matter C) 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j .aml. 2003.09.012
Typeset by ~4A4S-TEX
1388
R. ZHANGAND C. WANG
/
in the case of n = 2 and n = 3 [3] and proved t h a t w h e n n = 3,
IIQ'(t)ll-<3o<~<2mIIC~+1ax C~ll m a x { a ,
A(m,M),
1}
(2)
where
a ~ { 02303}1/3 m : min {~1, ~32},
(3)
022
021
2/3 1/3 020 tD3
Wl-
W2-
OJ0 023
- m)M - 1
{
1 + 2M if
3-'--'M---< m <
1; (6)
3+IOM+3mM} M,
max
(5)
1/3 2/3'
M (5m (-~-2-m~-~-l~/~f - 1'
A(m,M)=
(4)
M : m a x {&x,&2},
-G ~ 3 - ~ ) 2
,
otherwise.
H e r m a n n pointed out t h a t if w0 = w3 t h e n (2) is b e t t e r t h a n F l o a t e r ' s f o r m u l a [4]
W2
IIQ'II ~ n-j
o
(7)
,
where w=
m i n wi, 0
W = m a x wi.
(8)
0
In the general case, he has conjectured t h a t the above result is t r u e as well, b u t he has been unable to prove it. W h e r e a s , the i m p o r t a n t effect of rational cubic B6zier curves, we have considered this p r o b l e m and affirmed his conjecture is right. Now we will give the p r o o f in this note. This conjecture can be w r i t t e n in essence as follows.
CONJECTURE. a, a1
max To prove LEMMA
the conjecture,
we need
A(m,M)< - - w- 2
the following
(9)
"
two lemmas.
i. M a --
W 2 < --
m~ -
w2
(i = 0 , 1 ) .
(10)
PROOF. Since the proof of case i = 0 is similar with t h a t of case i = 1, we only prove the case i = 1. We distinguish two cases. CASE (1). If &l >-- &2, t h e n by (4) and (8) have
~
M
a - -7Tt = CASE
(2).
$1/3 1/3 2/3 023 Wl 020 ~03 2/3 W31 / 3 [W- -0 ) W2 WO
{W}5/3
W2 ~ t,o2 ----"
(11)
If o~1 < &2, t h e n b y (4) and (8) have
M a - -
m
Combining
W1 co2/3 ~ 2/3W2 WO
(Ii) and
1 1/3 022 Wo2/3 c°31/3 W2 02 2 W 1/3 2/3 -< -< - w0 w l w o w3 W l - - ~ - w 2" w3
=
I
--
(12), (i0) holds.
(12)
Proof of Hermann's Conjecture LEMMA 2. IfC~l ±c~2
=
1389 J
- k a 4 < 2 (0 < a i < 2, i = 1,2, 3,4), then
33
w i31 wj4-32 a3 0~4 COk 021
W2 <--
(13)
( i , j , k , l e {0, 1,2,3}).
~2
PROOF. Noticing (4) and (8), this l e m m a is simple. We o m i t it's proof. T h e conjecture's proof. W i t h o u t loss of generality we can s u p p o s e w3 _> coo, t h e n
{1}
max
a, a
~
(14)
a.
Now we distinguish two cases.
CASE (1). (1 +
2M)/3M <
m < 1. Noticing (9) and (6), we need to prove
M
(2 - ~)M (5m-2m
-
2-1)
< M
1
M-1
-
(15)
m
E q u a t i o n (15) is equivalent to m[(2-m)M-1]_<
(5m-2rn
2-1)
M-1,
or
3Mrn-
m2M-
M + m-
1 Z O.
Noticing 3 M m >_ 2 M + IL and m < 1, we have 3Mm-
rn2M-
M + m-
1 _> M ( 1 - m 2) + m > 0.
So in this case (15) holds. CASE (2). rn _> 1 or 0 < m _< (1 +
2M)/3M.
Since W 2
(16)
a M <_ c°2
is (10) w h e n i = 0, we only need to prove 3 + 10M + 3raM W 2 (l+3rn) 2 - co2"
(17)
To prove (17), we need to distinguish two cases again. , , 2/3 1/3, t , 1/3 2/3, C A S E A . (&l-->&2)- T h a t i s M = c o l / I . w o w3 ) , m = w 2 / i W o w a ) , t h e n we have 3 + 10M + 3rnM (1 + 3m) 2
= (w~ ~1/3
( , ,
kwo J
= [w_3a ~2/3 kWo J
2/3
1/3"~
3 + 10 t,~l/coo co3 ) + 3 (colco2/~oco3) 1+3 3w0w3 +
/"
t~ 1/3
iuw 0
.
1/3
2/3"~ 2
t co21coo w3
2/3
021093
)
A
(18)
-~- 3colW 2
'f 0201/3 CO3 2/3 _~ 3w212f
I
A p p l y i n g average inequality, we have 1/3
2/3
w0 w 3
1/3
2/3
+ 3 w 2 = c o o w3
/" 1/3
+co2+co2+co2>_4~coo
2/3
.
. ~1/4
w3 co2~2~2)
~
1/12
=~COo
3/4
1/6
w2 w3
-
(19)
1390
R . Z H A N G AND G . W A N G
!
Substituting (19) into (18) and using (13), we have ~2/3 ~ 1/3 2/3 3 + 10M + 3raM < 023 3020023 + ,UWo wlw3 + 3021022 -,~ 1/6 3/2 1/3 (1 + 3m) 2 L02o ) -t°02o 022 023 1/6 .4/3 1/3 W2 3 w0 w3 10 Wl023 3 021023 " -- 16 023/2 + 16 020 1/2 022 3/2 "~- 16 wo~/60221/2' ~ - 022 -
(20)
, , 1/3 2/3 . . . . 2/3 1 / 3 ~ CASE B. (Wl < ~2). T h a t is M = 022/[w o 023 ), m = ,~ll~W o w 3 ), t h e n we have
a
3 + 10M + 3mM =
02_~a/1"
, 1/3 2/3h
{
3 + 10
[ 020 )
) + 3 (021022/020023)
/
(1 + 3m) 2
{
, 2/3
1/3"~ }2
1 + 3 [Wl/W o w3 )
~,~ 213 .U3 3W0W3 ~- Iu02 0 021~3 ~- 3021w2
(21)
0202/3 0231/3+3021}2 A p p l y i n g average inequality, we have 2/3 1/3 2/3 1/3 [' 2/3 1/3 020 w3 + 3 W l =020 W3 -~-021Ar-W1-~-021>__4 023
~020
-'~1/4 W1021021]
~ 1/6 3/4 1/12 =-4Wo Wl 023 .
(22)
Substituting (22) into (21) and using (13), we have . n 2/3
a
.1/3
3 + 10M + 3raM < 3WOW3 -}- IUW0 ~lW 3
(1 + 3 )2
-~- 3W1~2 1/3 3/2 1/6 tOW0 021 023 2/3 5/6 1/3 1/6 3 022 W2 3 w0 w3 10 ~0 w 3 + 1/3 1/2 1/6 < 02i"" =--16 w13/2 + 1--6" w~/2 16 020 021 023
--
t.
(23)
B y (20) and (23), we know (17) holds. Combining (15)-(17), we k n o w 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 B6zier surfaces, Computer Aided Geometric Design 14, 377-381, (1997). 3. T. Hermann, On the derivatives of second and third degree rational B6zier curves, Computer Aided Geometric Design 16, 157-163, (1999). 4. M.S. Floater, Derivatives of rational B~zier curves, Computer Aided Geometric Design 9, 161-174, (1992).
Available online at www.sciencedirect.com
:SCIENCE ~DIFtECT ELSEVIER
e
Applied Mathematics Letters 17 (2004) 1387-1390
www.elsevier.com/locat e/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 CO, Zhejiang University Hangzhou, 310027, P.R. China and China Jiliang University, Zhejiang Hangzhou, 310018, P.R. China renj iang©mail, hz. z j . 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. z j . cn
(Received and accepted July 2003) Communicated by R. P. Agarwal A b s t r a c t - - T h i s note proves Thomas Hermann's conjecture on the comparison between two boundaries of the derivatives of rational cubic B~zier curves. The result is valuable for computer aided geometric design. (~) 2004 Elsevier Ltd. All rights reserved.
Keywords--Computer
aided geometric design, Rational B~zier curve, Derivatives, Boundary.
In the geometric design, estimating the boundary of the derivatives of rational B~zier curves is an important subject [1,2]. In [3], Hermann has investigated the boundaries of the derivatives of rational B~zier curves
Ciwibn,i(t) Q(t) = i=o
,
(1)
i=O This work was supported by the National Natural Science Foundation of China (No. 60373033 and No. 60333010), the National Natural Science Foundation for Innovative Research Groups (No. 60021201) and the Foundation of State Key Basic Research 973 Item (No. 2002CB312101). 0893-9659/04/$ - see front matter C) 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j .aml. 2003.09.012
Typeset by ~4A4S-TEX
1388
R. ZHANGAND C. WANG
/
in the case of n = 2 and n = 3 [3] and proved t h a t w h e n n = 3,
IIQ'(t)ll-<3o<~<2mIIC~+1ax C~ll m a x { a ,
A(m,M),
1}
(2)
where
a ~ { 02303}1/3 m : min {~1, ~32},
(3)
022
021
2/3 1/3 020 tD3
Wl-
W2-
OJ0 023
- m)M - 1
{
1 + 2M if
3-'--'M---< m <
1; (6)
3+IOM+3mM} M,
max
(5)
1/3 2/3'
M (5m (-~-2-m~-~-l~/~f - 1'
A(m,M)=
(4)
M : m a x {&x,&2},
-G ~ 3 - ~ ) 2
,
otherwise.
H e r m a n n pointed out t h a t if w0 = w3 t h e n (2) is b e t t e r t h a n F l o a t e r ' s f o r m u l a [4]
W2
IIQ'II ~ n-j
o
(7)
,
where w=
m i n wi, 0
W = m a x wi.
(8)
0
In the general case, he has conjectured t h a t the above result is t r u e as well, b u t he has been unable to prove it. W h e r e a s , the i m p o r t a n t effect of rational cubic B6zier curves, we have considered this p r o b l e m and affirmed his conjecture is right. Now we will give the p r o o f in this note. This conjecture can be w r i t t e n in essence as follows.
CONJECTURE. a, a1
max To prove LEMMA
the conjecture,
we need
A(m,M)< - - w- 2
the following
(9)
"
two lemmas.
i. M a --
W 2 < --
m~ -
w2
(i = 0 , 1 ) .
(10)
PROOF. Since the proof of case i = 0 is similar with t h a t of case i = 1, we only prove the case i = 1. We distinguish two cases. CASE (1). If &l >-- &2, t h e n by (4) and (8) have
~
M
a - -7Tt = CASE
(2).
$1/3 1/3 2/3 023 Wl 020 ~03 2/3 W31 / 3 [W- -0 ) W2 WO
{W}5/3
W2 ~ t,o2 ----"
(11)
If o~1 < &2, t h e n b y (4) and (8) have
M a - -
m
Combining
W1 co2/3 ~ 2/3W2 WO
(Ii) and
1 1/3 022 Wo2/3 c°31/3 W2 02 2 W 1/3 2/3 -< -< - w0 w l w o w3 W l - - ~ - w 2" w3
=
I
--
(12), (i0) holds.
(12)
Proof of Hermann's Conjecture LEMMA 2. IfC~l ±c~2
=
1389 J
- k a 4 < 2 (0 < a i < 2, i = 1,2, 3,4), then
33
w i31 wj4-32 a3 0~4 COk 021
W2 <--
(13)
( i , j , k , l e {0, 1,2,3}).
~2
PROOF. Noticing (4) and (8), this l e m m a is simple. We o m i t it's proof. T h e conjecture's proof. W i t h o u t loss of generality we can s u p p o s e w3 _> coo, t h e n
{1}
max
a, a
~
(14)
a.
Now we distinguish two cases.
CASE (1). (1 +
2M)/3M <
m < 1. Noticing (9) and (6), we need to prove
M
(2 - ~)M (5m-2m
-
2-1)
< M
1
M-1
-
(15)
m
E q u a t i o n (15) is equivalent to m[(2-m)M-1]_<
(5m-2rn
2-1)
M-1,
or
3Mrn-
m2M-
M + m-
1 Z O.
Noticing 3 M m >_ 2 M + IL and m < 1, we have 3Mm-
rn2M-
M + m-
1 _> M ( 1 - m 2) + m > 0.
So in this case (15) holds. CASE (2). rn _> 1 or 0 < m _< (1 +
2M)/3M.
Since W 2
(16)
a M <_ c°2
is (10) w h e n i = 0, we only need to prove 3 + 10M + 3raM W 2 (l+3rn) 2 - co2"
(17)
To prove (17), we need to distinguish two cases again. , , 2/3 1/3, t , 1/3 2/3, C A S E A . (&l-->&2)- T h a t i s M = c o l / I . w o w3 ) , m = w 2 / i W o w a ) , t h e n we have 3 + 10M + 3rnM (1 + 3m) 2
= (w~ ~1/3
( , ,
kwo J
= [w_3a ~2/3 kWo J
2/3
1/3"~
3 + 10 t,~l/coo co3 ) + 3 (colco2/~oco3) 1+3 3w0w3 +
/"
t~ 1/3
iuw 0
.
1/3
2/3"~ 2
t co21coo w3
2/3
021093
)
A
(18)
-~- 3colW 2
'f 0201/3 CO3 2/3 _~ 3w212f
I
A p p l y i n g average inequality, we have 1/3
2/3
w0 w 3
1/3
2/3
+ 3 w 2 = c o o w3
/" 1/3
+co2+co2+co2>_4~coo
2/3
.
. ~1/4
w3 co2~2~2)
~
1/12
=~COo
3/4
1/6
w2 w3
-
(19)
1390
R . Z H A N G AND G . W A N G
!
Substituting (19) into (18) and using (13), we have ~2/3 ~ 1/3 2/3 3 + 10M + 3raM < 023 3020023 + ,UWo wlw3 + 3021022 -,~ 1/6 3/2 1/3 (1 + 3m) 2 L02o ) -t°02o 022 023 1/6 .4/3 1/3 W2 3 w0 w3 10 Wl023 3 021023 " -- 16 023/2 + 16 020 1/2 022 3/2 "~- 16 wo~/60221/2' ~ - 022 -
(20)
, , 1/3 2/3 . . . . 2/3 1 / 3 ~ CASE B. (Wl < ~2). T h a t is M = 022/[w o 023 ), m = ,~ll~W o w 3 ), t h e n we have
a
3 + 10M + 3mM =
02_~a/1"
, 1/3 2/3h
{
3 + 10
[ 020 )
) + 3 (021022/020023)
/
(1 + 3m) 2
{
, 2/3
1/3"~ }2
1 + 3 [Wl/W o w3 )
~,~ 213 .U3 3W0W3 ~- Iu02 0 021~3 ~- 3021w2
(21)
0202/3 0231/3+3021}2 A p p l y i n g average inequality, we have 2/3 1/3 2/3 1/3 [' 2/3 1/3 020 w3 + 3 W l =020 W3 -~-021Ar-W1-~-021>__4 023
~020
-'~1/4 W1021021]
~ 1/6 3/4 1/12 =-4Wo Wl 023 .
(22)
Substituting (22) into (21) and using (13), we have . n 2/3
a
.1/3
3 + 10M + 3raM < 3WOW3 -}- IUW0 ~lW 3
(1 + 3 )2
-~- 3W1~2 1/3 3/2 1/6 tOW0 021 023 2/3 5/6 1/3 1/6 3 022 W2 3 w0 w3 10 ~0 w 3 + 1/3 1/2 1/6 < 02i"" =--16 w13/2 + 1--6" w~/2 16 020 021 023
--
t.
(23)
B y (20) and (23), we know (17) holds. Combining (15)-(17), we k n o w 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 B6zier surfaces, Computer Aided Geometric Design 14, 377-381, (1997). 3. T. Hermann, On the derivatives of second and third degree rational B6zier curves, Computer Aided Geometric Design 16, 157-163, (1999). 4. M.S. Floater, Derivatives of rational B~zier curves, Computer Aided Geometric Design 9, 161-174, (1992).