ETNA
Electronic Transactions on Numerical Analysis.
Volume 26, pp. 312-319, 2007.
Copyright  2007, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
POLYNOMIAL BEST CONSTRAINED DEGREE REDUCTION IN STRAIN
ENERGY
GERMAIN E. RANDRIAMBELOSOA
Abstract. We exhibit the best degree reduction of a given degree polynomial by minimizing the strain energy
of the error with the constraint that continuity of a prescribed order is preserved at the two endpoints. It is shown
that a multidegree reduction is equivalent to a step-by-step reduction of one degree at a time by using the Fourier
coefficients with respect to Jacobi orthogonal polynomials. Then we give explicitly the optimal constrained one
degree reduction in Bézier form, by perturbing the Bézier coefficients.
Key words. reduction, polynomials, approximation, Bézier curves
AMS subject classifications. 41A10, 65D05, 65D17
1. Introduction. Degree reduction of polynomials consists in approximating a given
polynomial by a lower degree polynomial by minimizing the error with respect to a certain measure. The most often used measure in degree reduction is the
-norm of the error for ; see [1, 2, 3, 4, 6, 7, 8, 9, 10, 12]. In this paper we
are concerned with the following problem: Given a degree polynomial , find a degree
!"# polynomial , %$'& ()*,+ , such that
- and have the same first .0/1 derivatives at %2( and the same first 34/1
derivatives at 56 , i.e.,
(1.1)
87:9<;(51=7:9<;,(>@?A1()*BCBCBD.E/F
34/G.IH KJ
7MLN; !O51 7PLN; !Q,SRT1()*BCBCB34/F
W"X
- minimizes the error strain energy UV
Y Z\[ [Z]
^[ [_N`ba for all such
that satisfy the endpoint constraints (1.1).
possible polynomials of degree H
This process is useful for many tasks in geometric modeling, such as data compression,
data comparison, rendering. Degree reduction is also needed to simplify some geometric or
graphical algorithms for intersection calculation of two polynomial curves or surfaces.
There have been many methods developed for degree reduction. As this is essentially a
problem of approximation, methods from classical approximation theory can be employed.
dc
Watkins and Worsey [12] used the Chebyshev economization to produce the best
-approximation of degree ef without constraint to a given degree polynomial. Later, Bogacki et
al. [1] achieve the best uniform approximation with endpoint interpolation by modifying the
economization procedure. The endpoints constraints that guarantee a prescribed order of continuity are often required in many applications and especially when degree reduction is combined with subdivision to generate continuous piecewise approximation. Lachance [7] and
c
Eck [3] made deeper the Chebyshev economization procedure for the best
-approximation
with prescribed order of continuity at the endpoints, but it seems that there was no explicit
formula for the degree reduced polynomial as pointed out in [4]. These difficulties can be
avoided by using the -norm. The degree reduction with endpoint interpolation that mini
`
studied by Eck [4]. His method used the inverse of a polynomial
mizes the -norm has been
`
degree elevation process in Bézier form [5] and obtained two sets of control points, then
g Received July 12, 2006. Accepted for publication February 12, 2007. Recommended by M. Gutknecht.
Departement of Mathematics and Informatic,
(grandri@univ-antananarivo.mg).
312
Tananarive
University,
Tananarive,
Madagascar
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POLYNOMIAL BEST CONSTRAINED DEGREE REDUCTION IN STRAIN ENERGY
313
considered a simple convex combination of these two sets of control points to generate the
control points for the degree reduced polynomial. Recently, Lutterkort et al. [8] discovered
a surprising result: finding the best - approximation in Bézier form without constraint is
`
equivalent to finding the best Euclidean approximation of Bézier coefficients. This result
can be extended to the multivariate case [9]. The best degree reduction of Bézier curves in
X
-norm with endpoint interpolation has been solved by Kim and Moon [6]. The optimal
degree reduction with respect to various norms was studied by Brunnett et al. [2] who have
also shown the separability of degree reduction into the different components of a parametric
curve.
2. Degree reduction method. The degree reduction is accomplished through two stages.
In the first stage, we construct a degree I1.A/h3/i polynomial jA interpolating the second
derivative [ [ at 51( up to the Z.0/kQ th order continuity and at 5l up to the 34/kQ th
order of continuity as follows
q X
q X
a
a
*
`
s
9
jAmonp 7:9<; (
/ouvp 7PLN; !Q `Qw L nQt u a nQ` t u
a `
Y
Y
9<r
LNr
where s
9 nOt u and w L nOt u are the so called Hermite polynomials of degree fx.T/3\/"y
defined by
azL s 9 Y
R€?
x} ()~ 
otherwise anQL t u { | r
a^ s 9 X
k(4
a nO t u { | r
?D_RTk()*BCBCBD.E/ƒ‚„(4CBCB*BN34/k~
and
azL w 9 X
R€?
x} ()~ 
otherwise a nQL t u { | r
a^ w 9 Y
k(4
a~ nQ t u { | r
?D_RTk()*BCBCB34/…‚„k(4CB*BCB.E/k~†
Both s
uniquely defined since they have .d/]3~/‡J degrees of freedom,
9 and w L nQt u asarewell.
and .E/ˆnQ34t u /ˆJ constraints
The second stage is to determine [ [ ‰1jA by minimizing the error strain energy
W"X
\ .
Y Z\[ [_]
[ [_N`#a~ , and then deduce the degree reduced polynomial
Note that the error strain energy can be expressed in terms of the -norm as
`
Ul‹ŠC [ [ ]
[ [ Š ` `
W"X
in the Hilbert space Œh& ()*,+ with the inner product 8!
2Ž
Y 
\!a5 , for a
convenient Borel positive measure 5 . For such a problem, choosing proper basis functions
Ul
often simplifies the computation. In our case, in order to allow a direct determination of
the polynomial without solving a linear system, the appropriate basis functions should
be orthogonal with respect the above inner product. Let us consider the Jacobi polynomials
‘
of degree ? that are orthogonal with respect to the inner product
9
’
8
“5
W"X
Y ` n !”K ` u 
\!aƒ.\N3A•()†
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314
G. RANDRIAMBELOSOA
They are defined by Rodrigues’ formula [11]
‘
(2.1)
with
‘Y
56
9 m
EQ
a>9
D— ` !”KD— `
9 !”K ` 9™ @?5•o
?D–
a
9 ˜ ` np
n
u E
uvp
and satisfy the orthogonality relation
W X
‘
‘
if ?AœR
Y ` n !”K ` u 9 L a~m }›š( 9 nu
if ?”œ
 R
?\/œ~.
?\/œ~.E/œz3
with
. From the identity (2.1), the first
š 9 z?8/œ~‘ .E/G~34/‘ ž ~. Ÿ¢¡£ž ‘ ~.
Ÿ
D
n
u
Y
few polynomials
9 are ¤ , X ¤ )_.¥/¦3i/§O'¨_~.¥/¦3_ ,
‘
m¥Z.E/G34/G,Z.E/G~3/Gy! ` ')Z.E/kQCZ~.E/œz3)/Gyb/1Z.0/kQ,_~.E/kQ†
` L EMMA 2.1. The functions [ [ 'jA and [ [ 'jA can be expressed as
‘
‘
[ [ 'jA©k !”K _ª Y Y #/kBCB*BQ/œª>« — — — « — — — (2.3)
`‘
`
‘
[ [ 'jA% n !”K u _¬ Y Y =/B*BCBO/œ¬­ — n — u — ­ — n — u — (2.4)
`
`
n
u
n u
n u
where ª and ¬ are the Fourier coefficients defined by
9
9
W X
‘
X
(2.5)
ª 9 ¥ š 9 — Y !”f Z\[ [®'jA 9 a~
n
u
nu
W"X
‘
X
(2.6)
¬ 9 ¥ š 9 — Y ”] ¯
[ [ZfjA 9 !a†
n
u
nDu
(2.2)
Proof. By construction the polynomial jA interpolates the second derivative [ [ at
up to the Z.\/hO th order continuity and at £6 up to the 3!/hO th order continuity. Then
the polynomials [ [ A'jA and [ [ A'jA have . -fold zeros at £k( and 3 -fold zeros at
56 . A common factor > can be factored out from [ [ *¢j= and [ [ *¢j= , thus
we can set [ [ °©jA%n 4Eu «
and [ [ z©jAm !²E ­ — — — .
`
n
u± « — n — u — ` and ­
n termsu³ of the
Now we express the polynomials
in
n u Jacobi
‘
—
—
—
—
—
—
`
`
±
³
polynomials , then using the orthogonality
n u relation (2.2),n we
u obtain the expressions (2.5)
9
and (2.6) for the Fourier coefficients ª and ¬ .
9
9
T HEOREM 2.2. The best degree reduced polynomial \ is such that
5k(
‘
‘
[ [ mkjA=/ˆ !”K _ª Y Y #/kBCBCBO/Gª^­ — — — ­ — —
`
(>5kZ(´
n !O5ku Q
n u
n
and ª Y ª X CB*BCBbª ­
are given by (2.5).
— — — error
` strain energy
Proof. Consider the
n u
WœX
Ul Y &<Z [ [ 'jAN£¯
[ [ £fjAN®+ ` a
­ —q — —
« —q — —
WœX
`
‘
`
‘
Y ` ”] `
ª 9
n u Zª 9 f¬ 9 9 #/
n u
Y
n
umµ
9<r
9<r ­ — — — X
n u
Differentiating it with respect to the coefficient ¬ gives
L
·
­ —q — —
« —q — —
W X
`
‘
‘
`
U·
¸‰ Y ` !”K `
n u Zª 9 f¬ 9 9 L #/
n u
¬L
Y
n
u µ
9<r
9:r ­ — — — X
n u
(2.7)
u
— ` 9 ¶
`
a~†
‘
‘
ª 9 9 L ¶ a†
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POLYNOMIAL BEST CONSTRAINED DEGREE REDUCTION IN STRAIN ENERGY
315
Taking into account the orthogonality relation (2.2), we obtain
·
· U 6
‰4ª L '¬ L š L SRT1()CB*BCB f
.T'38f)†
¬L
Dn u
To minimize U , we equate this derivative to zero and obtain ¬ kª for ?1()*BCB*B '.T
9
9
3\f . From (2.4), we deduce the expression of [ [ in (2.7).
Equation (2.7) shows that if the decomposition (2.3) is available for the second derivative
of the given polynomial, then the second derivative [ [ of the constrained approximation in strain energy can be immediately obtained by just removing the last ¹ terms
in the square bracket of (2.3). This means that a multidegree reduction is equivalent to a stepby-step reduction of one degree at a time. In the next section we give explicitly the optimal
constrained one degree reduction in Bézier form [5], by perturbing the Bézier coefficients.
[ [ 3. Coefficients perturbation in Bézier form. To obtain the coefficients of the reduced
degree polynomial without computing the Fourier coefficients we give a direct method based
on perturbing the coefficients of the initial polynomial.
«
«
«
Let º IS» !9!© — 9 , ?e½(4CB*BCB , be the degree Bernstein polynomials
9
N9 ¼
A degree ¾
«q — X
X
º « — , can be expressed in terms
basis.
Bézier polynomial ‰
YÀ¿ 9 9
9:r
of Bernstein polynomials of higher degree !Ž¥‡1Q . In particular we can write ¢
«q
9 º 9 « , with the new Bézier coefficients Y Y F 9 6? # 9 — X /kNi]?! # 9 , ?A
¡ ¿
¡ ¿
¿
Y
~9<r *BCBCBN¹~FZ²«i « — X . However, the converse is generally not true unless the coefficient
¿ of vanishes, which implies
of the th degree term
(3.1)
q«
9<r
Y
!EO 9
k()†
ž ?Ÿ 9
It has been proved (see [3]) that under the condition (3.1), we have
(3.2)
!EO!9 q 9 9
@?A1()*BCB*Biœ
X
Y ž R Ÿ L
¿
»« —
9 ¼ LNr
Now, the coefficients perturbation method consists in finding a perturbation vector
Á Y Á X CB*BCBÁ«) , such that given a degree polynomial 56Â « Y 9 º 9 « , the perturbed
9<r
polynomial
zÃ,5
q«
Y
v 9 G
/ Á 9 º 9 « 9:r
«
Y !EO!9 » « P /KÁ 51(
satisfies the condition (3.1), i.e., Â
:
9
r
9¼ 9 9
energy
, and minimizes the error strain
W X
W X «q — `
`
Ul Y Z \[ [_'\Ã [ [ ` a£ Y
Á 9 º 9 « — ` !¶ a
`
Y Ä
µ Å
9:r
with ` Á kÁ
9 ` 'zÁ 9 X /ˆÁ 9 .
Ä 9
p
p
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316
G. RANDRIAMBELOSOA
The continuity constraints (1.1) at the endpoints (>Åk Y and !OÅk8« have a very
simple formulation,
L EMMA 3.1. If à is required to match up to the _.¢/Q, th derivative at 51( ,
X ›BCB*B\‹ÁD«¾¥( guarantees Œ X continuity
then Á Y ÆB*BCB\‹Á X ¥( . Similarly, Á«
—
—
uvp
between à and np at £l .
u
«
Y ‚ º « at the
Proof. The derivatives of order Ç of a degree polynomial ‚=À Â
9 9
:
9
r
Y
endpoints ‚ and ‚²« are given by [5]
>a È
£–
Ȃ
‚=(>5
a È
IKÇz– Ä
aÈ
£–
Ȃ « —
‚#Q5
a~ È
É]Çz– Ä
where
Ä
È
Y
È
is the iterated forward difference operator defined by
Y
È ‚ 9 È — X ‚ 9 X È — X ‚ 9 @ÇEl~D*BCBCB\
‚ 9 1
‚ 9
Ä
Ä
Ä
Ä
p
X
Y
Y
Y
X
we list a few examples
‚ k‚ ¢‚ , ` ‚ k‚ E~‚ X /©‚ Y , ‚ Y 1‚ ¢y‚ /0y>‚ X E‚ Y .
`
Ä
Ä at an endpoint
Ä€Ê only onÊ the Ç¢` /2 Bézier
Thus the Ç -th derivative of a Bézier curve
depends
coefficients near (and including) that endpoint.
« Y !EOz» « Ë( , then the given polynomial is of degree less
Note that, if Â
9<r
9 ¼ 9 with the degree reduction, by introducing a Lagrange’s
than ‡1 . Otherwise, we proceed
multiplier Ì and then, including the constraints, our problem is equivalent to
(3.3)
with
(3.4)
Í Ã Î<ÔzÏ ÕÖ:Õ~Ò
Z Á *BCBCB#NÁ « — — D̲,
Ã7 ZÐNÑ~Ò €
`
`
;
,n p
u
tMÓMÓMӏt
t×
W X « q—
CB*BCBÁ « — — Ì8m Y
`
`
µ
n p
u
9<r
C B*BCBbÁD«
The right hand side of Á
`
by Lemma 3.1.
np
Á
u
— `
Á º « — ¶
Ä ` 9 9 `
`
np
— — ` D ̲
u
`
a~'Ì
q«
9<r
Y
!EO
ž ?OŸ
P 9 ˆ
/ Á 9 ,†
takes into account the values of the Á ’s given
9
T HEOREM 3.2. The minimization problem (3.3) has a unique solution given by the
system of linear equations
Ø
 « — — ` Á 9 ` š L9 'Ì=!EQ®L » «
(4SRT1.0/G)CBCB*B\ÉK3\'
9<r « u ` Ä « Y
L¼
«
X
«
 — — ` !EO!9*» np Á 9 /"Â
!EQ9 » 9 (
Ù
9<r u `
9<r
9¼
p 9¼
np
«
«
«
«
«
«
«
where L « ` — » — 9 ` ¼0Ú » —L ` ¼O¡ » ` 9 —8L Û ¼ f » 9 — L ` ¼O¡ » 9 ` L —²— Û X ¼ / » L — — ` ¼Q¡ » 9 ` L —8— Û ¼,Ü Ý?D®R§
š 9
`
`
`
.%/¾CB*BCBN0„3I ,Ê š L9 k( for ?mH.%/Kp or ?5•"0p „3¾p and ` š L9 š L9 p I š 9 X / š 9 .
Ä
Proof. By Theorem 2.2, the constrained degree reduction problem
has p a` uniquep solution
that implies the same property for (3.3).
Taking the partial derivative of
ZÁ *BCB*BNÁ « — — D̲ defined by (3.4), with respect to Á
CBCB*BNÁ « — — Ì , and set`
`
`
`
ting the derivatives
equal
np
u to zero leads to
np
u
· W X « q— —
`
· o Y
Á 9 º 9 « — ` ,º L « — ` £'~º L « — X ` =/ˆº L « — ` N ¶ a‰Ì#EQ L
1()
u
`
ž R Ÿ
ÁL
—
— `
Ä
µ
9<r `
np
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317
POLYNOMIAL BEST CONSTRAINED DEGREE REDUCTION IN STRAIN ENERGY
2
3
1.8
2.5
1.6
2
1.4
1.5
1.2
1
1
0
0.2
0.4
0.6
0.8
1
0
0.2
ÞÅßiàßiá
0.4
0.6
0.8
â
ã
F IG . 3.1. (left,
), (solid) degree Bézier curve, (cross) reduced degree curve, (right,
(solid) degree Bézier curve, (cross) reduced degree curve.
â
1
t
t
å
ޔßIáQä®à~ßÉá
),
for RTo.E/G)CB*BCBÉf38f ,
· « q— —
q«
`
· Á9 /
EQ 9 X
()†
u !EQ 9
ž ?Ÿ
Ì
Y
p ž ?Ÿ 9
9<r
9<r `
np
« « » « º « , and the formula [5], æ Y X º
«
«
Then by the identity º !º % » »
9
L
9 ¼ L ¼ ¡ 9 ` L ¼ 9` L
X , we get
p
p
« X
p
· « q— —
`
· k()SRT1.0/G)CB*BCB\Éf3\f)
(3.5)
u š L9 ` Á 9 fÌ=!EO L
ž R Ÿ
ÁL
Ä
9:r `
n,p
» « — ` » « — ` » ` « —8Û f » « — ` » ` « —²Û X / » « — ` » ` « —²Û ,
with L `
«
š 9
9 ¼ Ú L ¼O¡ 9 L ¼
9 L ¼°¡ 9 L — ¼
L — ` ¼Q¡ 9 L — ` ¼ Ü
—
.E/G)CBCB*BiK` 3\'Ê .
p
p
p
p
«
9 !a£
?D_R
Rearranging the terms of the sum in (3.5), we can write
« q— —
`
· u Á 9 ` š L9 'Ì=!EQ L
k(4¨RTo.¢/œ*BCB*BiK3\'†
ž R Ÿ
ÁL
— `
Ä
9:r `
np
• if3\" and ` š L9 š L9 ' š 9 X / š 9 .
where L k( for ?mH.E/ or ?mœ
š 9
Ä
`
Finally, the following linear system of K2_.e/k3b/o~ p equations
p gives the unknowns
Á *BCB*BbNÁD« — — Ì ,
`
`
np Ø
u
 « — — ` Á 9 ` š L9 fÌ=!EO_L²» «
(4SRTo.E/G)CBCB*BiK3\'
9<r « u ` Ä « Y
(3.6)
L¼
«
X
«
—
—
Â
!EO!9Q» np Á 9 /"Â
!EQ9 » 9 (4†
9<r u ` `
9<r
9¼
p 9¼
np
« Y !EQ » « o( ),
R EMARK 3.3. When the polynomial is of degree , (i.e. Â
9
9<r ( , which
9 ¼ implies
system (3.6) has the unique solution Á
TCBCB*BDSÁ « — — SÌç¨
`
`
np
u
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318
G. RANDRIAMBELOSOA
1.7
1.7
1.6
1.6
1.5
1.5
1.4
1.4
1.3
1.3
1.2
1.2
1.1
1.1
1
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
t
á
0.6
ޔßiáQä_àß]è
é
F IG . 3.2. (left,
), (solid) degree Bézier curve, (cross) reduced degree
), (solid) degree Bézier curve, (cross) reduced degree curve.
é
0.8
1
t
ã
â
curve, (right,
ޔ߾èä®à~ß
à êë . Thereby, the method of coefficient perturbation reproduces the initial polynomial whenever the exact degree reduction exists. On the other hand, the solution of the
system (3.6) produces, in general, a polynomial of reduced degree Hœi" .
R EMARK 3.4. If we would like ~ÃC to be of degree !É"Q , the conditions (3.1)
must be replaced by i constraints
7 « q — — ` LN;
u t !EO 9
ž
Y
9<r
9<r `
np Ì X CB*BCBÌ
Introducing i Lagrange’s multipliers
W X « q—
Á
CB*BCBÁ« — — FÌ X CBCB*B̲« — ­E5 Y
`
`
µ
np
u
9<r
ì
L qL
!EQ 9
R
ˆ
/ í£î ï
ž ?Ÿ 9
then solving the minimization problem
Á 1()¨RT /k~CB*BCBN£†
R8Ÿ 9
« — ­ , we construct the fonctional
`
q«
— `
ì
Ì L— ­ L
u Á 9 º 9 « !¶ aA
LNr ­ X
`
p
np
Ô°Õ~ñ ; Z Á ` * BCB*BNÁ « — — ` FDÌ X CB*BCBDÌ « — ­ n p
u
tMÓMÓMӏt
t ×°ðtMÓMÓMӏt ×
gives the perturbation coefficients Á with obviously more complicated expressions.
9
7 Ã ÐÑÒ
à ԰ÕÍh
Ö:Õ~Î<Ò Ï
R EMARK 3.5. It is worth mentioning that the techniques developed in this paper can be
applied to compute the degree reduction of parametric Bézier curves.
3.1. Particular case. For the one degree reduction, if .T/I‹¾ˆ3#G , then all the
that is given by the last equation in (3.6)
Á 9 ’s are equal to zero, except Á
np
`
«
`
»
np
`¼
np
The reduced degree polynomial 5¸Â
!EQ9 q 9 9
X
¿
Y ž R Ÿ
»« —
9 ¼ LNr
Á
q«
!EO 9
†
p#np Ê ž ? Ÿ 9
9<r
« — Y X º « — X , according to (3.2), is such that
:9 r ¿ 9 9
Y
P L G
/ Á L ,@?1()*BCBCB\Ni"~†
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POLYNOMIAL BEST CONSTRAINED DEGREE REDUCTION IN STRAIN ENERGY
319
For k›J , Z.'Æ3‰›(> , the constraints (3.1) correspond to the endpoints and endtangent
vectors interpolation and we have Á Y ½Á X ë()Á Â Û Y !EO!9 » Û Á ½Á ( .
9:r
`
p Ê 9 ¼ 9 Ê \ Û are of
Fig. 3.1 shows two numerical examples where the reduced degree polynomials
degree y and . For Éoò , we have .£/ê3bl , which implies Z.\3Z5¥(4CQ or Z.\3Z5¥!~( .
Fig. 3.2 shows two examples where for Z.\3ZÅx(4CQ the reduced degree curve is of degree
J , and for Z.N3Zi !N(> the reduced degree curve is of degree y . Note that the relative
position of the reduced degree curve with respect to the initial curve depends on the sign of
Á ; indeed we have _A' Ã mkÁ º « .
np
`
`
`
n,p hull
np property of Bernstein polynomials, the rate
4. Error estimation. Using the convex
of approximation for the one degree reduction, is given by
ó ô4Y õ X®ø ŠCAfzÃC*ŠÝH
óNô)õ
ŠCÁ 9 Š
« — —
9
`Cù ù
`
|®ö÷ t
np
u
When the approximation error between and Ã, is larger than the prescribed tolerance,
we can subdivide the interval & (4CC+ and perform constrained degree reduction on each subinterval. For in the particular case .e/k¾›]"3A" , if we subdivide the given curve at parameter values (Gë Y  X úBCB*BdÆ û , then the error estimation (4.1) is de« where kÍeý°þ X  K . We get finally a continuous, piecewise
creased by a factor 9 9
9
°
¡
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approximation of of lower degree.
p
(4.1)
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in the
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