An Example of a Stable and Symmetric Cuve

An Example of a Stable and Symmetric Cuve

著者 Yokoyama Misako

journal or

publication title

Reports of Faculty of Science, Shizuoka University

volume 30

page range 1‑11

year 1996‑02‑25

出版者 Shizuoka University. Faculty of Science URL http://doi.org/10.14945/00002566

Reports of the Faculty of Sciencg SHIZUOKA IINITERSITY, Vol. ^30, p.

to p.

An Example of a Stable and Symmetric Cuve

Misako Yoroyeue

Defartment of Mathematics Facalty of Science, Shizuoka Uniaersity, 836 Ohya, Sh,izuoka 422, taPan

(Receiaed Sept. 18, 1995)

Abstract.

Here we have obtained an example of

stable closed plane curve with

crossing points which is homeomorphic to a given cunre and satisfies some technical conditions. That curve is also symmetrig which means the class has a symmetric representative. To compute an example we use spline functions and nonlinear optimization techniques. The ocample has no computational error. The technical condition would be considered to be an approximation that we have done.

l.Introduction

In recent yeanr, numerical approaches have been s<tensively done in finding exact answers to analytical ^problems in a number of fields ^(e.S. f l, [9] ). In the present paper, we consider a numerical procedure for obtainingverified results on computation of the stable plane cunre, by the use of spline functions and optimrzatian techniques. The motivation of this paper is the speech about knot energy t10l ^{. See} also t3l and t8l

Let N be a closed plane _cunre with crossing points. What is a "stable" curve F like? Is it

possibly symmetric? By "stable" we mean that F is an isolated local minimum in the class q(N) with respect to the "energy" P.

A Cz-mapping F:[0, 1] *R'z _belongs to 9(N) if F satisfies some conditions ^(see Definition

4. 5).

The set T(N) is an infinite dimensional space. By the minimal ^property of spline functions we reduce the problem in an infinite dimensional space to that in a finite dimensional space.

Then we use a method in the nonlinear programming theory.

In this paper\re present asuccessful example of a closed planecunrewittr 3 crossing points.

The resulting solution of that ocample is errorfree and invariant with

rotation through(Z/S) n

(see Corollary 6.2).

Misako Yorov.eue

2. heparation I : Splines

We recall some basic definitions and properties on splines.

Ileftrition 2.1 (APeriodic _Spline Frmction) A function f

[0, 1]-R is a periodi,c sflirufunction

of degree J with ^nodes tt, "' *o where 0 : tr

(1) f ^(t) is a polynomial of degree at most 3 on each subintenral [tr, tr+r] , i : 1, "', n-li (2)f e c' ^{[0, 1]}

(3) f(') (0) :

0), i ^:0,L,2.

Prolnsition 2.2 (T\e Existence and Uniquenees [1] ) For grven te,xrcR, i:1, "' , n, with 0 :

t, < ^... 1 tn :

there ocists one and only one periodic spline function of degree 3 f : [0, ^1]

- ^{R with nodes tt,;" , t" which} satisfies the conditions

f(t;:xr,i:1,"',tr.

The following theorem is well known. See t5l ^{for the natural} spline version.

fireorem 2.$ (Ihe Mininal Property) Letf:10, 1l -R ^be the periodi,c spliru function ^{of degree} 3 utith nodes tr, "' ,b(A: ^tr ( "' ( _t" : _L) which satisfies the cortditions

f (t;:xf i:1, "',tr'

Thea for any C-feriodi,c functiong: ^[0, L] -R ^whhh sati.sftes the condi'tions

g(ta)-)9,i:1,"',r,

the following ^holds:

J

ltrrtlF ^dts

lgttlrat

The eqaali.ty hold,s if and onU if f : ^g.

Ileftrition 2,4 (AParanetric Periodic Spline Mapping) A mapping F:(f, ^g) : [0, 1] * ^{R2 is}

a parametric

sfline mabPtng of degree Swith nodes tr, "' _,tz where 0 : _tr ("'( tz :

f. if the functions t [0,1] - ^{B and g: [0, 1]} * _{R are} periodic spline functions of degree 3

with nodes t4 "',ta.

3. Preparartion II ^: Nonlinear Programming

NCIil,werecall somenonlinearprogrammingtechniques. Let o :R - ^{Rand Fa: R"} r _&

i : _{1, "' , m be} C'tappings. Define the set ^O as follows:

O : _{X :(Xr,... _X") eR" _I Er(X) 30,i:1, "',m}

Deftrition 3,1 (An Isolat€d Local Minimum) X* e R" is an isolated lacal minimam inthe set O with respect to o if there exists 6 > ^{0 such} that

o (x*)<o (x),)s + v Xe,o n B(x*;0),

where ^B (X*; 6): {X e R" | (.> (X*,-X)T/2< 6 }

An Example of a Stable and Symmetic Curve

助山曖饉

m3。

K¨

α

^l,… ,XnD∈

1,…

R̀È∞

È∞

If the above condition holds9 R Is called the Kuhn‐

尭 σ ∞ ^+謄 _lL尭 鴫 ^=Qk=L… Ⅲ

∂

∂

∞

^R̀È∞

el)

Let X∈

as

I∞ = IIα

Э

=

r/

6.カ

Ω and let R∈ Rπ bethe Kuhn‐ TuckeF VeCtOr associated with X.Dttne the sets

{i∈ {1,・・・

,m}IE″

{i∈ {1,"。 ,m}IR̀>0},

{Y∈

Èoo Y=Q Vi∈

CЮ

▽È(DY≦ Q Vi∈ ^{{j lj∈ I(D,j￠}

^I100}}, where▽ È∞

_棺

^。

^{È∞ ].}

Theorem 8.8 (lbe $econd Order $ufficlent Condition) [2] ^{Sufgose thatY (Xr, "' , X") e R"}

andV ^(R*r, "',R*.) ^{G Bn"} satisfytheKahn-TorckerCondi,tion(3.1).LetH6",R) bethenn.matrir

whose (k, f)umiorwnt ^is

Yι

Hα

Y>0, 0≠

Vα

thenY is an isolatd, local rninimam in the set A u,ith respect to

4. Ilefinitions

Ihfinition ^4.1 (A Closed Plane Cbve) A continuous mapping F:6, ^g): [0, 1] * _{R' is a closd}

frtawcan)eif fe:[0,L] *BareC2-periodicfunctionswith(f(t)'+ Gi(t))'*0for ^{anyt €}

[0, 1] . A ^{point (A y)} of Rz is a crossing goint of F if tlere exist tr, tz e [0, ].), tt * tz such

thatF(tr):F(tz):(:;y).ApointofF([0,1])whichisnotacrossingpointisaregalarfioint.

A crossing point (a y) is a double crossing foint if the set F*l ^(x,y) n [0,1) contains o<actly

two elernents.

Misako Yoroyeue

We assume that any pomt of F([0,1])iS either a renar pOint or a double crossing point v7here F([0,1])intersects transversdy(io e.fOr each double crossing pomtは

,,Of F=CD

such thatに

^∈

^≠し ,the vectors lfelゝ 」

0)

are linearly independenO。

D面

^m4。 21AnArcl LetF:10,1]―

∞ mponent of{F([0,1])一 {CrOSSing points of F}}is an  ^{α π of F。}

輸 価腱43 1A Specinc Pointl Let F:10,1]―

α鵬亀

ぜ

十

Й例レタα

aψ _{蒻θ夕蒻グ} of F ifitis a crossing point or a middle point of F.

珈 血

問哺

劇れ

磁″′ 客 απ″

if it is a circle which is the boundary of the closure of a bounded,connected component of

{R2̲F([0,11).

D面 壼億

,n‑1}st F(iめ ^≠

R2and F=CD:[0,1]一

Put

{tl,・

・ ・

}=F 1({all Specific points of F}) where O=伍 <¨ ^・

Ψω  if the following holds:

(1)t̀=(i‑1)/1n‑1),i=1,…

,n;

(atherettsts anorientationpreservinghom∞ morphismh:R2̲R2suChthathO([0,1]))=

F([0,1]);

431 the area Of each o五

di宙

141 F o)=10,0.Put lXo,yol=Flt

Note thatf and g are C2̲periodicfunctions by the definitiOn Of a dosed plane curve.

D面

面仙 m461The Enewl We define the̲ρ

[0,1]―

R2aS fO1lows:

ρC)=∫lば^{'(0)2 dt+∫} lば'ctl)2&

沢物%α滋

47 LetF,Fl,F2:[0,1]‐

R2be C2̲mappings defined by

Fo=CO,2tl),

Flltl=C(0+X9go+yl,

F20=(∞

^{Sin  θ}

θ  fttl+∞

Ю

Then  ρ  C)=ρ

 lFD.

Am EXample of a Stable and Symmetric Curve

Ddh山

ねは4.81The Metricl Let F=cl,F2),G=(Gl,G2):[0,1]一 R2be C2̲mttpings.

We define the%ι ′π

d(F,G)of F and G by

d lF,G)=maxt=1,2 maXP=0,1,2 Supた Q」

カ )ltl一

の

ltl l。

Dtton4.9(Stable)Let N:[0,1]―

R2be a given dosed plane curveo A dosed plane curve F:[0,1]― R2is sれ み顔 ″

(1)F∈ ^Ψ

2)F is an isolated local minimumin the set 

ΨINl宙

ρ

>O such that

ρ lF)<ρ

(G),F≠

INl∩ B IF;ε

where B OL 

^ε )={G:[0,1]一

C2[0,1],dlF,G)<ε }.

5.Aュ Exalnple

Let N be as Fな 。 1.We find a stable dosed plane curve Fx*for N as follows:

Fむ。

P10

Sメ″ I:Putthe dmdy ordered specific points as Plメ 。,P13;See F尊1.Puttheirx‐

y∞

as follows:

P■ =P7=P13=10,0,P2=tXl,L),P3=P9=偽 ,0),

P4=偽 ,Xlo),P5=Pll=後

,X11),P6=偽

P8=偽 ,X13),P10=lX7,X14),P12=偽,X15),

where L>0。

Put X=α l,…

R=P7=P・

Misako Yoroyeue

S′″

2:COmpute the parametnc pe■

Fx((i‑1)/121=P̀,i=1,… ,13.

Put _σ∞

=165/51841 

 C⇒ ,then 

)is aS A.lin Append破

sttp 3:Put

Ml∞

=lthe area of the oriented polygon<PlP12P5P4P9P3>),

M2∞ =(the area of the o五ented polygon<PlP8P9P2>),

M3∞

=(the area of the odented polygon<P9P4P5P10>), M400=(the area of the o五

By Definition 45 131 we haVe

M′∞ ≧

1,i=1,23,4.

Put

E,(Ю

+2, i=1,2,3,4,

then E′ lXl≦ ^0。E′ ∞

^are aS

El∞

Xメ

E2∞

E3∞

XX14 X3Xll+X4X10 X4X■ 4+X,Xll, E4∞

Put

Ω={X=α

^l,…

R151Eグ

0,i=1,2,3,4}。

Sι″ イ

:The Kuhn―

th respectto X=α

R15,R=cl,・

R41∈ R4for ^σ

 lXl and E′

,i=1,2,3,4is as A2 in Appendk

Sι″5:We∞nsiderwhether there exists a sttmetric solution ofA.2.Let the three triangles

△ PlP5P9,△ P8P12P4and△Pr6P10 be all regular triangles which have the same bary centtr, and the three points P2,P8,P5be On the same line.Then we substitute the following in A.2:

Xl=X6=X4,X2=2X4,X5=2X4 X7,

草三 滋 ^1'=littf警 象

讐 F二 iJ3X3,

h this situation,we have only one solution X*and R*of A.2.Put a=21/2,b=131/2,c=

651μ,d=31ノ

ち

X*1=X*4=X*6=aC/b, r2=2ac/b,r3= (aC3d/

b)+(15ac/8b),

(ac/8b),XX7=(aC3d/40b)+(15ac/8b),

X*8=(aC3d/40b)+(aC/8b),X*9=(acd/4b)+(aC3/20b),

X*10=X*15= 15acd/8b)十 (ac3/4b),X*11=― (acd/b),

X*12=X*14= 15acd/8b)―

An D【

Step 6: Fx* is as A3 in Appendix It contains no computational error because we did not do approximation. To verify Fx* c ^q(N), it is sufficient to see that Fx* is equivalent to N. Fr*

( _t0, 1l ₎ is as Fig.2. Since Fx* is a parametric spline mapping, we can say Fr* e ^q(N).

Fig。

2

S″夕

7:We check whether the Second Su亜

R*1=0,R*2=R*3=R*4=20」^65。

I+α

り

り

Vα *)={Y∈ _{R151▽ L cЮ}

▽

L∞

20Y9‑20Y13+√ 65Y2=0,

‑20Y10+20Y■4+20√ 3Y3 20√

3Y7+√ 3√

65Y2+√65Y4=0,

3Y8+√ ^3√

^65Y4=0.

Since

YιH 

αち ^Rり Y≧

Vαり

YオHば ,P)Y=O implies Y=0,

6. Proof

Theorem 6.1 The aboue Fx*

a stable closd blane curae forN.

By definition

Then we have

and

We have obtained Fx*

Misako Yoroyaue

Praグ Put n=15.By Teorem 3.3 there mists  δ =O such that σ

∞

Ω ∩Bσ倅

Put 

ε   δ

n.Takeany G∈ ΨO)∩ BCx*;ε

by the specttic points of G。

If r=xt by the factthat G≠ Fx*and Theorem 24 we have

ρ

rr≠

σα')=ρ (Fxり ≦ ρ

We daim that r∈

Bα ^*;δ

Bcx*;ε _),we have

lX'′

― Xち

≦d Cx*,G)<ε ,  ∀ j=1,…

Therefore

‖

_α

^{― X*め}

1=1

<Σ   ^ε

_ε2=δ

1=1

Since G∈

ρ

 αP)<1518″

σ  α

ρ

This completes the proo■

oorouary 6.2二 ι′

_{助 工}

潔形

お α ^s″ み ルσ Jas″

″

̀π

η′

λ′ ε λ なれπ

′就

"″ ^減,%厖

^λ

^。

Rcttα tt aθ

Ach… 色.

The author had done this work when she bdonged to Kyushu University.She thanks

Hiroshi Ohtsuka for his introducing her the sofhare mu MATH,Hidefumi Kawasaki and

An Enanrple of a Stable and Symmetric Curye

1.Ahl挽 喀,J.H"Nilson,ENeandWalsLJoLデ

New YoFk and London,Academic Press 1967。

2.Fiacco,A.V.and McCo..鳳ick,G.P。

,'Nonlinear Programmmg9 Sequential Unconstraind

3.Fukuharap S。 ,Energy of a knot ln'切

■ le Fa℃

じademic Press l激 、 ^48‑451。

4.Grebile,T◆ N.B,"Spline Functions and Applications,"Wisconsin Universiけ

,1"0。

5◆ Holladayp J◆ C。

,Sm00thest curve approxmation,in"Mathematical Tables and Aids to Computation,Ⅱ

6.Kuhn,■ 。 w.and Tuに ,Ao W"Nonlinear Programmi略

^{疵 dey}

Symposium on Mathematical Statistics and Probabiliけ

7.い ,Ko Ro and Schmidt Do S。 (editOrs),"COmputer Aided PF00fS in Analysis,"SpFinger―

Verlag9 1991.

3.Sakumら 耽

^En釧

9.Ullrich Co leditOrl,"COmputer Arithmetic and Sdf― Validating Numerical Methods,"

Academic Press:1990。

100■

J"Family of energy Jmctionals of knots,Topology Appl.4819%147‑161.

Misako Yoroyelae

Appendix

σα

)=‑468Xう

^+1例

3 416X分

2XJ酔+14XX8 468XX5 104XX6 468XX7 468XX3+194X5X6 14Xよ 7+2XX8+

194X:X7+14X6X3+194X7X3+194X9X■

9X15 468X10X■ 1+194X10X12+14X10X13+2X10X14+14XloX■ 5 468XlIX12‑104XllX13

468X■ lX14 468XllX15+194Xl♪ Q3+14XlX14+X12X15+194XlX14+14XlX15+194X14

ttx42+31lX52̲31lX62 311X72+31lX82+31lX92+

31lX102+622112+31lX122̲31lX132+31lX142+31lX151 JL2

621‑468L+194L‑1嘔 +14L+2X6+14し

+1例

一 晨渇Xl+1248X2 468X3+410L‑1 臨

‑468X6 468X7 1創

X9R2+X10Rl―

XloR3 X13RI+X13L+X14R3=0,

194Xl一

XllR3=0,

一IMXl+416X2‑468L+1248X4 銀

^1創

‑468X7 468X8 X10Rl+X■

X12R4 X14R3+X15RI― X■

5R4=0,

14L‑1創脇

+1創

2X■ ‑468L+14L‑1創 脇

+1例

lαl‑468X2+2L‑468X4+14L+1例 臨 +622X7+194X8+XllR3=0,

194Xl二

XllR4=0, 622X9+IMX10‑104Xll+14X12+2X13+14X14+194X15 X2R2=0,

194X9+622X10‑468X■ 1+194X12+14X13+2X14+14X15+X沢

X沢

0,

‑1

_X10+12蟷

X7R3 X3RI+X8R4=0,

14聰

+194LO‑468X11+622X12+19Ⅸ 13+14脇 4+2X■

4+14X15 X2Rl十

14聰

+2Xl。 ‑4渇

13+194X■

2Rl tt X2X10RI―

X2X13RI+X3X1lRI― LX10RI+X4X15RI― X8X1lRl=0, 2R2‑X2X8R2+X2X13R22=0,

2R3 X2XloR3+】

R′ ≧0,i=1,…

_,4 È∞

0,i=1,…

JL3

(abcd/6)fx*￨【

i‑1)/12,i/12]

=‑18t+2dc2t+36dc2t2̲144dc2t3+324t'一 LD6tt ifi=1,

An EXample of a Stable and Symmetric Curve      ll

=‑18t+2d♂

^‑144d♂ f+34♂ ^‑1以

, if i=2,

=‑23+96t一

dc2+20dFt‑72d♂

+72d♂

=112‑1224t一

+20d♂t‑72d♂

+72drf+4320F‑4968F, if i=4,

=‑256+488t一 ^d♂

+20d♂t‑72d♂

+72d♂

=369‑2412t一 ^d♂

+20d♂t‑72d♂

+72d♂

+5184/‑3672f, if i=6,

=‑22+1314t+26dd‑142d♂

^t+252d♂

^‑144d♂

2268F+126t3, if i=7,

252+1314t+26dP‑142d♂ t+252d♂

_, if i=8,

=1220‑5310t‑38d♂

+146d♂t‑180d♂

+7ぽ

F″

_, if i=9,

=‑2425+9270t‑38d♂ ^{+146d♂ t‑180d♂}

^+72d♂

=3325‑11430t‑38d♂

+146d♂t‑180d♂

+72d♂f+13068f‑4968f, if i=11,

=― 認∞+10mt― ^優通♂+146d&‑180d♂^′

+72d♂

10697+3672t3, if i=12.

(abcd/61 gx*￨[(1‑1)/1a i/121

=6dt‑324d♂ ‑2880df+6c2t― ^{野 ′}

, if i=1,

=lχttt‑1116dF+2880d♂

‑36J′

, if i=2,

=‑288dt+1368d′ ^{― 卿 ず}

+2に

‑144P′

^2t3+K59/助

,if i=3,

=252dt‑792df+792dt3+2に

̲144P′ +216ぽ

d―

=252dt‑792dザ +792df‑120Ft+288J′ ^‑216♂

, if i=5,

=‑1248dt+2808dP‑208d♂

2t3+183d+15♂ ,if i=6,

=2478dt一

‑4Bd‑11ば _,if i=7,

=‑802dt+5436d′

‑3

,if i=&

=30dt〒 4500dt2+2088dt3+ulCt‑468c■ 2+216c2t3̲r2300/31d‑76ct ifi=9,

=‑lmdt+1980dt2‑792dt3+330c%‑468c■ 2+216c%3+(1投

=‑lmdt+1980dt2‑792dt3̲570c先

^■

=562dt‑5940dt2+2088dt3̲570c■

2‑216c■ 3̲1770d+174ct Fi=12.