曲線デザイン指標としての曲率積分

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Curvature

lntegration

as

a

Curve

pm

wa

7ny-

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y]s

en

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KeioUniversity

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Design-lndex

lncurve

design,

controllingmacroscopic featurethat emerg-es

from

thetotalof shape elements

is

important.

However,

trialand error isrequired inorder to control the curved _profile

to

be

setreflectingmacroscopic

feature

imaged

by

designer,

becausethereisno usefulmethod forcontrollingmacroscopic

feature

intheconventional computer aided designsystem. We proposedthe method

for

representation ofmacroscopic

fea-tureusing curvature integrationand multi-resolution

represen-tation.Thismethod was applied toshape-generation

for

the designof automobile side-view.As a result,itwas confirmed

thatthecontrol of macroscopic

feature

was

_possible.

In

the

present

study,itwas shown thatthe

_possibility

ofnew

design-support

in

curved

_profile.

1.

Introduction

Currently,acurve and curved surface shape are widely used

invarious industrial_products.Inconsidering new curved

pro-files,controlling theoveral[ shape featurethatemerges from

thetotalofshape elements isimportant.The main reason for

thisappears tobethathuman beingstendtoperceivethe

overall shape feature

_(such

as Gesta]t)macroscopically as

pointedout

in

cognitive psychology

[1].

However,

trialand errorisrequired inorder tocontrol curved

_profiles

tobe set reflectingmacroscopic featureimaged bydesigner,because

there isno usefulmethod forcontro]lingmacroscopic

feature

intheconventiona] computer aided designsystem.

Quantitative

representation ofmacroscopic

feature

is

impoF

tanttocontrol macroscopic

feature.

Howevec representation of macroscopic

feature

is

difficult

using conventional micro-scopic shape information

(such

as

dimension

and cuwature),

because

themicroscopic shape

information

represents the partialfeatureof curved _profi[e.If_quantitative representa-tion of macroscopic featurei$ rehlized, a

designer

can control

macroscopic featureindirectlybyshape-generation that

uti-lizedsearch algorithm

(such

as _geneticalgorithm) including method forrepresentation of it,Therefore,incurve design,

themethod forrepresentation of macroscopic featureand the

design-supportsystem thatcan control macroscopjc feature

by

designer

is

desired.

Inour _paststudy, we had _proposedthemethod for repre-sentation ofmacroscopic feature[`complexity"

using curvature

integration,and the effectiveness of themethod had been confjrmed invarious curved profiles

(such

as basiccurved

profilesand existingautomobile side-views)

[2-5].

"Complex-ity"'affectsevaluation ofthe

important

itemon

design,

such as "beauty" _and "similarity",

Moreover,

it

is

possiblethatthe

quantification

of "complexity" _{using the amount of}

physics

computed

from

curved profile,

because

there

is

litt[e

individual

differenceofevaluation for`"complexity.i'

ln

the

present

study,

firstly,

we proposed themethod

for

representation of macroscopic feature`tcomplexity'' _using cur-vature integrationand multi-resolution representation.

Multi-resolution representation was utilized for_preventingthe

geneF

ationofcurved

_profiles

containing theswellthataman cannot recognize, Next,thismethod was applied toshape-generation

forthedesignofautomobile s]de-view describedbycubic

Beziercurve, and thepossibilityofcontrolling macroscopic

feature"comp[exity" _{was verified.}

2.

Curvature

lntegration

lntheknowledgeofthestudy about the

`'complexity''

in

out-lineshapes, the number ofvertices isclted as one of impoF

tantfactorsofthe

`'complexity"

[6,7],

A

vertex isthe feature point

for

astraight-Iineprofile,

The

feature

point

in

thecurved

profile

is

equivalent toa

high

curvature point

[8],

Therefore,it

is

considered thatthenumber of

high

curvature pointscause

`[complexity"

_inthecurved

profile,However,inorder that cur-vature changes continuously, the thresholdthatdividesa high curvature _pointand the other _pointisneeded as a _parameter.

Inour paststudy,thenumber of highcurvature _pointswas not computed using threshold,buttheintegrationof theabsolute curvature was computed as curvature integration.Thisvalue is

known one of the_{globalproperties}of curved _profileinthe

dif-ferential_geometry

_[9].

7Tfft)\msk=e

specialissueofjapamesesocietytorthesciemceefdesign

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Fig.1.Curvatureintegration

l

Curvatureintegrationfromcurvature functioninthecurved

profileiscalculated inthefollowingmanner. InFig.1,the

verti-cal axis

is

curvature K,the horizontalaxis jsthecurve iength

4

the curvature functionisK(l)and thetotallengthof cuwed

profile

is

L

Curvature

integrationiscalculated using the

fol-lowing

equation :

I=

i.

J,L

nc(l) dl

_(I)l)

_(1}

3.

Multi-Resolution

Representation

Smoothing was utilized for_preventingthe _generationof curved _profilescontaining theswell thata man cannot recog-nize,lnthjsmethod, _parametercontrols the size of the swell removed. Moreover,itiscalled multi-resolution representa-tionofshape tochange a

_parameter

tomany stages and to acquire theshape ofvarious reso]utions

_[1O].

Thereisstudy thatanalyzes the

_property

of shape basedon change of the amount of_physicsinmulti-resolutionrepresentation

_[1

1-13]. Themulti-resolution representation incurved _profileisbased on theview ofscale space _proposed byWitkin

_[14].

Inthis

method,

Gaus$ian

kernel

G(u,u)ofwidth a :

G(u,a)=SEa

exp

(-

i2,)

(2}

isused forsmoothing. o isthe

_parameter

for$moothing. A two-dimensjonal_planercurve

is

defined

in

the

following

equa-tion:

C(u)=(x(u),y(u))

(3)

Then,smoothed curve, X(u, u) and Y(u,aL arecomputed

bytheconvolution ofC(u)and G(u,a),and are

defined

as:

X(u,a)=x(u)XG(u,a)

=

l:

X(V)

I511;a

eXPC("iaV2)2

)

dv

(4)

32fifrf>\ffxksee

specialis$ueetjapanesesocietyforthescienceefdesign vel.15-4 no.60 2008 Y(u,a)=y(u)opG(u,a)

=

J-pm"s

_Y(V)

G;a

_exp

(-

("iaV2)2

)

_ch,

(5)

lnsmoothing byGaussian kernel,no new inflection_points are created at highersmoothing

[1

5].Therefore,thecurvature

integrationfunctionJ(o) computed

by

multi-resolution repre-sentationisamonotonicaliy decreasingfunction.

Forpreventingthegenerationofcurved profilescontaining

theswell thata man cannot recognize, thefollowingtwo meth-ods were _proposedinthe_presentstudy.

One

is

thatcurvature

integrationiscomputed after smoothing with arbitrary

param-eter. Inthismethod, multi-resolution representation isutilized

foradjustment of _parameter.The other isthat

index

S_that shows the robustness of

"complexity"

represented by

curva-tureintegration

is

newly proposed,

ln

thismethod, thecurved

profi]ewith theswellofthesizethatishardtoberecognized

is

removed using Sas

index.

S

is

defjned

as

following

equation using I'(a)

_by

multi-resolutionrepresentation.

S=II"(a) da

₍₆₎

Here,

I(u) was standardized as following:

J(a)-1

I*(a) =

(7)

I(O)-1

4. Applicationto

Shape-Generation

4.1

.

Constructionef

Shape-Generation

Method

The algorithm of _proposed shape-generation method is shown inFig.2

Based on $tudies of Tian

[l

6],the automobile side-view was describedas a _{polygonalprofile}consisting ofeight

basic

points

{Fig.

3),and defined

junction

_points

(Fjg.

4}for

descrip-tionbycurved _profile.Inconsideration of thefreedomofshape

descrip±ion,and thesimplicityof control, cubjc Beziercurve was used as descriptionofcurved _profileintheshape

genera-tionmethod

{Fig.

5),The automobile side-view

(Sedan)

was used as theinitialshape, and thecurve control variables

(the

position

of thebasic_point,thedirectionof a tangentvector, and thesizeofa tangentvector)were changed inthe

shape-generation.

Then,

movable ranges of curve control variables were definedfor

_preventing

shape

_generation

from_generating

acurved _profile

having

a self-intersectionand cusp. The mov-able range ofbasic

_points

isshown inFig,6as an example.

NII-Electronic Library Service Descr]ptionefmTtialshape Gg.n.,.e,..!j,.g.,.a..!,g.g.,F.L/.h.,]p.v

lIII･

l･l･l･l'

Preparationofinitialpopulat{on Genotype-Phenotype (Deferrnationefinltialshape) CaieulationofCurvaturelntegr:itlenl CalcutatToneffitnessA MeetendeonditiofiNe Yes Selection,Cressovcr,MutationEnd

'li･l-l･lI･l'l

'

I

'

Fig.2,Algorithmofshape-generation method

E, F., 4F, 4, 4e l"s Fig.3.Basicpoints Grqk Jg i'1 4 4 rlFig.4. A･4 rs hr] JLe J]1rGJEI_J/ JlJunction points G hJrdo Ju Jis J'1J]aJisrs J,4

･h4･h

Ji2JisJ14Jis Fig,5.

,J/lnterpolation

byacubic Beziercurve

Fig,6.

-InitiaTShape

ShapeafterDeformation BasicPointTransferVecter

Jlfi

Movable range ofbasic_points

[ntheshape

generation

method, a geneticalgorithm

(GA}

was used as a search algorithm. GA isa search algorithm

imitatingtheevo[ution _processofalivingthing.GIobalsearch

isattained inorder that _parallelsearch by many indMduals

is_performed,The curve control variables were manipulated

inthe shape _generation.The chromosome forGA was

com-posed oian arrangement ofthenumerical vafiues forthis ma-nipulation. Thefitnesswas theabsolute va]ue ofthedifference

between

curvature

integration

of an individualand curvature

integrationthatthedesignerset. Crossoverwas handledinthe manner

described

by

Obayashi

[1

7].The random weighted mean ofthereal number variable was used, OtherGA param-eterswere referredto

DeJong's

standard parameter

[1

8].

4.2.

Shape-Generation

lnshape-generation, the amount of change incurvature

integration

was setas

four

levels

of

-O.25,

+O.25, +O,50, and +O.75 based on the result of analysisabout therange of change

in

curvature

integration.

The

presentationofsamples

isshown inFig.7.As a resultofanalyzing the relationship

between

curvature

integration

and F`complexityi',

it

_was

found

thatbotharehighcorrelation.

However,

inacertaingenerated

shape, although theva]ue ofcurvature integrationwas high, thevalue of `tcomplexity''

was

low

(Fig.

8).

The

generated

shape

(a)

towhich curvature integrationand

"complexity''

correspond was extracted, and itwas compared with the

_generated

shape

(b)

towhich curvature integration

and [`complexity" _{not correspond}

(although

_curvature

integra-tion

is

equal, evaluations of

"complexity"

differgreatly).Asa result,

it

was confirmed thatthegeneratedshape

(a}

contained the swell of the size recognized enough, and, on the other

hand,thegeneratedshape

(b)

contained theswell of the size which ishardto berecognized

(Fig.

9).Thistendency was

foundinthe whole sample. Therefore,itwas confirmed that

the

importance

ofpreventingthegenerationofcurved profiles

containing the swell thataman cannot recognize.

4.3. Mutti-ResolutionAnalysis

Multi-resolutionrepresentation was applied tothe_generated shapes

(a)

and

(b)

shown inFig.9,and J(a)was computed. In

Fig.1O,the vertical axis isI{cr),thehorizontalaxis iso. More-over, Fig.11shows thesituation of smoothing intherange of o towhich J(o) decreases_greatly.The swell of thesize which can berecognized easily issmoothed intherange of cr=O.O1

too=O.1 inthe

generated

shape

_(a).

On theother hand,the

swell of thesize which ishardtoberecognized issmoothed in

T-vayvffnk=･g speciatissueefjapanesesocietyforthescienceofdes[gn vol.15-4 ne.6e 2008 NII-Electronic Mbra

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Japanese Society for the Science of Design JapaneseSociety for the Science of Design

AI--O,25 _{Al!!+O.25 Initiai}shape

di = + O.75 zilz+O.50 At=+O.75

di!=+O.25

Fig,7.Presentationofsamples

At=-O.2S Al=+O.SO

Regressioncurve:y!=3.g5kiCx) + I.37

Contributjonratio :Ri ; e.62

Signifieancelevel:e.OO

s se.

(a)xx'

(b) -X,"'"i 5.ij 4tif382 1 is N''ttts. iji _i

i''

''"''

'ttttt

tttttttttttl/

SweU(largescale)

"'tT･c,"/11'11---

-l,

,i" i/ /...1/ O I 2 3 CurL,atureintegratien

Fig.8.Reiationshipbetween curvature integrationand complexity Fig,9,Swell

(large

scale and small scale)

Swell(smailseale)

therange of o=O.OOI tou=O.Ol inthe_generatedshape

(b).

Fig.12 expands a _partof the_generatedshape

(b)

inFig.11.

Aboutthe1stmethod _proposed in

Chapter

3,

in

order to

calculatethevalue of suitable o, thecorrelation coefficient R

between thenatural logarithmof l(a) and "complexjty" _in

generated

shapes was computed toevery a

(Fig.

13}.Asa re-sult,R

became

thehighestwhen a was settoO.02,as shown

jnFig.

13,

Howeve4 thisknowledgeisrestrictedtothe experi-ment conditions inthepresentstudy. Inorder toutilizethjs method, itisnecessary tonewiy buildthemodel thatoutputs

thevalueofsuitableo

.

About

the

2nd

method

_proposed

inChapter3,inorder to calculate thevalue ofsuitable S,thecorrelationcoefficientR

between

thenatural

logarithm

of Iand "complexity" _in

gener-ated shapes removed inthesmall orderof Swas computed to

every S

(Fig.

14},

As

a result,

it

was confirmed thatR becomes highas _generatedshape was removed inorder withthesmall value of S.Ifthe value ofS

is

setmore highly,therobustness of

the`tcomplexity'' _represented

_by

_{curvature integration}

wiil be-come

highen

Therefore,

theadjustment ofSforremove is con-sidered to

be

easy as compared withthatofsuitableo. How-evec itisnecessary tosmoothing repeatedly forcalculationofS, and calculationcost becomes high.

About the comparison of above-mentioned two methods,

itwill verify invarious application

from

now on, ]nthe

pres-ent study,themethod

for

preventingthegenerationofcurved

profilescontaining the sweMhat aman cannot recognize was

proposed,and the effectiveness ofthismethod

in

the

shape-generationwas confirmed.

5. VehicleDesign

lnthepresentstudy, the _proposedshape generatjon meth-od was applied tothedesignof anew vehicle fora wheelchair user at KeioAdvanced DesignSchool.First,considering the

human space, theinitialshape was set as shown inFjg.16,

Next,as shown inFig.17,we created diversecurved _profiles

34T-ifo\ffxkke

specialissueofjapanesesocietyforthescienceofdesign

NII-Electronic Library Service

(a)

o)

Abw 2.001.751,SOL251,OO Fig.10.

/''/''{'Ell''''

///''''/''iIl

'

'''/''''

''''''''ll}1/li

･1tt

111''iI '''l/ ttr4lo.a10-210-t a

Multi

-

resolution analysis 1 Ioo 2.ee:.7StL.b 1,5enyl.25l.oolo41tr110-2ffle-1feo

(a)

(b)

_(b)

.f--Y'

/1'.v/ o

==

o.oee

,

J= 1.7so a=o.ofo _,I

±

1.6ss a == e.ooo_, I=1,7so cr

==

O.OOI, I=.l,642

..ta{-ii

.-'/

ftt f' gl l'i s' sri o=e.040,l=t.436 cr = O.070_.I= 1.27e cr = e.O04

,

I= 1.348 i : cr sc o.eooI=: !.750 a

==

o.oo7, J=1.17s

l""'""""""""'I

aaO.OO1i==1.642

ii'-"""""ii

i lr""""""",

/

1 1/ I

t

/

/

/

/

/

I i i i Fig.lla

!.

e.loe

.

I

±

1.17s

.

Multi

-

re$olution analysis 2a=oolo,f=Los3 Fig.12.

a

==

e.O04l=1.34Sa=e.eo7I=

l.175

Multi

-

resolution analysis3

a

==

O.e1Ot=1.083 Fig,13. I.O O.9ec e.s e.7 O.6 1/'''''''t/'/1/1 /1/1''{''''/'/1/t11''tt//

'

''I' '''''''1'//1'll//

''''''tl/1 tl''''''/'/'/////1/1//t/tt

''''''':il''''''''1/l,

'''''''''''''//'111'11'''''//11''''1'/'/1t//1t/

'''''''//'''''''/11////'I'''''//'//11'''1t//1t/

o.eeol o.eole e.oloe o.'loool.oooo cr

Relationshipbetweeno and correlation coefficientR Fig15,

1.0 O.9ce

O.8 O.7 e.6

o.oel o.olo o,loe

s

RelationshipbetweenSand corre[ation coefficient

(a)

(b)

l,O O,9 O,8". D,7b Q,6w e,5k o,4 O.3 O.2 O,1 o.o o,o e,] ff

Fig,14.Multi

-

reso[ution analysis 4

' '' ll{I/ -{ / IL/

lll

ii

!･iiSttO,065,

l/

///

/

I/ll'

/'{-Nv/ /1･･l/1//e.2 ]..o e.g o.srt. O.7b O.6ve.s"-' . O.4 O.3 O.2 O.1 o.e [t/{li /{//iI1' ''/''1' I" " /='i/ls--o.oosl/n1// / _li

''Ii

/ lIili ' 'I/ll

O.D O,la O.2

i=lfly\mRkeeg

specialissueeflapanesesocietyferthescienceofdesign

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35

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Japanese Society for the Science of Design JapaneseSociety for the Science of Design

using the shape _generationmethod. AtKeioAdvanced De$ign

School,one shape

U

±1

.50,

"1 :i,

in

Fig.

1

7)

_{was seiected}

from

the _generatedshapes, Based on the se[ected shape, rough model

by

3D

CG

was carriedout asshown

in

Fig,

18

and the 1flOscale mock-up was _producedas shown inFig.

I

9.

6.

Conclusions

lnthepresentstudy,themethod forrepresentation of mac-roscopic

feature

"complexity"

using curvature integrationand multi-resolutionrepresentation was _proposed.And,this meth-od was applied toshape-generation forthedesignof

automo-bileside-view.Asa result,itwas confirmed thatthecontrolof macroscopic feature"complexity"

was _possiblebyuse of this method as shape-generation index.Itwas shown thatthe

pos-sibilityofnew design-supportincurved _profile.

Thiswork was supported byGrant-in-AidforResearch Fel-lowof theJapan SocietyforthePromotionof Science.

References

1

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PolanyiM. ThelacitDimension,Routledge&Kegan Paul

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966}.

2. Ujiie

_Y

Matsuoka Y Shape-Generation Method

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4. Ujiie

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Matsuoka Y MacroscopicShape-lnformationas A

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Ujiie

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MatsuokaY fotalAbsoluteCuwature toRepresent

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E

gs

.---F---+---F---t---:I :I lI Fig.]6, H

/

/

/

/

/

/

/ 53e / 1..'oo /4oo

/ 1 /

'Movable

range ofbasic_pointson initialshape

I!1.00 J;1.2j 1 iii: Initiaishape(i=1,OO} 2 3 JiT.50 1 2 3 i Examples of_generatedshapes

? Fig,17. Generated$hape(selectedi) Fig,18, il･l" Hg.19.

>

Top /t

t

t/tftt

tttttt

Froml Roughmodelling by3DCG Mock-up ix' 3 L)erspectivc '

c.fi='ge!l

Sidc t.

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_E

Mackworth A.K.Scale-Based Description

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_E

Mackworth

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992).

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A.R

Scale

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TianM,Sugiyama K,Kamaike M, Watanabe M.A Car

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specialissueotjapanesesocietyforthescienceotdesign

vol,15-4 no.60 2008NII-ElectronicMbra

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