西工大数模公园内道路设计问题.docx

西工大数模公园内道路设计问题.docx

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西工大数模公园内道路设计问题

FormversusFunction:

WalkingtheLine

AmrishDeshmukh,NikoStahl,TarunChitra

November18,2008

1

Abstract

TheArtsquadisaspacethatissubjecttoseveralcompetinginterests.Itsdesignisexpected

tooffertheopportunityforanindividualtomoveconvenientlyfromoneendtoanother,tokeep

generalmaintenancecostsfeasibleandtoallowforactivitiesrangingfromlecturestosnowballfights.

Thispaperproposesamathematicalmodel,whichprovidesawayofintelligentlydesigningthe

pathnetworkofthequad.Thecenterpieceofourmodelisacostfunction,whichevaluatesthe

feasibilityofagivenpathconfiguration.

Toexplorethesetoffeasiblepathconfigurationwewroteanalgorithmthatrandomlygenerates

samplesofthisset.Wethenimprovedonthissearchbyconstructinganoptimizationalgorithm

inspiredbyMarkovchainMonteCarlomethods.Webelievethisimprovedsearchhasfoundalocal

minimumpathconfiguration,asitappearsstableunderperturbation.

2

Contents

I

II

III

IV

V

VI

ProblemStatement

TheArtsQuadrangleasaGraph

TheLengthofaPath

UnofficialPathInducedbyHumanBehavior

TheCostFunction

Assumptions:

4

4

6

7

8

9

VII

VIII

IX

AlgorithmsforFindingOptimalPathConfigurations

Results:

RecommendedSolution

9

10

11

X

XI

FutureWork

Bibliography

3

15

16

PartI

ProblemStatement

ThetaskistoredesigntheArtsquadwalkwaysusingamathematicalmodelthatwillhelpus

determineapreferreddesign.Beyondthegeneralfactthatminimizingthetotallengthofthepaths

andmaximizingtheareasofcontiguouslawnsarepreferable,weareaskedtoconsiderthefollowing

criteria:

·Pathmaintenancecosts

·Landscapingcosts

·Pedestriantrafficandbehavior

·Thecreationofunofficialpathsanditsimpactonthelawn

·Thegeneralappealofthequad

Toimplementthesecriteriainourmodel,weareprovidedwiththefollowingprinciples:

·Thepathmaintenancecostisproportionatetothetotalpathlength.

·Thelandscapingcostdependsonthenumberofcontiguouslawns,thecreationofunofficial

paths（asaresultofpedestriansleavingthepavedpathstoarriveattheirdestinationmore

quickly）andthegeometryofcontiguouslawn.

·Ifthepathbetweentwopointsis15%longerthanthestraightlineconnectingthepoints,a

pedestrianwillleavethepathandcutacrossthequad.

·Anaveragepedestrianmightleavethepathifitimpliessavingmorethan10%ofthetotal

lengththepath.

PartII

TheArtsQuadrangleasaGraph

Graphtheoryhasbeenanimportanttoolinexploringproblemswhichrangefromdeterminingthe

neuralnetworkofnematodeC.eleganstofindingthecauseoffailureinelectricalpowergrids1.

Byframingourwalkwaydesignprobleminthelanguageofgraphs,wecanreadilyextractthekey

relationshipsbetweenstructureandfunction.

WedescribetheArtsquadrangle（herebyreferredtoastheArtsquadorsimplyquad）

asagraphof10nodes,whichrepresentthemostcommonpointsofentryandexittothequad

（seefigure1below）.

1See

[1]

4

Figure1:

CornellArtsQuad

LetthesetofnodesbeA={x|x∈{1,2,...,10}}.Nowwecandefineapathtobean

orderedpair（a,b）andthesetofallpathsastherelation

R={（a,b）|a,b∈A,a=b}

（1）

sinceeverypairofdistinctnodeswilldefinealinesegment,orone-waypath,intheplane.Thenthe

setofallpossibleconfigurationsofpathsisgivenbythepowersetP（R）.Thissethas290elements

（thecardinalityofapowersetofasetwith90elements）

Thispresentsanoverwhelmingsetofpossibilities,butfortunatelytherearethreeconstraints,

whichweimposedtomakeoursetlessunwieldy.Wewillonlymodel:

1.Non-directedgraph:

Currentlythespace（1,3）isdistinctfromthepath（3,1）.Wefindthisto

beunreasonableaspedestrianpathsareveryrarely“one-way”

2.Connectedgraph:

Aestheticallyandfunctionallyitmakeslittlesensetoallowabuildingtobe

surroundedcompletelybygrass.Furthermore,wepickedthetennodesbecauseweconsidered

themtobeessentialcirculationpointsofthequad.Therefore,havingoneofthemdisconnected

fromthenetworkwouldbeunreasonable

3.Graphsincludingtheperimeter:

Thisisagainchoseninlinewithouropinionsonaesthetics

andutility.Whilepedestriansarelikelytoacceptlongerdistancesthanastraightlineto

remainontheofficialpath,itseemsunlikelythatapersongoingfromAtoBwillabidewith

apaththatstrictlyincreasesthedistancetoBbeforeallowingthepedestriantoactually

approachB（seefigure2below）.

5

Figure2:

Inthefirstgraph,weseethattravellingfrom（0,1）→（1,0）neverincreasesthedistance

from（1,0）,whereasinthesecondgraph,g