mathematica80制造的美丽图案doc.docx
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mathematica80制造的美丽图案doc.docx
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mathematica80制造的美丽图案doc
IntroductiontoManipulate
ThesinglecommandManipulateletsyoucreateanastonishingrangeofinteractiveapplicationswithjustafewlinesofinput.ManipulateisdesignedtobeusedbyanyonewhoiscomfortableusingbasiccommandssuchasTableandPlot:
itdoesnotrequirelearninganycomplicatednewconcepts,noranyunderstandingofuserinterfaceprogrammingideas.
TheoutputyougetfromevaluatingaManipulatecommandisaninteractiveobjectcontainingoneormorecontrols(sliders,etc.)thatyoucanusetovarythevalueofoneormoreparameters.Theoutputisverymuchlikeasmallappletorwidget:
itisnotjustastaticresult,itisarunningprogramyoucaninteractwith.
ThistutorialisdesignedforpeoplewhoarefamiliarwiththebasicsofusingtheMathematicalanguage,includinghowtousefunctions,thevariouskindsofbracketsandbraces,andhowtomakesimpleplots.Someoftheexampleswillusemoreadvancedfunctions,butitisnotnecessarytounderstandexactlyhowtheseworkinordertogetthepointoftheexample.
Despitethelengthofthistutorial,itisonlyhalfthestory."AdvancedManipulateFunctionality"providesfurtherinformationaboutsomeofthemoresophisticatedfeaturesofthisrichcommand.
ManipulateIsasEasyasTable
Atitsmostbasic,thesyntaxofManipulateisidenticaltothatofthehumblefunctionTable.ConsiderthisTablecommand,whichproducesalistofnumbersfromonetotwenty.
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SimplyreplacethewordTablewiththewordManipulate,andyougetaninteractiveapplicationthatletsyouexplorevaluesofnwithaslider.
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IfyouarereadingthisdocumentationinsideMathematica,youcanclickanddragtheslidertoseethedisplayedvaluechangeinrealtime(meaningthatitchangeswhileyouaredraggingtheslider,notjustwhenyoureleaseit).Ifyouarereadingastaticformofthedocumentation,youwillseetheslidermovedtoanarbitraryposition.(Bydefault,itstartsoutontheleftside,butinthefollowingexamplesthesliderhastypicallybeenmovedawayfromitsinitialposition.)
InbothTableandManipulate,theform
isusedtospecifyan"iterator",givingthenameofthevariableandtherangeoverwhichtovaryit.
OfcoursethewholepointofManipulate(andTableforthatmatter)isthatyoucanputanyexpressionyoulikeinthefirstargument,notjustasimplevariablename.MovingthesliderinthisverysimpleoutputalreadystartstogiveanideaofthepowerofManipulate.
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Again,ifyouarereadingthisinastaticformyouwillhavetotrustthatthegraphchangesinrealtimewhenthesliderismoved.
Notethatthesliderhasanextraiconnexttoitwhich,whenclicked,opensasmallpanelofadditionalcontrols.Here,thepanelfromthepreviousexampleisopened.
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Thepanelallowsyoutoseethenumericalvalueofthevariable,aswellassetitinmotionusingtheanimationcontrols.
Ifyouwanttoseethevalueofthevariablewithouthavingtoopenthesubpanel,youcanaddtheoptionAppearance
tothevariablespecification.(Notethenumberdisplayedtotherightoftheplussign,whichisupdatedinrealtimeasthesliderismoved.)
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ThisisalsothefirsthintthatManipulategoesfarbeyondtherelativesimplicityofTable,bothinitsoutputandintheflexibilityandrangeofwhatcanbespecifiedinthelistofvariables.
JustlikeTable,Manipulateallowsyoutogivemorethanonevariablerangespecification.
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Youcanhaveasmanyvariablesasyoulike,includingsomanythatasimilarTablecommandwouldtrytoenumerateanunreasonablylargenumberofentries.
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Youcanopenanyorallofthesubpanelstoseenumericalvalues,andyouarefreetoanimatemanydifferentvariablesatthesametimeifyoulike.
OnewaytothinkofManipulateisasawaytointeractivelyexplorealargeparameterspace.Youcanmovearoundthatspaceatwill,exploringinterestingdirectionsastheyappear.Asyouwillseeinlatersections,Manipulatehasmanyfeaturesdesignedtomakesuchexplorationeasierandmorerewarding.
SymbolicOutputandStepSizes
Thepreviousexamplesaregraphical,andindeedthemostcommonapplicationforManipulateisproducinginteractivegraphics.ButManipulateiscapableofmakinganyMathematicafunctioninteractive,notjustgraphicalones.
Oftenthefirstissueinexamplesinvolvingsymbolic,ratherthangraphical,outputisthatyouwanttodealwithintegers,ratherthancontinuouslyvariablerealnumbers.InTablethedefaultstepsizeis1,soyounaturallygetintegers,whileinManipulatethedefaultistoallowcontinuousvariation(whichyoucouldthinkofasastepsizeofzero).Comparethesetwoexamples,andnotethatManipulateallowsvaluesinbetweenthosereturnedbyTable.
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Functionsinvolvingalgebraicmanipulations,forexample,oftendonothinginterestingwhengivennonintegerparametervalues.ThisExpandfunctionneverexpandsanything.
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Fortunatelyitistrivialtoaddanexplicitstepsizeof1totheManipulatecommand,yieldingexactlythesamesetofpossiblevaluesinManipulateasisreturnedbyTable.
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Withanexplicitstepsize,theExpandexampleismuchmoreinteresting.
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ThefactthatonlyonevalueisdisplayedatatimeallowsyoutocreateexamplesthatgofarbeyondwhatwouldbepracticalinaTablecommand.AnimportantpropertyofManipulateoutputisthatthereisnofixedpanelsizeorarbitrarylimitastohowlargetheoutputpanelcangrow.
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(Inprintedformsofthisdocumentation,thesliderissetfairlylowtoavoidwastingpaper,butwhenmovedallthewaytotheright,theoutputsmoothlygrowstocovermanypagesworthofverticalspace.)
AswithTable,ifyouuserationalnumbersfortheminimumandstep,youwillgetperfectrationalnumbersinthevariable,notapproximaterealnumbers.HereisanexamplethatusestheformattingfunctionRowtocreateasimpleexampleofaddingfractions.
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Youcanevenuseendpointsandstepsizesthataresymbolicexpressionsratherthanjustplainnumbers.
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TypesofControls
Manipulatesupportsawiderangeofalternatewaysofspecifyingvariables,whichgeneratedifferentkindsofcontrolsforthosevariables.Thisincludescheckboxes,popupmenus,andothersinadditiontosliders.
Theprincipleisthatforeachvariable,youaskforaparticularsetofpossiblevalues,andManipulateautomaticallychoosesanappropriatetypeofcontroltomakethosevaluesconvenientlyavailable.ForatypicalnumericalTable-likeiterator,aslideristhemostconvenientinterface.
Youmight,ontheotherhand,wanttospecifyadiscretelistofpossiblevalues(numericorsymbolic)ratherthanarange.Thisisdonewithaniteratoroftheform
.
(Notetheextraleveloflistcomparedtotherangespecification.)Ifyouaskforasmallnumberofseparatevalues,youwillgetarowofbuttons.
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Ifyouaskforalargernumberofdiscretevalues,Manipulatewillswitchtousingapopupmenu.
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IfyouusethespecificvaluesTrueandFalse,youwillgetacheckbox.
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Thesechoicesareofcoursesomewhatarbitrary,buttheyaredesignedtobeconvenient,andyoucanalwaysoverridetheautomaticchoiceofcontroltypeusingaControlTypeoptioninsertedintothevariablespecification.(ThefulllistofpossiblecontroltypesisgiveninthedocumentationforManipulate.)
Forexample,youcanaskforarowofbuttonseveniftheautomaticbehaviorwouldhavechosenapopupmenu,usingtheoptionControlType->SetterBar.
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Sliderscanbeusedtoscanthroughdiscretesymbolicvalues,notjustthroughnumericalranges(andthisallowsyoutoanimatethroughthemaswell).TheoptionControlType->ManipulatorasksforthedefaultcontrolusedbyManipulate,whichisasliderplusanoptionalcontrolpanelwithnumericalvalueandanimationcontrols(seethepreviousexample).ControlType->Sliderasksforaplainslider.
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Itisevenpossibletousetwodifferentcontrolstoadjustthevalueofthesamevariable.Herebothapopupmenuandasliderareconnectedtothevalueofthe
variable.Ifthesliderisusedtoselectavaluethatdoesnotappearinthepopupmenu,thepopupwillappearblank,butremainsfunctional.Whenavalueischosenfromthepopupmenu,thesliderismovedtothecorrespondingposition.Bothcontrolscanthusbeusedinterchangeablytoadjustthesamevalue,andeachonefollowsalongwhentheotherisbeingused.
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ThisisnotanexhaustivelistofthepossiblecontroltypesinManipulate.SeetheManipulatedocumentationforamoredetailedlisting.Oneofthemostimportantcontroltypes,Locator,whichallowsyoutoplacecontrolpointsinsidegraphicaloutputinaManipulate,isdiscussedin"Locator";Slider2Disdiscussedinthe"2DSliders"section.
InitialValuesandLabels
Hereisafunexample
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