GeoGebraHelpOfficialManual3.2

MarkusHohenwarterandJudithHohenwarterwww.geogebra.org

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GeoGebraHelp3.2

Lastmodified:April22,2009AuthorsMarkusHohenwarter,markus@geogebra.orgJudithHohenwarter,judith@geogebra.orgGeoGebraOnline:http://www.geogebra.org

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Contents

1. WHATISGEOGEBRA?..........................................................................................................6

1.1. MultipleViewsforMathematicalObjects.......................................................................................61.1.1. GraphicsView...................................................................................................................................61.1.2. AlgebraView....................................................................................................................................71.1.3. SpreadsheetView.............................................................................................................................8

1.2. GeoGebraasaToolforTeachingandLearningMathematics...........................................................81.2.1. CustomizingtheUserInterface........................................................................................................81.2.2. ThePropertiesDialog.....................................................................................................................101.2.3. TheContextMenu..........................................................................................................................10

1.3. GeoGebraasaPresentationTool..................................................................................................111.3.1. TheNavigationBar.........................................................................................................................111.3.2. TheConstructionProtocol..............................................................................................................111.3.3. CustomizetheSettings...................................................................................................................12

1.4. GeoGebraasanAuthoringTool....................................................................................................131.4.1. PrintingOptions.............................................................................................................................131.4.2. CreatingPicturesoftheGraphicsView...........................................................................................131.4.3. CreatingInteractiveWebpages......................................................................................................14

2. GEOMETRICINPUT............................................................................................................16

2.1. GeneralNotes..............................................................................................................................16

2.2. ConstructionTools........................................................................................................................162.2.1. GeneralTools.................................................................................................................................172.2.2. PointTools.....................................................................................................................................182.2.3. VectorTools...................................................................................................................................192.2.4. SegmentTools................................................................................................................................192.2.5. RayTool..........................................................................................................................................202.2.6. PolygonTools.................................................................................................................................202.2.7. LineTools.......................................................................................................................................202.2.8. ConicSectionTools.........................................................................................................................212.2.9. ArcandSectorTools.......................................................................................................................222.2.10. NumberandAngleTools............................................................................................................232.2.11. BooleanVariableTool................................................................................................................252.2.12. LocusTool..................................................................................................................................252.2.13. GeometricTransformationTools...............................................................................................252.2.14. TextTool....................................................................................................................................262.2.15. ImageTool.................................................................................................................................28

3. ALGEBRAICINPUT.............................................................................................................30

3.1. GeneralNotes..............................................................................................................................30

3.2. DirectInput..................................................................................................................................32

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3.2.1. NumbersandAngles......................................................................................................................323.2.2. PointsandVectors..........................................................................................................................333.2.3. LinesandAxes................................................................................................................................333.2.4. ConicSections................................................................................................................................343.2.5. Functionsofx.................................................................................................................................343.2.6. PredefinedFunctionsandOperations...........................................................................................353.2.7. BooleanVariablesandOperations.................................................................................................363.2.8. ListObjectsandOperations............................................................................................................373.2.9. MatrixObjectsandOperations.......................................................................................................383.2.10. ComplexNumbersandOperations............................................................................................38

3.3. Commands...................................................................................................................................393.3.1. GeneralCommands........................................................................................................................403.3.2. BooleanCommands.......................................................................................................................403.3.3. NumberCommands.......................................................................................................................413.3.4. AngleCommand.............................................................................................................................453.3.5. PointCommands............................................................................................................................453.3.6. VectorCommands..........................................................................................................................473.3.7. SegmentCommand........................................................................................................................483.3.8. RayCommand................................................................................................................................483.3.9. PolygonCommand.........................................................................................................................493.3.10. LineCommands.........................................................................................................................493.3.11. ConicSectionCommands...........................................................................................................513.3.12. FunctionCommands..................................................................................................................523.3.13. ParametricCurveCommand......................................................................................................533.3.14. ArcandSectorCommands.........................................................................................................543.3.15. TextCommands.........................................................................................................................553.3.16. LocusCommand........................................................................................................................583.3.17. ListandSequenceCommands...................................................................................................583.3.18. GeometricTransformationCommands......................................................................................613.3.19. StatisticsCommands..................................................................................................................633.3.20. SpreadsheetCommands............................................................................................................673.3.21. MatrixCommands.....................................................................................................................67

4. MENUITEMS........................................................................................................................69

4.1. FileMenu.....................................................................................................................................69

4.2. EditMenu.....................................................................................................................................71

4.3. ViewMenu...................................................................................................................................73

4.4. OptionsMenu..............................................................................................................................74

4.5. ToolsMenu..................................................................................................................................76

4.6. WindowMenu..............................................................................................................................77

4.7. HelpMenu...................................................................................................................................77

5. SPECIALGEOGEBRAFEATURES.....................................................................................79

5.1. Animation....................................................................................................................................795.1.1. AutomaticAnimation.....................................................................................................................795.1.2. ManualAnimation..........................................................................................................................79

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5.2. ConditionalVisibility.....................................................................................................................80

5.3. UserDefinedTools.......................................................................................................................81

5.4. DynamicColors.............................................................................................................................82

5.5. JavaScriptInterface......................................................................................................................82

5.6. KeyboardShortcuts......................................................................................................................83

5.7. LabelsandCaptions......................................................................................................................87

5.8. Layers...........................................................................................................................................87

5.9. Redefine.......................................................................................................................................88

5.10. TraceandLocus............................................................................................................................88

6. INDEX.....................................................................................................................................90

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1. WhatisGeoGebra?

GeoGebraisdynamicmathematicssoftwarethatjoinsgeometry,algebraandcalculus.ItisdevelopedforlearningandteachingmathematicsinschoolsbyMarkusHohenwarterandaninternationalteamofprogrammers.

1.1. MultipleViewsforMathematicalObjects

GeoGebraprovidesthreedifferentviewsofmathematicalobjects:aGraphicsView,a,numericAlgebraView,andaSpreadsheetView.Theyallowyoutodisplaymathematicalobjectsinthreedifferentrepresentations:graphically(e.g.,points,functiongraphs),algebraically(e.g.,coordinatesofpoints,equations),andinspreadsheetcells.Thereby,allrepresentationsofthesameobjectarelinkeddynamicallyandadaptautomaticallytochangesmadetoanyoftherepresentations,nomatterhowtheywereinitiallycreated.

1.1.1. GraphicsView

UsingtheconstructiontoolsavailableintheToolbaryoucandogeometricconstructionsintheGraphicsViewwiththemouse.SelectanyconstructiontoolfromtheToolbarandreadtheToolbarHelp(nexttotheToolbar)inordertofindouthowtousetheselectedtool.AnyobjectyoucreateintheGraphicsViewalsohasanalgebraicrepresentationintheAlgebraView.

AlgebraView

GraphicsView

SpreadsheetView

InputBar

Toolbar ToolbarHelp

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Note:Afteractivatingthetool MoveyouareabletomoveobjectsintheGraphicsViewbydraggingthemwiththemouse.Atthesametime,theiralgebraicrepresentationsaredynamicallyupdatedintheAlgebraView.EveryiconintheToolbarrepresentsatoolboxthatcontainsaselectionofsimilarconstructiontools.Inordertoopenatoolbox,youneedtoclickonthesmallarrowinthelowerrightcorneroftheToolbaricon.Hint:Constructiontoolsareorganizedbythenatureofresultingobjectsorthefunctionalityofthetools.YouwillfindtoolsthatcreatedifferenttypesofpointsinthePointToolbox(defaulticon )andtoolsthatallowyoutoapplygeometrictransformationsintheTransformationToolbox(defaulticon ).

1.1.2. AlgebraView

UsingtheInputBaryoucandirectlyenteralgebraicexpressionsinGeoGebra.AfterhittingtheEnterkeyyouralgebraicinputappearsintheAlgebraViewwhileitsgraphicalrepresentationisautomaticallydisplayedintheGraphicsView.Example:Theinputf(x) = x^2givesyouthefunctionfintheAlgebraViewanditsfunctiongraphintheGraphicsView.IntheAlgebraView,mathematicalobjectsareorganizedasfreeanddependentobjects.Ifyoucreateanewobjectwithoutusinganyotherexistingobjects,itisclassifiedasafreeobject.Ifyournewlycreatedobjectwascreatedbyusingotherexistingobjects,itisclassifiedasadependentobject.Hint:IfyouwanttohidethealgebraicrepresentationofanobjectintheAlgebraView,youmayspecifytheobjectasanauxiliaryobject:Rightclick(MacOS:Ctrlclick)onthecorrespondingobjectintheAlgebraViewandselectPropertiesfromtheappearingContextMenu.OntabBasicofthePropertiesDialogyoumayspecifytheobjectasanAuxiliaryObject.Bydefault,auxiliaryobjectsarenotshownintheAlgebraView,butyoucanchangethissettingbyselectingtheitemAuxiliaryObjectsfromtheViewmenu.NotethatyouareabletomodifyobjectsintheAlgebraViewaswell:Makesurethatyouactivatethe MovetoolbeforeyoudoubleclickonafreeobjectintheAlgebraView.Intheappearingtextboxyoucandirectlyeditthealgebraicrepresentationoftheobject.AfterhittingtheEnterkey,thegraphicalrepresentationoftheobjectwillautomaticallyadapttoyourchanges.IfyoudoubleclickonadependentobjectintheAlgebraView,adialogwindowappearsallowingyoutoRedefinetheobject.GeoGebraalsooffersawiderangeofcommandsthatcanbeenteredintotheInputBar.YoucanopenthelistofcommandsintherightcorneroftheInputBarbyclickingonthebuttonCommand.Afterselectingacommandfromthislist(ortypingitsnamedirectlyintotheInputBar)youcanpresstheF1keytogetinformationaboutthesyntaxandargumentsrequiredtoapplythecorrespondingcommand.

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1.1.3. SpreadsheetView

InGeoGebrasSpreadsheetVieweverycellhasaspecificnamethatallowsyoutodirectlyaddresseachcell.Forexample,thecellincolumnAandrow1isnamedA1.Note:Thesecellnamescanbeusedinexpressionsandcommandsinordertoaddressthecontentofthecorrespondingcell.Inthespreadsheetcellsyoucanenternotonlynumbers,butalltypesofmathematicalobjectsthataresupportedbyGeoGebra(e.g.,coordinatesofpoints,functions,commands).Ifpossible,GeoGebraimmediatelydisplaysthegraphicalrepresentationoftheobjectyouenteredinaspreadsheetcellintheGraphicsViewaswell.Thereby,thenameoftheobjectmatchesthenameofthespreadsheetcellusedtoinitiallycreateit(e.g.,A5,C1).Note:Bydefault,spreadsheetobjectsareclassifiedasauxiliaryobjectsintheAlgebraView.YoucanshoworhidetheseauxiliaryobjectsbyselectingAuxiliaryObjectsfromtheViewmenu.

1.2. GeoGebraasaToolforTeachingandLearningMathematics

1.2.1. CustomizingtheUserInterface

TheuserinterfaceofGeoGebracanbecustomizedbyusingtheViewmenu.Forexample,youcanhidedifferentpartsoftheinterface(e.g.,theAlgebraView,SpreadsheetView,orInputBar)bycheckingoruncheckingthecorrespondingmenuitemintheViewmenu.

ShowingandHidingObjects

YoumayshoworhideobjectsintheGraphicsViewindifferentways. Youmayusetool Show/HideObjecttoshoworhideobjects. OpentheContextMenuandselectitem ShowObjecttochangethevisibility

statusoftheselectedobject. IntheAlgebraView,theicontotheleftofeveryobjectshowsitscurrentvisibility

state( shownor hidden).Youmaydirectlyclickonthelittlemarbleiconinordertochangethevisibilitystatusofanobject.

Youcanalsousethetool CheckBoxtoShow/HideObjectsinordertoshoworhideoneorseveralobjects.

CustomizingtheGraphicsView

InordertoadjustthevisiblepartofthedrawingpadintheGraphicsView,youcandragthedrawingpadbyusingtool MoveDrawingPadandusethefollowingwaysofzooming:

Youmayusethetools ZoomInand ZoomOutinordertozoomintheGraphicsView.Note:Thepositionofyourclickdeterminesthecenterofzoom.

YoumayusethescrollwheelofyourmouseinordertozoomintheGraphicsView.

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Youmayusekeyboardshortcutstozoomin(Ctrl+)andtozoomout(Ctrl). Afterrightclicking(MacOS:Ctrlclick)onanemptyspotonthedrawingpada

ContextMenuappearswhichallowsyoutoZoom. Youmayspecifyazoomrectanglebyrightclicking(MacOS:Cmdclick)onanempty

spotintheGraphicsViewanddraggingthemousetotheoppositecornerofyourdesiredzoomrectangle.Releasethemousebuttoninordertofinishthezoomrectangle,whichwillthenautomaticallyadjusttofillallthespaceintheGraphicsView.

YoucanalsoshoworhidethecoordinateaxesandacoordinategridintheGraphicsViewbyusingtheViewmenu.Note:Anotherwayofshowingorhidingtheaxesandthegridisbyrightclicking(MacOS:Ctrlclick)onthedrawingpadandselectingthecorrespondingitems Axesor GridfromtheappearingContextMenu.

CustomizingCoordinateAxesandGrid

ThecoordinateaxesandgridcanbecustomizedusingthePropertiesDialogoftheGraphicsView.Afterrightclicking(MacOS:Ctrlclick)onthedrawingpad,youcanopenthisdialogwindowbyselectingPropertiesfromtheappearingContextMenuoftheGraphicsView.

OntabAxes,youcan,forexample,changethelinestyleandunitsofthecoordinateaxes,andsetthedistanceofthetickmarkstoacertainvalue.Notethatyoucancustomizebothaxesindividually,byclickingontabsxAxisoryAxis.Furthermore,youcanalsochangetheratiobetweentheaxesandhideorshowtheaxesindividually.

OntabGrid,youcan,forexample,changethecolorandlinestyleofthecoordinategrid,andsetthedistanceforgridlinestoacertainvalue.Inaddition,youmayalsosetthegridtobeIsometric.

Note:ScalingtheaxesispossibleineverymodebypressingandholdingtheShiftkey(PC:alsoCtrlkey)whiledraggingtheaxis.Note:ThePropertiesDialogoftheGraphicsViewisdifferentfromthePropertiesDialogforobjects.

CustomizingtheToolbar

TheToolbarcanbecustomizedbyselectingCustomizeToolbarfromtheToolsmenu.SelectthetoolortoolboxyouwanttoremovefromtheToolbarinthelistonthelefthandsideoftheappearingdialogwindowandclickbuttonRemove>inordertoremovethetool/toolboxfromtheToolbar.Note:YoucanrestorethedefaultToolbarbyclickingonthebuttonRestoreDefaultToolbarintheleftlowercornerofthedialogwindow.

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1.2.2. ThePropertiesDialog

ThePropertiesDialogallowsyoutomodifypropertiesofobjects(e.g.,size,color,filling,linestyle,linethickness,visibility).YoucanopenthePropertiesDialoginseveralways:

Rightclick(MacOS:Ctrlclick)onanobjectandselect PropertiesfromtheappearingContextMenu.

Selectitem PropertiesfromtheEditmenu. Selectthe MovetoolanddoubleclickonanobjectintheGraphicsView.Inthe

appearingRedefinedialogwindow,clickonthebuttonProperties.InthePropertiesDialogobjectsareorganizedbytypes(e.g.,points,lines,circles)inthelistonthelefthandside,whichmakesiteasiertohandlelargenumbersofobjects.Youneedtoselectoneormoreobjectsfromthislistinordertochangeits/theirproperties.Note:Byclickingonaheadinginthelistofobjects(e.g.,Point)youcanselectallobjectsofthistypeandtherefore,quicklychangethepropertiesforalltheseobjects.Youcanmodifythepropertiesofselectedobjectsusingthetabsontherighthandside(e.g.,Basic,Color,Style,Advanced).Note:Dependingontheselectionofobjectsinthelist,adifferentsetoftabsmaybeavailable.ClosethePropertiesDialogwhenyouaredonewithchangingpropertiesofobjects.

1.2.3. TheContextMenu

TheContextMenuprovidesaquickwaytochangethebehaviororadvancedpropertiesofanobject.Rightclick(MacOS:Ctrlclick)onanobjectinordertoopenitsContextMenu.Forexample,itallowsyoutochangetheobjectsalgebraicnotation(e.g.,polarorCartesiancoordinates,implicitorexplicitequation)andtodirectlyaccessfeatureslike Rename, Delete, TraceOn,AnimationOn,or CopytoInputBar.Note:IfyouopentheContextMenuforapointintheGraphicsView,itgivesyoutheoptionTracetoSpreadsheet(onlyiftheSpreadsheetViewisactive).Onceselected,thisfeature

allowsyoutorecordthecoordinatesofthepointintheSpreadsheetViewifitismoved.Note:Selecting PropertiesintheContextMenuopensthePropertiesDialog,whereyoucanchangethepropertiesofallobjectsused.

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1.3. GeoGebraasaPresentationTool

1.3.1. TheNavigationBar

GeoGebraoffersaNavigationBarthatallowsyoutonavigatethroughtheconstructionstepsofapreparedGeoGebrafile.SelectitemNavigationBarforConstructionStepsintheViewmenuinordertodisplaytheNavigationBaratthebottomoftheGraphicsView.TheNavigationBarprovidesasetofnavigationbuttonsanddisplaysthenumberofconstructionsteps(e.g.,2/7meansthatcurrentlythesecondstepofatotalof7constructionstepsisdisplayed):

button:gobacktostep1 button:gobackstepbystep button:goforwardstepbystep button:gotothelaststep Play:automaticallyplaytheconstructionstepbystep

Note:Youmaychangethespeedofthisautomaticplayfeatureusingthetextboxtotherightofthe Playbutton.

Pause:pausetheautomaticplayfeatureNote:ThisbuttononlyappearsafteryouclickonthePlaybutton.

button:ThisbuttonopenstheConstructionProtocol.

1.3.2. TheConstructionProtocol

YoucanaccesstheinteractiveConstructionProtocolbyselectingitemConstructionProtocolfromtheViewmenu.Itisatablethatshowsallconstructionsteps.TheConstructionProtocolallowsyoutoredoapreparedconstructionstepbystepusingtheNavigationBaratthebottomoftheConstructionProtocoldialog.

NavigatingandModifyingtheConstructionProtocol

YoumayusethekeyboardtonavigateintheConstructionProtocol: Usetheuparrowofyourkeyboardtogotothepreviousconstructionstep. Usethedownarrowofyoukeyboardtogotothenextconstructionstep. UsetheHomekeytogotothebeginningoftheConstructionProtocol. UsetheEndkeytogototheendoftheConstructionProtocol. UsetheDeletekeyinordertodeletetheselectedconstructionstep.

Note:Thismayalsoaffectotherobjectsthatdependontheselectedobject/constructionstep.

YoumayalsousethemouseinordertonavigateintheConstructionProtocol:

Doubleclickarowinordertoselectaconstructionstep. DoubleclicktheheaderofanycolumninordertogotothestartoftheConstruction

Protocol.

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DraganddroparowinordertomoveaconstructionsteptoanotherpositionintheConstructionProtocol.Note:Thisisnotalwayspossibleduetothedependenciesbetweendifferentobjects.

RightclickarowinordertoopentheContextMenufortheobjectofthisconstructionstep.

Note:Youcaninsertconstructionstepsatanyposition.Selecttheconstructionstepbelowyouwouldliketoinsertanewconstructionstep.LeavetheConstructionProtocolwindowopenwhileyoucreateanewobject.ThisnewconstructionstepisimmediatelyinsertedintotheselectedpositionoftheConstructionProtocol.UsingthecolumnBreakpointintheViewmenuoftheConstructionProtocolwindow,youareabletodefinecertainconstructionstepsasBreakpoints.Thisallowsyoutogroupseveralobjectstogether.WhennavigatingthroughyourconstructionusingtheNavigationBar,groupsofobjectsareshownatthesametime.Note:YoumayswitchthedifferentcolumnsoftheConstructionProtocolonandoffbyusingtheViewmenuoftheConstructionProtocolwindow.

ExportingtheConstructionProtocolasaWebpage

GeoGebraallowsyoutoexporttheConstructionProtocolasawebpage.First,youneedtoopentheConstructionProtocolusingtheViewmenu.Then,youcanopentheFilemenuoftheappearingConstructionProtocolwindowandselectitemExportasWebpage.IntheexportwindowoftheConstructionProtocolyoucanenterTitle,Author,andaDatefortheconstructionandchoosewhetherornotyouwanttoincludeapictureoftheGraphicsViewandtheAlgebraView.Inaddition,youcanalsochoosetoexportaColorfulConstructionProtocol.ThismeansthatobjectsintheConstructionProtocolwillmatchthecolorofthecorrespondingobjectsintheconstruction.Note:TheexportedHTMLfilecanbeviewedwithanyInternetbrowser(e.g.Firefox,InternetExplorer)andeditedwithmanytextprocessingsystems(e.g.OpenOfficeWriter).

1.3.3. CustomizetheSettings

GeoGebraallowsyoutochangeandsavesettingsusingtheOptionsmenu.Forexample,youmaychangetheAngleUnitfromDegreetoRadians,orchangethePointStyle,CheckboxSize,andRightAngleStyle.Inaddition,youmaychangehowCoordinatesaredisplayedonscreenandwhichobjectsarelabeled(Labeling).PleaseseethesectionabouttheOptionsmenuformoreinformation.Youcansaveyourcustomizedsettingsbyselectingitem SaveSettingsfromtheOptionsmenu.Afterdoingso,GeoGebrawillrememberyourcustomizedsettingsandusethemforeverynewGeoGebrafileyoucreate.Note:YoumayrestorethedefaultsettingsbyselectingRestoreDefaultSettingsfromtheOptionsmenu.

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Note:IfyouuseGeoGebraasapresentationtool,youmightwanttoincreasetheFontSize(Optionsmenu)soyouraudiencecaneasilyreadtextandlabelsofobjects.

1.4. GeoGebraasanAuthoringTool

1.4.1. PrintingOptions

PrintingtheGraphicsView

GeoGebraallowsyoutoprinttheGraphicsViewofyourconstructions.YoucanfindthecorrespondingitemPrintPreviewintheFilemenu.IntheappearingPrintPreviewdialogwindow,youcanspecifytheTitle,Author,andaDatefortheconstruction.Inaddition,youcansettheScaleofyourprintout(incm)andchangetheOrientationofthepaperused(portraitorlandscape).Note:InordertoupdatethePrintPreviewafteryoumadechangestothetextorlayoutoftheprintout,youneedtopresstheEnterkey.

PrintingtheConstructionProtocol

IfyouwanttoprinttheConstructionProtocol,youfirstneedtoopentheConstructionProtocoldialogwindowbyusingtheViewmenu.Then,youcanopenthePrintPreviewwindowoftheConstructionProtocolfromtheFilemenuofthisnewwindow.Again,youmayenterTitle,Author,andaDateorchangetheScaleorpaperOrientationbeforeprintingyourConstructionProtocol.Note:YoumayswitchthedifferentcolumnsName,Definition,Command,Algebra,andBreakpointoftheConstructionProtocolonandoffbyusingtheViewmenuoftheConstructionProtocoldialogwindow.

1.4.2. CreatingPicturesoftheGraphicsView

SavingtheGraphicsViewasaPicture

YoucansavetheGraphicsViewofyourconstructionsasapicturefileonyourcomputer.Note:ThefullGraphicsViewwillbesavedasapicture.IfyourconstructiondoesnotusealltheavailablespaceintheGraphicsView,youmightwantto

usetools MoveDrawingPad, ZoomIn,and/or ZoomOutinordertoplaceyourconstructionintheupperleftcorneroftheGraphicsView.Afterwards,youmayreducethesizeoftheGeoGebrawindowbydraggingoneofitscornerswiththemouse.

usetheselectionrectangleinordertospecifywhichpartoftheGraphicsViewshouldbeexportedandsavedasapicture.

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YoumaycreatepointscalledExport_1andExport_2,whichwillbeusedtodefinediagonallyoppositecornersoftheexportrectangle. Note:PointsExport1andExport2mustbewithinthevisibleareaoftheGraphicsView.

IntheFilemenu,selectitemExportbeforeclickingonitem GraphicsViewasPicture.IntheappearingdialogwindowyoumayspecifytheFormat,Scale(incm),andtheResolution(indpi)oftheoutputpicturefile.Note:Thetruesizeoftheexportedimageisshownatthebottomoftheexportwindowjustabovethebuttons,bothincentimetersandpixel.PleasefindmoreinformationaboutthedifferentpicturefilesavailableinsectionExport GraphicsViewasPicture(png,eps).

CopyingtheGraphicsViewtoClipboard

TherearedifferentwaysofcopyingtheGraphicsViewtoyourcomputersclipboard: IntheEditmenu,youmayselectitem GraphicsViewtoClipboard. IntheFilemenu,youfirstneedtoselectitemExport,beforeyoucanclickonitem

GraphicsViewtoClipboard. IntheExportGraphicsViewasPicturedialogwindow(menuFileExport

GraphicsViewasPicture(png,eps))youmayclickonthebuttonClipboard.ThisfeaturecopiesascreenshotoftheGraphicsViewtoyoursystem'sclipboardasaPNG(seePNGformat)picture.Thispicturecanbepastedintootherdocuments(e.g.awordprocessingdocument).Note:InOrdertoexportyourconstructionatacertainscale(incm)pleaseusethemenuitem GraphicsViewasPictureintheFilemenu,Export.

1.4.3. CreatingInteractiveWebpages

GeoGebraallowsyoutocreateinteractivewebpages,socalledDynamicWorksheets,fromyourfiles.IntheFilemenu,youneedtoselectitemExportbeforeyoucanclickonitemDynamicWorksheetasWebpage(html).Thisopenstheexportdialogwindowfor

DynamicWorksheets: AtthetopoftheexportwindowyoucanentertheTitle,Author,andaDateforyour

DynamicWorksheet. TabGeneralallowsyoutoaddsometextaboveandbelowthedynamicconstruction

(e.g.,adescriptionoftheconstructionandsometasks).Youcanalsodetermineiftheconstructionitselfmaybeincludeddirectlyintothewebpageorifitcanbeopenedbyclickingonabutton.

TabAdvancedallowsyoutochangethefunctionalityofthedynamicconstruction(e.g.,showareseticon,doubleclickshouldopentheGeoGebraapplicationwindow)aswellastomodifytheuserinterfaceshownintheinteractiveapplet(e.g.,showtheToolbar,modifyheightandwidth).

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Note:Ifthesizeofyourappletistoobigtofitonacomputerscreenwithstandardresolution(1024x768),youmaywanttoresizeitbeforetheactualexportasaDynamicWorksheet.

Note:SeveralfilesarecreatedwhenyouexportaDynamicWorksheet:

HTMLfile(e.g.circle.html)thisfileincludestheworksheetitself GGBfile(e.g.circle.ggb)thisfileincludesyourGeoGebraconstruction JAR(severalfiles)thesefilesincludeGeoGebraandmakeyourworksheet

interactiveAllthesefiles(e.g.circle.html,circle.ggbandthegeogebra.jarfiles)havetobeinonefolder(directory)toletthedynamicconstructionwork.TheexportedHTMLfile(e.g.circle.html)canbeviewedwithanyInternetbrowser(e.g.Mozilla,InternetExplorer,Safari).Inordertoletthedynamicconstructionwork,Javahastobeinstalledonthecomputer.YoucangetJavafromhttp://www.java.comwithoutcharge.IfyouwanttouseyourDynamicWorksheetinyourschool'scomputernetwork,askyourlocalnetworkadministratortoinstallJavaonthecomputers.Note:YoucanedittheDynamicWorksheet'stextwithmanywordprocessingsystems(e.g.FrontPage,OpenOfficeWriter)byopeningtheexportedHTMLfile.YoumayalsoedittheDynamicWorksheet'sappletbyopeningtheGGBfileinGeoGebraandsavingitwiththesamenameafterwards.

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2. GeometricInput

2.1. GeneralNotes

TheGraphicsViewshowsthegraphicalrepresentationofmathematicalobjects(e.g.,points,vectors,segments,polygons,functions,curves,straightlines,conicsections).Wheneverthemouseismovedoveroneoftheseobjectsadescriptionappearsasarollovertextandtheobjectishighlighted.Thereareseveraltools/modestotellGeoGebrahowitshouldreacttomouseinputintheGraphicsView(seesectionConstructionTools).Forexample,clickingonthedrawingpadcancreateanewpoint(seetool NewPoint),intersecttwoobjects(seetool IntersectTwoObjects),orcreateacircle(see Circletools).

2.2. ConstructionTools

ThefollowingconstructiontoolsormodescanbeactivatedbyclickingonthebuttonsoftheToolbar.YoucanclickonthesmallarrowinthelowerrightcornerofanicontoopenaToolboxwithsimilarothertools.Note:Withmostconstructiontoolsyoucaneasilycreatenewpointsbyclickingonemptyspacesonthedrawingpad.

SelectingObjects

Toselectanobjectmeanstoclickonitwiththemouseafterselectingthe Movetool.Ifyouwanttoselectseveralobjectsatthesametime,youcoulddrawaselectionrectangle:Selectthe Movetoolandclickonthepositionofthefirstcornerofyourdesiredselectionrectangle.Holdtheleftmousekeypresseddownandmovethepointertothepositionofthediagonallyoppositecornerofyourdesiredselectionrectangle.Afterreleasingthemousebutton,allobjectswithintheselectionrectangleareselected.Note:ByholdingtheCtrlkey(MacOS:Cmdkey)whileclickingondifferentobjects,youcanselectseveralobjectsatthesametime.

FastRenamingofObjects

ToquicklyrenameaselectedornewlycreatedobjectjuststarttypingtoopentheRenamedialogforthisobject.Then,typeinthenewnameoftheselectedobjectandclickontheOKbutton.

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2.2.1. GeneralTools

CopyVisualStyle

Thistoolallowsyoutocopyvisualproperties(e.g.,color,size,linestyle)fromoneobjecttooneormoreotherobjects.Todoso,firstselecttheobjectwhosepropertiesyouwanttocopy.Then,clickonallotherobjectsthatshouldadopttheseproperties.

DeleteObject

Clickonanyobjectyouwanttodelete(alsoseecommandDelete).Note:Youcanusethe Undobuttonifyouaccidentallydeletedthewrongobject.

Move

Draganddropfreeobjectswiththemouse.IfyouselectanobjectbyclickingonitinMovemode,youmay deletetheobjectbypressingtheDeletekey movetheobjectbyusingthearrowkeys(seesectionManualAnimation)

Note:Youcanquicklyactivatethe MovetoolbypressingtheEsckeyofyourkeyboard.

MoveDrawingPad

DraganddropthedrawingpadintheGraphicsViewtochangeitsvisiblearea.Note:

YoucanalsomovethedrawingpadbypressingtheShiftkey(MSWindows:alsoCtrlkey)anddraggingitwiththemouseinanymode.

Inthismodeyoucanalsoscaleeachoftheaxesbydraggingitwiththemouse.

RecordtoSpreadsheet

ThistoolallowsyoutomoveanobjectandtorecordasequenceofitsvaluesintheSpreadsheetView.Thistoolworksfornumbers,points,andvectors.Note:GeoGebrawillusethefirsttwoemptycolumnsoftheSpreadsheetViewtorecordthevaluesoftheselectedobjects.

Relation

Selecttwoobjectstogetinformationabouttheirrelationinapopupwindow(alsoseecommandRelation).

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RotatearoundPoint

Selectthecenterpointoftherotationfirst.Then,youmayrotatefreeobjectsaroundthispointbydraggingthemwiththemouse(alsoseecommandRotate).

Show/HideLabel

Clickonanobjecttoshoworhideitslabel.

Show/HideObject

Selecttheobjectyouwanttoshoworhideafteractivatingthistool.Then,switchtoanothertoolinordertoapplythevisibilitychangestothisobject.Note:Whenyouactivatethistool,allobjectsthatshouldbehiddenaredisplayedintheGraphicsViewhighlighted.Inthisway,youcaneasilyshowhiddenobjectsagainbydeselectingthembeforeswitchingtoanothertool.

ZoomIn

Clickonanyplaceonthedrawingpadtozoomin(alsoseesectionCustomizingtheGraphicsView).Note:Thepositionofyourclickdeterminesthecenterofzoom.

ZoomOut

Clickonanyplaceonthedrawingpadtozoomout(alsoseesectionCustomizingtheGraphicsView).Note:Thepositionofyourclickdeterminesthecenterofzoom.

2.2.2. PointTools

IntersectTwoObjects

Intersectionpointsoftwoobjectscanbecreatedintwoways(alsoseecommandIntersect). Selectingtwoobjectscreatesallintersectionpoints(ifpossible). Directlyclickingonanintersectionofthetwoobjectscreatesonlythissingle

intersectionpoint.Note:Forsegments,rays,orarcsyoumayspecifywhetheryouwanttoAllowoutlyingintersectionsontabBasicofthePropertiesDialog.Thiscanbeusedtogetintersectionpointsthatlieontheextensionofanobject.Forexample,theextensionofasegmentorarayisastraightline.

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MidpointorCenter

Youmayclickoneithertwopointsoronesegmenttogetitsmidpoint.Youcanalsoclickonaconicsection(circleorellipse)inordertocreateitscenterpoint(alsoseecommandsCenterandMidpoint).

NewPoint

ClickonthedrawingpadintheGraphicsViewinordertocreateanewpoint.Thecoordinatesofthepointarefixedwhenthemousebuttonisreleased.Note:

Byclickingonasegment,straightline,polygon,conicsection,function,orcurveyoucancreateapointonthisobject(alsoseecommandPoint).

Clickingontheintersectionoftwoobjectscreatesthisintersectionpoint(alsoseetool IntersectTwoObjectsandcommandIntersect).

2.2.3. VectorTools

VectorbetweenTwoPoints

Selectthestartingpointandthentheendpointofthevector(alsoseecommandVector).

VectorfromPoint

SelectapointAandavectorvtocreatethenewpointB=A+vaswellasthevectorfromAtoB(alsoseecommandVector).

2.2.4. SegmentTools

SegmentbetweenTwoPoints

SelecttwopointsAandBinordertocreateasegmentbetweenAandB(alsoseecommandSegment).Note:IntheAlgebraView,thesegment'slengthisdisplayed.

SegmentwithGivenLengthfromPoint

ClickonapointAthatshouldbethestartingpointofthesegment.Specifythedesiredlengthaofthesegmentintheappearingwindow(alsoseecommandSegment).Note:ThistoolcreatesasegmentwithlengthaandendpointBwhichmayberotatedaroundthestartingpointAbyusingtool Move.

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2.2.5. RayTool

RaythroughTwoPoints

SelectingtwopointsAandBcreatesaraystartingatAthroughB(alsoseecommandRay).Note:IntheAlgebraViewtheequationofthecorrespondinglineisdisplayed.

2.2.6. PolygonTools

Polygon

Successivelyselectatleastthreepointswhichwillbetheverticesofthepolygon.Then,clickthefirstpointagaininordertoclosethepolygon(alsoseecommandPolygon).Note:IntheAlgebraView,thepolygon'sareaisdisplayed.

RegularPolygon

SelecttwopointsAandBandspecifythenumbernofverticesinthetextfieldoftheappearingdialogwindow.ThisgivesyouaregularpolygonwithnverticesincludingpointsAandB(alsoseecommandPolygon).

2.2.7. LineTools

AngleBisector

Anglebisectorscanbedefinedintwoways(alsoseecommandAngleBisector): SelectingthreepointsA,B,andCproducestheanglebisectoroftheenclosedangle,

wherepointBistheapex. Selectingtwolinesproducestheirtwoanglebisectors.

Note:Thedirectionvectorsofallanglebisectorshavelength1.

BestFitLine

Createthebestfitlineforasetofpointsinthefollowingways(alsoseecommandFitLine): Createaselectionrectanglethatcontainsallpoints. Selectalistofpointstocreatetheircorrespondingbestfitline.

LinethroughTwoPoints

SelectingtwopointsAandBcreatesastraightlinethroughAandB(alsoseecommandLine).Note:Thelinesdirectionvectoris(BA).

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ParallelLine

SelectingalinegandapointAdefinesastraightlinethroughAparalleltog(alsoseecommandLine).Note:Thelinesdirectionisthedirectionoflineg.

PerpendicularBisector

ClickoneitherasegmentsortwopointsAandBinordertocreateaperpendicularbisector(alsoseecommandPerpendicularBisector).Note:ThebisectorsdirectionisequivalenttotheperpendicularvectorofsegmentsorAB(alsoseecommandPerpendicularVector).

PerpendicularLine

SelectingalinegandapointAcreatesastraightlinethroughAperpendiculartolineg(alsoseecommandPerpendicularLine).Note:Thelinesdirectionisequivalenttotheperpendicularvectorofg(alsoseecommandPerpendicularVector).

PolarorDiameterLine

Thistoolcreatesthepolarordiameterlineofaconicsection(alsoseecommandPolar). Selectapointandaconicsectiontogetthepolarline. Selectalineoravectorandaconicsectiontogetthediameterline.

Tangents

Tangentstoaconicsectioncanbeproducedinseveralways(alsoseecommandTangent): SelectingapointAandaconiccproducesalltangentsthroughAtoc. Selectingalinegandaconiccproducesalltangentstocthatareparalleltolineg. SelectingapointAandafunctionfproducesthetangentlinetofinx=x(A).

Note:x(A)representsthexcoordinateofpointA.IfpointAliesonthefunctiongraph,thetangentrunsthroughpointA.

2.2.8. ConicSectionTools

CirclewithCenterandRadius

SelectthecenterpointMandentertheradiusinthetextfieldoftheappearingdialogwindow(alsoseecommandCircle).

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CirclewithCenterthroughPoint

SelectingapointMandapointPdefinesacirclewithcenterMthroughP(alsoseecommandCircle).Note:ThecirclesradiusisthedistanceMP.

CirclethroughThreePoints

SelectingthreepointsA,B,andCdefinesacirclethroughthesepoints(alsoseecommandCircle).Note:Ifthethreepointslieononestraightline,thecircledegeneratestothisline.

Compass

UKEnglish:CompassesSelectasegmentortwopointstospecifytheradius.Then,clickonapointthatshouldbethecenterofthenewcircle.

ConicthroughFivePoints

Selectingfivepointsproducesaconicsectionthroughthesepoints(alsoseecommandConic).Note:Iffourofthesefivepointslieonaline,theconicsectionisnotdefined.

Ellipse

Selectthetwofocioftheellipse.Then,specifyathirdpointthatliesontheellipse(alsoseecommandEllipse).

Hyperbola

Selectthetwofociofthehyperbola.Then,specifyathirdpointthatliesonthehyperbola(alsoseecommandHyperbola).

Parabola

Selectapointandthedirectrixoftheparabola(alsoseecommandParabola).

2.2.9. ArcandSectorTools

Note:InGeoGebra,thealgebraicvalueofanarcisitslength.Thevalueofasectorisitsarea.

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CircularArcwithCenterbetweenTwoPoints

First,selectthecenterpointMofthecirculararc.Then,selectthestartingpointAofthearc,beforeyouselectapointBthatspecifiesthelengthofthearc(alsoseecommandCircularArc).Note:WhilepointAalwaysliesonthecirculararc,pointBdoesnothavetolieonit.

CircularSectorwithCenterbetweenTwoPoints

First,selectthecenterpointMofthecircularsector.Then,selectthestartingpointAofthesectorsarc,beforeyouselectapointBthatspecifiesthelengthofthesectorsarc(alsoseecommandCircularSector).Note:WhilepointAalwaysliesonthesectorsarc,pointBdoesnothavetolieonit.

CircumcircularArcthroughThreePoints

SelectingthreepointsA,B,andCcreatesacirculararcthroughthesepoints.Thereby,pointAisthestartingpointofthearc,pointBliesonthearc,andpointCistheendpointofthearc(alsoseecommandCircumcircularArc).

CircumcircularSectorthroughThreePoints

SelectingthreepointsA,B,andCcreatesacircularsectorthroughthesepoints.Thereby,pointAisthestartingpointofthesectorsarc,pointBliesonthearc,andpointCistheendpointofthesectorsarc(alsoseecommandCircumcircularSector).

Semicircle

SelecttwopointsAandBtocreateasemicircleabovethesegmentAB(alsoseecommandSemicircle).

2.2.10. NumberandAngleTools

Angle

Withthistoolyoucancreateanglesindifferentways(alsoseecommandAngle): Clickonthreepointstocreateananglebetweenthesepoints.Thesecondpoint

selectedisthevertexoftheangle. Clickontwosegmentstocreatetheanglebetweenthem. Clickontwolinestocreatetheanglebetweenthem. Clickontwovectorstocreatetheanglebetweenthem.

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Clickonapolygontocreateallanglesofthispolygon. Note:Ifthepolygonwascreatedbyselectingitsverticesincounterclockwiseorientation,theAngletoolgivesyoutheinterioranglesofthepolygon.

Note:Anglesarecreatedincounterclockwiseorientation.Therefore,theorderofselectingtheseobjectsisrelevantfortheAngletool.Ifyouwanttolimitthemaximumsizeofanangleto180,uncheckAllowReflexAngleontabBasicofthePropertiesDialog.

AnglewithGivenSize

SelecttwopointsAandBandtypetheanglessizeintothetextfieldoftheappearingwindow(alsoseecommandAngle).Note:ThistoolcreatesapointCandanangle,whereistheangleABC.

Area

Thistoolgivesyoutheareaofapolygon,circle,orellipseasanumberandshowsadynamictextintheGraphicsView(alsoseecommandArea).

DistanceorLength

Thistoolgivesyouthedistancebetweentwopoints,twolines,orapointandalineasanumberandshowsadynamictextintheGraphicsView.Itcanalsogiveyouthelengthofasegment,thecircumferenceofacircle,ortheperimeterofapolygon(alsoseecommandsDistanceandLength).

Slider

ClickonanyfreeplaceintheGraphicsViewtocreateasliderforanumberoranangle.TheappearingdialogwindowallowsyoutospecifytheName,Interval[min,max],andIncrementofthenumberorangle,aswellastheAlignmentandWidthoftheslider(inpixel).Note:IntheSliderdialogwindowyoucanenteradegreesymbolorpi()fortheintervalandincrementbyusingthefollowingkeyboardshortcuts:

AltO(MacOS:CtrlO)forthedegreesymbol AltP(MacOS:CtrlP)forthepisymbol

ThepositionofaslidermaybeabsoluteintheGraphicsView(thismeansthatthesliderisnotaffectedbyzooming,butalwaysremainsinthevisiblepartoftheGraphicsView)orrelativetothecoordinatesystem(seePropertiesDialogofthecorrespondingnumberorangle).Note:InGeoGebra,aslideristhegraphicalrepresentationofafreenumberorfreeangle.YoucaneasilycreateasliderforanyexistingfreenumberoranglebyshowingthisobjectintheGraphicsView(seeContextMenu;seetool Show/HideObject).

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Slope

ThistoolgivesyoutheslopeofalineandshowsaslopetriangleintheGraphicsView(alsoseecommandSlope).

2.2.11. BooleanVariableTool

CheckBoxtoShow/HideObjects

ClickingintheGraphicsViewcreatesacheckbox(seesectionBooleanVariablesandOperations)thatallowsyoutoshowandhideoneormoreobjects.Intheappearingdialogwindowyoucanspecifywhichobjectsshouldbeaffectedbythecheckbox.Note:Youmayselecttheseobjectsfromthelistprovidedinthedialogwindoworselectthemwiththemouseinanyview.

2.2.12. LocusTool

Locus

SelectapointBthatdependsonanotherpointAandwhoselocusshouldbedrawn.Then,clickonpointAtocreatethelocusofpointB(alsoseecommandLocus).Note:PointAhastobeapointonanobject(e.g.line,segment,circle).Example:

Typef(x) = x^2 2 x 1intotheInputBarandpresstheEnterkey. PlaceanewpointAonthexaxis(seetool NewPoint;seecommandPoint). CreatepointB = (x(A), f'(x(A)))thatdependsonpointA. Selecttool LocusandsuccessivelyclickonpointBandpointA. DragpointAalongthexaxistoseepointBmovingalongitslocusline.

2.2.13. GeometricTransformationTools

Thefollowinggeometrictransformationsworkforpoints,lines,conicsections,polygons,andimages.

DilateObjectfromPointbyFactor

UKEnglish:EnlargeObjectfromPointbyFactorSelecttheobjecttobedilated.Then,clickonapointtospecifythedilationcenterandenterthedilationfactorintothetextfieldoftheappearingdialogwindow(alsoseecommandsDilate(US)andEnlarge(UK)).

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ReflectObjectaboutLine

UKEnglish:ReflectObjectinLineSelecttheobjectyouwanttoreflect.Then,clickonalinetospecifythemirror/lineofreflection(alsoseecommandReflect).

ReflectObjectaboutPoint

UKEnglish:ReflectObjectinPointSelecttheobjectyouwanttoreflect.Then,clickonapointtospecifythemirror/pointofreflection(alsoseecommandReflect).

ReflectPointaboutCircle

UKEnglish:ReflectPointinCircleThistoolallowsyoutoinvertapointinacircle.Selectthepointyouwanttoinvert.Then,clickonacircletospecifythemirror/circleofinversion(alsoseecommandReflect).

RotateObjectaroundPointbyAngle

Selecttheobjectyouwanttorotate.Then,clickonapointtospecifythecenterofrotationandentertherotationangleintothetextfieldoftheappearingdialogwindow(alsoseecommandRotate).

TranslateObjectbyVector

Selecttheobjectyouwanttotranslate.Then,clickonthetranslationvector(alsoseecommandTranslate).

2.2.14. TextTool

InsertText

WiththistoolyoucancreatestaticanddynamictextorLaTeXformulasintheGraphicsView(alsoseesectionTextCommands).Atfirst,youneedtospecifythelocationofthetextinoneofthefollowingways:

ClickintheGraphicsViewtocreateanewtextatthislocation. Clickonapointtocreateanewtextthatisattachedtothispoint.

Then,adialogappearswhereyoumayenteryourtext.

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Note:YoumayspecifythepositionofatextasabsoluteonscreenorrelativetothecoordinatesystemontabBasicofthePropertiesDialog.Statictextdoesnotdependonanymathematicalobjectsandisusuallynotaffectedbychangesoftheconstruction.Dynamictextcontainsvaluesofobjectsthatautomaticallyadapttochangesmadetotheseobjects.Mixedtextisacombinationofstaticanddynamictext.Inordertocreateamixedtextyoumayenterthestaticpartofthetextusingthekeyboard(e.g.,Point A =).Then,clickontheobjectwhosevalueyouwanttodisplayinthedynamicpartofthetext.Note:GeoGebraautomaticallyaddsthesyntax("Point A = " + A)necessarytocreateyourmixedtext:quotationmarksaroundthestaticpartofthetextandaplus(+)symboltoconnectthedifferentpartsofthetext.

Input DescriptionThis is static text StatictextA Dynamictext(ifpointAexists)"Point A = " + A TwopartmixedtextusingthevalueofpointA"a = " + a + "cm" Threepartmixed textusingthevalueof

numberaNote:Ifanobjectwiththenamexxalreadyexistsandyouwanttocreateastatictextusingtheobjectsname,youneedtoenteritwithquotationmarks("xx").Otherwise,GeoGebrawillautomaticallycreateadynamictextthatgivesyouthevalueofobjectxxinsteadofitsname.However,youcantypeanytextthatdoesntmatchanyexistingobjectsnamewithoutthequotationmarks.Note:Withinamixedtext,thestaticpartneedstobeinbetweenapairofquotationmarks.Differentpartsofatext(e.g.,staticanddynamicparts)needtobeconnectedusingplus(+)symbols.

LaTeXFormulas

InGeoGebrayoucanwriteformulasaswell.Todoso,checktheboxLaTeXformulainthedialogwindowofthe InsertTexttoolandenteryourformulainLaTeXsyntax.Note:InordertocreatetextthatcontainsaLaTeXformulaaswellasstatictextyoumayenterthestaticpartofthetextandthenaddtheLaTeXformulainbetweenasetofdollarsymbols($).Example:The length of the diagonal is $\sqrt{ 2 }$.YoucanselectthesyntaxforcommonformulasymbolsfromthedropdownmenunexttotheLaTeXcheckbox.ThisinsertsthecorrespondingLaTeXcodeintothetextfieldandplacesthecursorinbetweenasetofcurlybrackets.Ifyouwouldliketocreatedynamictextwithintheformula,youneedtoclickonanobjectcausingGeoGebratoinsertitsnameaswellasthesyntaxformixedtext.

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SomeimportantLaTeXcommandsareexplainedinfollowingtable.PleasehavealookatanyLaTeXdocumentationforfurtherinformation.

LaTeXinput Resulta \cdot b ba \frac{a}{b}

ba

\sqrt{x} x \sqrt[n]{x} n x \vec{v} vr\overline{AB} ABx^{2} 2x a_{1} 1a \sin\alpha + \cos\beta cossin + \int_{a}^{b} x dx ba xdx \sum_{i=1}^{n} i^2 =ni i1 2

2.2.15. ImageTool

InsertImage

ThistoolallowsyoutoinsertanimageintotheGraphicsView.First,specifythelocationoftheimageinoneofthefollowingtwoways:

ClickintheGraphicsViewtospecifythepositionoftheimageslowerleftcorner. Clickonapointtospecifythispointasthelowerleftcorneroftheimage.

Then,afileopendialogappearsthatallowsyoutoselecttheimagefilefromthefilessavedonyourcomputer.Note:Afterselectingthetool InsertImage,youcanusethekeyboardshortcutAltclickinordertopasteanimagedirectlyfromyourcomputersclipboardintotheGraphicsView.

PropertiesofImages

Thepositionofanimagemaybeabsoluteonscreenorrelativetothecoordinatesystem.YoucanspecifythisontabBasicofthePropertiesDialogoftheimage.YoumayspecifyuptothreecornerpointsoftheimageontabPositionofthePropertiesDialog.Thisgivesyoutheflexibilitytoscale,rotate,andevendistortimages(alsoseecommandCorner).

Corner1:positionofthelowerleftcorneroftheimage

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Corner2:positionofthelowerrightcorneroftheimage Note:ThiscornermayonlybesetifCorner1wassetbefore.Itcontrolsthewidthoftheimage.

Corner4:positionoftheupperleftcorneroftheimage Note:ThiscornermayonlybesetifCorner1wassetbefore.Itcontrolstheheightoftheimage.

Example:CreatethreepointsA,B,andCtoexploretheeffectsofthecornerpoints.

SetpointAasthefirstandpointBasthesecondcornerofyourimage.BydraggingpointsAandBin Movemodeyoucanexploretheirinfluence.

Now,removepointBasthesecondcorneroftheimage.SetpointAasthefirstandpointCasthefourthcornerandexplorehowdraggingthepointsnowinfluencestheimage.

Finally,youmaysetallthreecornerpointsandseehowdraggingthepointsdistortsyourimage.

Example:Youalreadysawhowtoinfluencethepositionandsizeofyourimage.IfyouwanttoattachyourimagetoapointAandsetitswidthto3anditsheightto4units,youcoulddothefollowing:

SetCorner1toA SetCorner2toA + (3, 0) SetCorner4toA + (0, 4)

Note:IfyounowdragpointAin Movemode,thesizeofyourimagedoesnotchange.YoumayspecifyanimageasaBackgroundImageontabBasicofthePropertiesDialog.Abackgroundimageliesbehindthecoordinateaxesandcannotbeselectedwiththemouseanymore.Note:Inordertochangethebackgroundsettingofanimage,youmayopenthePropertiesDialogbyselecting PropertiesfromtheEditmenu.TheTransparencyofanimagecanbechangedinordertoseeobjectsoraxesthatliebehindtheimage.YoucansetthetransparencyofanimagebyspecifyingaFillingvaluebetween0%and100%ontabStyleofthePropertiesDialog.

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3. AlgebraicInput

3.1. GeneralNotes

Thealgebraicrepresentationsofmathematicalobjects(e.g.,values,coordinates,equations)areshownintheAlgebraView.YoucancreateandmodifyobjectsbyusingtheInputBaratthebottomoftheGeoGebrawindow(seesectionsDirectInputandandCommands).Note:AlwayspresstheEnterkeyaftertypingalgebraicinputintotheInputBar.Note:PressingtheEnterkeyatanytimetogglesthefocusbetweentheInputBarandtheGraphicsView.ThisallowsyoutoenterexpressionsandcommandsintotheInputBarwithouthavingtoclickonitwiththemousefirst.

NamingObjects

YoucanassignacertainnametoanobjectwhenyoucreateitusingtheInputBar: Points:InGeoGebra,pointsarealwaysnamedusinguppercaseletters.Justtypein

thename(e.g.,A,P)andanequalsigninfrontofthecoordinatesorcommands.Examples: C = (2, 4),P = (1; 180),Complex = 2 + i

Vectors:Inordertodistinguishbetweenpointsandvectors,vectorsneedtohavealowercasenameinGeoGebra.Again,typeinthename(e.g.,v,u)andanequalsigninfrontofthecoordinatesorcommands.Examples:v = (1, 3),u = (3; 90),complex = 1 2i

Lines,circles,andconicsections:Theseobjectscanbenamedbytypinginthenameandacoloninfrontoftheirequationsorcommands.Examples:g: y = x + 3,c: (x-1)^2 + (y 2)^2 = 4,hyp: x^2 y^2 = 2

Functions:Youcannamefunctionsbytyping,forexample,f(x) =org(x)=infrontofthefunctionsequationorcommands. Examples:h(x) = 2 x + 4,q(x) = x^2,trig(x) = sin(x)

Note:

Ifyoudontmanuallyassignanametoanobject,GeoGebraassignsthenamesofnewobjectsinalphabeticalorder.

Youcancreateindiceswithinthenamesofobjectsbyusinganunderscore.ForexampleA1isenteredasA_1andsABisenteredass_{AB}.

ChangeValues

Therearetwowaysofmanipulatingafreeobjectsvalue: ChangethevalueoftheobjectbyenteringitsnameandthenewvalueintheInput

Bar(seesectionDirectInput).

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Example:Ifyouwanttochangethevalueofanexistingnumbera=3,typea = 5intotheInputBarandpresstheEnterkey.

Editthealgebraicrepresentation:Activatetool MoveanddoubleclickontheobjectintheAlgebraView.Thisopensatextboxwhereyoucanedittheobjectsvalue.PresstheEnterkeytoapplyyourchanges.

Note:Whilefreeobjectsvaluescanbechangeddirectly,thevaluesofdependentobjectscanonlybeinfluencedbychangingtheirparentobjectsorbyredefiningthedependentobject.

DisplayInputBarHistory

AfterplacingthecursorintheInputBaryoucanusetheupanddownarrowkeysofyourkeyboardinordertonavigatethroughpriorinputstepbystep.Note:Clickonthelittlequestionmark totheleftoftheInputBarinordertodisplaythehelpfeaturefortheInputBar.

InsertName,Value,orDefinitionofanObjectintotheInputBar

Insertthenameofanobject:Activatetool MoveandselecttheobjectwhosenameyouwanttoinsertintotheInputBar.Then,presstheF5keyonyourkeyboard.Note:ThenameoftheobjectisappendedtoanyexpressionyoutypedintotheInputBarbeforepressingtheF5key.Insertthevalueofanobject:Therearetwowaysofinsertinganobjectsvalue(e.g.,(1,3),3x5y=12)intotheInputBar.

Rightclick(MacOS:Ctrlclick)ontheobjectandselectitem CopytoInputBarfromtheappearingContextMenu.

Activatetool MoveandselecttheobjectwhosevalueyouwanttoinsertintotheInputBar.Then,presstheF4keyonyourkeyboard.Note:ThevalueoftheobjectisappendedtoanyexpressionyoutypedintotheInputBarbeforepressingtheF4key.

Insertthedefinitionofanobject:Therearetwowaysofinsertinganobjectsdefinition(e.g.,A=(4,2),c=Circle[A,B])intotheInputBar.

AltclickontheobjecttoinserttheobjectsdefinitionanddeletewhateverinputmighthavebeenintheInputBarbefore.

Activatetool MoveandselecttheobjectwhosedefinitionyouwanttoinsertintotheInputBar.Then,presstheF3keyonyourkeyboard.Note:ThedefinitionoftheobjectreplacesanyexpressionyoutypedintotheInputBarbeforepressingtheF3key.

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3.2. DirectInput

GeoGebracanworkwithnumbers,angles,points,vectors,segments,lines,conicsections,functions,andparametriccurves.YoucanentertheseobjectsintotheInputBarbyusingtheircoordinatesorequationsandpressingtheEnterkey.

3.2.1. NumbersandAngles

Numbers

YoucancreatenumbersbyusingtheInputBar.Ifyouonlytypeinanumber(e.g.,3),GeoGebraassignsalowercaseletterasthenameofthenumber.Ifyouwanttogiveyournumberaspecificname,youcantypeinthenamefollowedbyanequalsignandthenumber(e.g.,createadecimalrbytypinginr = 5.32).Note:InGeoGebra,numbersandanglesuseaperiod(.)asadecimalpoint.YoucanalsousetheconstantandtheEulerconstanteforexpressionsandcalculationsbyselectingthemfromthedropdownmenunexttotheInputBarorbyusingkeyboardshortcuts.Note:Ifthevariableeisnotusedasanameofanexistingobjectyet,GeoGebrawillrecognizeitastheEulerconstantifyouuseitinnewexpressions.

Angles

Anglesareenteredindegree()orradians(rad).Theconstantisusefulforradianvaluesandcanalsobeenteredaspi.Note:Youcanenteradegreesymbol()orthepisymbol()byusingthefollowingkeyboardshortcuts:

AltO(MacOS:CtrlO)forthedegreesymbol AltP(MacOS:CtrlP)forthepisymbol

Example:Youcanenteranangleindegree(e.g., = 60)orinradians(e.g., = pi/3).Note:GeoGebradoesallinternalcalculationsinradians.Thedegreesymbol()isnothingbuttheconstant/180usedtoconvertdegreeintoradians.Examples:

Ifa=30isanumber,then = aconvertsnumberatoanangle=30,withoutchangingitsvalue.

Ifyoutypeinb = / ,theangleisconvertedbacktothenumberb=30,withoutchangingitsvalue.

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SlidersandArrowKeys

FreenumbersandanglescanbedisplayedasslidersintheGraphicsView(seetoolSlider).Usingthearrowkeys,youmaychangethevalueofnumbersandanglesinthe

AlgebraViewtoo(seesectionManualAnimation).

LimitValuetoInterval

Freenumbersandanglesmaybelimitedtoaninterval[min,max]byusingtabSliderofthePropertiesDialog(seealsotool Slider).Note:FordependentanglesyoucanspecifywhethertheymaybecomereflexornotontabBasicofthePropertiesDialog.

3.2.2. PointsandVectors

PointsandvectorsmaybeenteredinCartesianorpolarcoordinates(seesectionNumbersandAngles).Note:Uppercaselabelsdenotepointswhereaslowercaselabelsrefertovectors.Examples:

ToenterapointPoravectorvinCartesiancoordinatesyoumayuseP = (1, 0)orv = (0, 5).

InordertousepolarcoordinatestypeinP = (1; 0)orv = (5; 90).Note:Youneedtouseasemicolontoseparatethetwocoordinates.Ifyoudonttypeinthedegreesymbol,GeoGebrawilltreattheangleasifenteredinradians.

InGeoGebra,youcanalsodocalculationswithpointsandvectors.Examples:

YoucancreatethemidpointMoftwopointsAandBbyenteringM = (A + B) / 2intotheInputBar.

Youmaycalculatethelengthofavectorvusinglength = sqrt(v * v)

3.2.3. LinesandAxes

Lines

YoucanenteralineasalinearequationinxandyorinparametricformintotheInputBar.Inbothcasespreviouslydefinedvariables(e.g.numbers,points,vectors)canbeusedwithintheequation.Note:Youcanenteralinesnameatthebeginningoftheinputfollowedbyacolon.Examples:

Typeing: 3x + 4y = 2toenterlinegasalinearequation. Defineaparametert(e.g.,t = 3)beforeenteringlineginparametricformusing

g: X = (-5, 5) + t (4, -3).

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Definetheparametersm = 2andb = -1.Then,youcanentertheequationh: y = m*x + btogetalinehinyinterceptform.

Axes

ThetwocoordinateaxesareavailableincommandsusingthenamesxAxisandyAxis.Example:ThecommandPerpendicular[A, xAxis]constructstheperpendicularlinetothexaxisthroughagivenpointA.

3.2.4. ConicSections

Youmayenteraconicsectionasaquadraticequationinxandy.Priordefinedvariables(e.g.,numbers,points,vectors)canbeusedwithintheconicsequation.Note:Theconicsectionsnamecanbeenteredatthebeginningoftheinputfollowedbyacolon.Examples:

Ellipseell: ell: 9 x^2 + 16 y^2 = 144 Hyperbolahyp: hyp: 9 x^2 16 y^2 = 144 Parabolapar: par: y^2 = 4 x Circlec1: c1: x^2 + y^2 = 25 Circlec2: c2: (x 5)^2 + (y + 2)^2 = 25

Note:Ifyoudefinetwoparametersa = 4andb = 3inadvance,youmayenterforexampleanellipseasell: b^2 x^2 + a^2 y^2 = a^2 b^2.

3.2.5. Functionsofx

Toenterafunctionyoucanusepreviouslydefinedvariables(e.g.numbers,points,vectors)aswellasotherfunctions.Examples:

Functionf: f(x) = 3 x^3 x^2 Functiong: g(x) = tan(f(x)) Namelessfunction: sin(3 x) + tan(x)

Note:Allavailablepredefinedfunctions(e.g.sin,cos,tan)aredescribedinsectionPredefinedFunctionsandOperations.InGeoGebrayoucanalsousecommandstogetforexample,theIntegralandDerivativeofafunction.Note:Youcanalsousethecommandsf'(x)orf''(x), inordertogetthederivativesofapreviouslydefinedfunctionf(x).Example:Definefunctionfasf(x) = 3 x^3 x^2.Then,youcantypeing(x) = cos(f' (x + 2))inordertogetfunctiong.

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Furthermore,functionscanbetranslatedbyavector(seecommandTranslate)andafreefunctioncanbemovedwiththemousebyusingtool Move.

LimitFunctiontoInterval

Inordertolimitafunctiontoaninterval[a,b],youcanusethecommandFunction.

3.2.6. PredefinedFunctionsandOperations

Tocreatenumbers,coordinates,orequations(seesectionDirectInput)youmayalsousethefollowingpredefinedfunctionsandoperations.Note:Thepredefinedfunctionsneedtobeenteredusingparentheses.Youmustnotputaspacebetweenthefunctionnameandtheparentheses.

Operation /Function InputAddition + Subtraction - Multiplication * orSpace key Scalarproduct * orSpacekey Division / Exponentiation ^ or 2 Factorial ! Gammafunction gamma( ) Parentheses ( ) xcoordinate x( ) ycoordinate y( ) Absolutevalue abs( ) Sign sgn( ) Squareroot sqrt( ) Cubicroot cbrt( ) Randomnumberbetween0and1 random( ) Exponentialfunction exp( ) or x Logarithm(natural,tobasee) ln( ) or log( ) Logarithmtobase2 ld( ) Logarithmtobase10 lg( ) Cosine cos( ) Sine sin( ) Tangent tan( ) Arccosine acos( ) Arcsine asin( ) Arctangent atan( ) Hyperboliccosine cosh( ) Hyperbolicsine sinh( ) Hyperbolictangent tanh( ) Antihyperboliccosine acosh( ) Antihyperbolicsine asinh( )

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Operation /Function InputAntihyperbolictangent atanh( ) Greatestintegerlessthanorequal floor( ) Leastintegergreaterthanorequal ceil( ) Round round( )

3.2.7. BooleanVariablesandOperations

YoucanusetheBooleanvariablestrueandfalseinGeoGebra.Justtype,forexample,a = trueorb = falseintotheInputBarandpresstheEnterkey.

CheckBoxandArrowKeys

FreeBooleanvariablescanbedisplayedascheckboxesintheGraphicsView(seetoolCheckBoxtoShow/Hideobjects).Byusingthearrowkeysofyourkeyboardyoumayalso

changeBooleanvariablesintheAlgebraView(seesectionManualAnimation).Note:YoumayalsouseBooleanvariableslikenumbers(value0or1).Thisallowsyoutouseacheckboxasthedynamicspeedofananimatedsliderallowingyoutostartandstoptheanimation.Inthiscase,theanimationbuttonisonlyshownintheGraphicsViewifthereisalsoananimatedsliderwithstatic(i.e.nondynamic)speed.

Operations

YoucanusethefollowingoperationsforBooleanvariablesandconditionsinGeoGebrabyeitherselectingthemfromthelistnexttotheInputBarorbyenteringthemusingthekeyboard:

List Keyboard Example Objecttypes

Equal == a b or a == b numbers,points,lines,conicsa,b

Unequal != a b or a != b numbers,points,lines,conicsa,b

Lessthan a > b numbersa,bLessorequalthan

= b numbersa,bAnd && a bora && b Booleansa,bOr || a bora || b Booleansa,bNot ! aor!a BooleanaParallel a b linesa,bPerpendicular a b linesa,b

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3.2.8. ListObjectsandOperations

Usingcurlybracesyoucancreatealistofseveralobjects(e.g.points,segments,circles).Examples:

L = {A, B, C}givesyoualistconsistingofthreepriordefinedpointsA,B,andC. L = {(0, 0), (1, 1), (2, 2)} producesalistthatconsistsoftheentered

points,aswellasthesenamelesspoints. Note:Bydefault,theelementsofthislistarenotshownintheGraphicsView.

CompareListsofObjects

Youcancomparetwolistsofobjectsbyusingthefollowingsyntax: List1 == List2:Checksifthetwolistsareequalandgivesyoutrueorfalseasa

result. List1 != List2:Checksifthetwolistsarenotequalandgivesyoutrueorfalseas

aresult.

ApplyPredefinedOperationsandFunctionstoLists

Note:Ifyouapplyoperationsandpredefinedfunctionstolists,youwillalwaysgetanewlistasaresult.AdditionandSubtractionexamples:

List1 + List2:Addscorrespondingelementsoftwolists. Note:Thetwolistsneedtobeofthesamelength.

List + Number:Addsthenumbertoeveryelementofthelist. List1 List2:Subtractstheelementsofthesecondlistfromcorresponding

elementsofthefirstlist. Note:Thelistsneedtobeofthesamelength.

List Number:Subtractsthenumberfromeveryelementofthelist.MultiplicationandDivisionexamples:

List1 * List2:Multipliescorrespondingelementsoftwolists. Note:Thelistsneedtobeofthesamelength.Ifthetwolistsarecompatiblematrices,matrixmultiplicationisused.

List * Number:Multiplieseverylistelementwiththenumber. List1 / List2:Divideselementsofthefirstlistbycorrespondingelementsofthe

secondlist. Note:Thetwolistsneedtobeofthesamelength.

List / Number:Divideseverylistelementbythenumber. Number / List:Dividesthenumberbyeveryelementofthelist.

Examplesusingfunctions:

List^2:Squareseveryelementofthelist. sin(List):Appliesthesinefunctiontoeveryelementofthelist.

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3.2.9. MatrixObjectsandOperations

GeoGebraalsosupportsmatrices,whicharerepresentedasalistofliststhatcontaintherowsofthematrix.

Example:InGeoGebra,{{1,2,3},{4,5,6},{7,8,9}}representsthematrix

987654321

.

MatrixOperations

Additionandsubtractionexamples: Matrix1 + Matrix2:Addsthecorrespondingelementsoftwocompatible

matrices. Matrix1 Matrix2:Subtractsthecorrespondingelementsoftwocompatible

matrices.Multiplicationexamples:

Matrix * Number:Multiplieseveryelementofthematrixbythegivennumber. Matrix1 * Matrix2:Usesmatrixmultiplicationtocalculatetheresultingmatrix.

Note:Therowsofthefirstandcolumnsofthesecondmatrixneedtohavethesamenumberofelements.Example:{{1, 2}, {3, 4}, {5, 6}} * {{1, 2, 3}, {4, 5, 6}}givesyouthematrix{{9,12,15},{19,26,33},{29,40,51}}.

2x2 Matrix * Point(or Vector):Multipliesthematrixwiththegivenpoint/vectorandgivesyouapointasaresult. Example:{{1, 2}, {3, 4}} * (3, 4)givesyouthepointA=(11,25).

3x3 Matrix * Point(orVector):Multipliesthematrixwiththegivenpoint/vectorandgivesyouapointasaresult. Example:{{1, 2, 3}, {4, 5, 6}, {0, 0, 1}} * (1, 2)givesyouthepointA=(8,20). Note:Thisisaspecialcaseforaffinetransformationswherehomogenouscoordinatesareused:(x,y,1)forapointand(x,y,0)foravector.Thisexampleisthereforeequivalentto:{{1, 2, 3}, {4, 5, 6}, {0, 0, 1}} * {1, 2, 1}.

Otherexamples(seealsosectionMatrixCommands):

Determinant[Matrix]:Calculatesthedeterminantforthegivenmatrix. Invert[Matrix]:Invertsthegivenmatrix Transpose[Matrix]:Transposesthegivenmatrix

3.2.10. ComplexNumbersandOperations

GeoGebradoesnotsupportcomplexnumbersdirectly,butyoumayusepointstosimulateoperationswithcomplexnumbers.

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Example:Ifyouenterthecomplexnumber3 + 4iintotheInputBar,yougetthepoint(3,4)intheGraphicsView.Thispointscoordinatesareshownas3+4iintheAlgebraView.Note:YoucandisplayanypointasacomplexnumberintheAlgebraView.OpenthePropertiesDialogforthepointandselectComplexNumberfromthelistofCoordinatesformatsontabAlgebra.Ifthevariableihasnotalreadybeendefined,itisrecognizedastheorderedpairi=(0,1)orthecomplexnumber0+1i.Thisalsomeans,thatyoucanusethisvariableiinordertotypecomplexnumbersintotheInputBar(e.g.,q = 3 + 4i).Additionandsubtractionexamples:

(2 + 1i) + (1 2i)givesyouthecomplexnumber31i. (2 + 1i) - (1 2i)givesyouthecomplexnumber1+3i.

Multiplicationanddivisionexamples:

(2 + 1i) * (1 2i)givesyouthecomplexnumber43i. (2 + 1i) / (1 2i)givesyouthecomplexnumber0+1i.

Note:Theusualmultiplication(2, 1)*(1, -2)givesyouthescalarproductofthetwovectors.Otherexamples:GeoGebraalsorecognizesexpressionsinvolvingrealandcomplexnumbers.

3 + (4 + 5i)givesyouthecomplexnumber7+5i. 3 - (4 + 5i)givesyouthecomplexnumber15i. 3 / (0 + 1i)givesyouthecomplexnumber03i. 3 * (1 + 2i)givesyouthecomplexnumber3+6i.

3.3. Commands

Usingcommandsyoucanproducenewandmodifyexistingobjects.Note:Acommand'sresultmaybenamedbyenteringalabelfollowedbyanequalsign(=).Intheexamplebelow,thenewpointisnamedS.Example:TogettheintersectionpointoftwolinesgandhyoucanenterS = Intersect[g, h](seecommandIntersect).Note:Youcanalsouseindiceswithinthenamesofobjects:A1isenteredasA_1whileSABiscreatedusings_{AB}.

AutomaticCompletionofCommands

WhenyoutypeacommandintoGeoGebrasInputBar,thesoftwaretriestoautomaticallycompletethecommandforyou.Thismeansthatafteryoutypedinthefirsttwolettersof

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thecommandintotheInputBar,GeoGebradisplaysthefirstcommandofanalphabeticallysortedlistthatstartswiththeseletters.

Inordertoacceptthissuggestionandplacethecursorinbetweenthebrackets,hittheEnterkey.

Ifthesuggestedcommandisnottheoneyouwantedtotypein,justkeeptyping.GeoGebrawilladaptitssuggestionstothelettersyouenter.

3.3.1. GeneralCommands

ConstructionStep

ConstructionStep[]:ReturnsthecurrentConstructionProtocolstepasanumber.ConstructionStep[Object]:ReturnstheConstructionProtocolstepforthegivenobject

asanumber.

Delete

Delete[Object]:Deletestheobjectandallitsdependentsobjects.Note:Alsoseetool DeleteObject

Relation

Relation[Object a, Object b]:Showsamessageboxthatgivesyouinformationabouttherelationofobjectaandobjectb.Note:Thiscommandallowsyoutofindoutwhethertwoobjectsareequal,ifapointliesonalineorconic,orifalineistangentorapassinglinetoaconic.

Note:Alsoseetool Relation

3.3.2. BooleanCommands

If

If[Condition, Object]:Yieldsacopyoftheobjectiftheconditionevaluatestotrue,andanundefinedobjectifitevaluatestofalse.

If[Condition, Object a, Object b]:Yieldsacopyofobjectaiftheconditionevaluatestotrue,andacopyofobjectbifitevaluatestofalse.

IsDefined

IsDefined[Object]:Returnstrueorfalsedependingonwhethertheobjectisdefinedornot.

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IsInteger

IsInteger[Number]:Returnstrueorfalsedependingwhetherthenumberisanintegerornot.

3.3.3. NumberCommands

AffineRatio

AffineRatio[Point A, Point B, Point C]:ReturnstheaffineratioofthreecollinearpointsA,B,andC,whereC=A+*AB.

Area

Area[Point A, Point B, Point C, ...]:CalculatestheareaofthepolygondefinedbythegivenpointsA,B,C,

Area[Conic c]:Calculatestheareaofaconicsectionc(circleorellipse).Note:

Inordertocalculatetheareabetweentwofunctiongraphs,youneedtousethecommandIntegral.

Alsoseetool Area

AxisStep

AxisStepX[]:Returnsthecurrentstepwidthforthexaxis.AxisStepY[]:Returnsthecurrentstepwidthfortheyaxis.Note:TogetherwiththeCornerandSequencecommands,theAxisStepcommandsallowyoutocreatecustomaxes(alsoseesectionCustomizingCoordinateAxesandGrid).

BinomialCoefficient

BinomialCoefficient[Number n, Number r]:Calculatesthebinomialcoefficientnchooser.

Circumference

Circumference[Conic]:Returnsthecircumferenceofacircleorellipse.

CrossRatio

CrossRatio[Point A, Point B, Point C, Point D]:CalculatesthecrossratiooffourcollinearpointsA,B,C,andD,where=AffineRatio[B,C,D]/AffineRatio[A,C,D].

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Curvature

Curvature[Point, Function]:Calculatesthecurvatureofthefunctioninthegivenpoint.

Curvature[Point, Curve]:Calculatesthecurvatureofthecurveinthegivenpoint.

Distance

Distance[Point A, Point B]:YieldsthedistanceoftwopointsAandB.Distance[Point, Line]:Yieldsthedistanceofthepointandtheline.Distance[Line g, Line h]:Yieldsthedistanceoftheparallellinesgandh.

Note:Thedistanceofintersectinglinesis0.Thus,thiscommandisonlyinterestingforparallellines.

Note:Alsoseetool DistanceorLength

GCD

UKEnglish:HCFGCD[Number a, Number b]:Calculatesthegreatestcommondivisorofnumbersaandb

(UKEnglish:HCF=highestcommonfactor).GCD[List of Numbers]:Calculatesthegreatestcommondivisorofthelistofnumbers

(UKEnglish:HCF=highestcommonfactor).

IntegerDivision

Div[Number a, Number b]:Calculatestheintegerquotientfordivisionofnumberabynumberb.

Integral

Integral[Function, Number a, Number b]:Returnsthedefiniteintegralofthefunctionintheinterval[a,b].Note:Thiscommandalsodrawstheareabetweenthefunctiongraphoffandthexaxis.

Integral[Function f, Function g, Number a, Number b]:Yieldsthedefiniteintegralofthedifferencef(x)g(x)intheinterval[a,b]. Note:Thiscommandalsodrawstheareabetweenthefunctiongraphsoffandg.

Note:AlsoseecommandforIndefiniteIntegral

Iteration

Iteration[Function, Number x0, Number n]:Iteratesthefunctionntimesusingthegivenstartvaluex0. Example:Afterdefiningf(x) = x^2thecommandIteration[f, 3, 2]givesyoutheresult(32)2=81.

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LCM

LCM[Number a, Number b]:Calculatestheleastcommonmultipleoftwonumbersaandb(UKEnglish:LCM=lowestcommonmultiple).

LCM[List of numbers]:Calculatestheleastcommonmultipleoftheelementsofthelist(UKEnglish:LCM=lowestcommonmultiple).

Length

Length[Vector]:Yieldsthelengthofthevector.Length[Point A]:Yieldsthelengthofthepositionvectorofthegivenpoint.Length[Function, Number x1, Number x2]:Yieldsthelengthofthefunctiongraph

intheinterval[x1,x2].Length[Function, Point A, Point B]:Yieldsthelengthofthefunctiongraph

betweenthetwopointsAandB. Note:Ifthegivenpointsdonotlieonthefunctiongraph,theirxcoordinatesareusedtodeterminetheinterval.

Length[Curve, Number t1, Number t2]:Yieldsthelengthofthecurvebetweentheparametervaluest1andt2.

Length[Curve c, Point A, Point B]:YieldsthelengthofcurvecbetweentwopointsAandBthatlieonthecurve.

Length[List]:Yieldsthelengthofthelistwhichisthenumberofelementsinthelist.Note:Alsoseetool DistanceorLength

LinearEccentricity

LinearEccentricity[Conic]:Calculatesthelineareccentricityoftheconicsection.Note:Thelineareccentricityisthedistancebetweenaconic'scenteranditsfocus,oroneofitstwofoci.

LowerSum

LowerSum[Function, Number a, Number b, Number n]:Yieldsthelowersumofthegivenfunctionontheinterval[a,b]withnrectangles.Note:Thiscommanddrawstherectanglesforthelowersumaswell.

MinimumandMaximum

Min[Number a, Number b]:Yieldstheminimumofthegivennumbersaandb.Max[Number a, Number b]:Yieldsthemaximumofthegivennumbersaandb.

ModuloFunction

Mod[Integer a, Integer b]:Yieldstheremainderwhenintegeraisdividedbyintegerb.

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Parameter

Parameter[Parabola]:Returnstheparameteroftheparabola,whichisthedistanceofdirectrixandfocus.

Perimeter

Perimeter[Polygon]:Returnstheperimeterofthepolygon.

Radius

Radius[Circle]:Returnstheradiusofthecircle.

Randomcommands

RandomBetween[Min Integer, Max Integer]:Generatesarandomintegerbetweenminandmax(inclusive).

RandomBinomial[Number n of Trials, Probability p]:Generatesarandomnumberfromabinomialdistributionwithntrialsandprobabilityp.

RandomNormal[Mean, Standard Deviation]:Generatesarandomnumberfromanormaldistributionwithgivenmeanandstandarddeviation.

RandomPoisson[Mean]:GeneratesarandomnumberfromaPoissondistributionwithgivenmean.

SemiMajorAxisLength

SemiMajorAxisLength[Conic]:Returnsthelengthofthesemimajoraxis(halfofthemajoraxis)oftheconicsection.

SemiMinorAxisLength

SemiMinorAxisLength[Conic]:Returnsthelengthofthesemiminoraxis(halfoftheminoraxis)oftheconicsection.

Slope

Slope[Line]:Returnstheslopeofthegivenline. Note:ThiscommandalsodrawstheslopetrianglewhosesizemaybechangedontabStyleofthePropertiesDialog.

Note:Alsoseetool Slope

TrapezoidalSum

UKEnglish:TrapeziumSumTrapezoidalSum[Function, Number a, Number b, Number n]:Calculatesthe

trapezoidalsumofthefunctionintheinterval[a,b]usingntrapezoids. Note:Thiscommanddrawsthetrapezoidsofthetrapezoidalsumaswell.

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UpperSum

UpperSum[Function, Number a, Number b, Number n]:Calculatestheuppersumofthefunctionontheinterval[a,b]usingnrectangles. Note:Thiscommanddrawstherectanglesoftheuppersumaswell.

3.3.4. AngleCommand

Angle

Angle[Vector v1, Vector v2]:Returnstheanglebetweentwovectorsv1andv2(between0and360).

Angle[Line g, Line h]:Returnstheanglebetweenthedirectionvectorsoftwolinesgandh(between0and360).

Angle[Point A, Point B, Point C]:ReturnstheangleenclosedbyBAandBC(between0and360),wherepointBistheapex.

Angle[Point A, Point B, Angle]:ReturnstheangleofsizedrawnfrompointAwithapexB. Note:ThepointRotate[A,,B]iscreatedaswell.

Angle[Conic]:Returnstheangleoftwistofaconicsectionsmajoraxis(seecommandAxes).

Angle[Vector]:Returnstheanglebetweenthexaxisandgivenvector.Angle[Point]:Returnstheanglebetweenthexaxisandthepositionvectorofthegiven

point.Angle[Number]:Convertsthenumberintoanangle(resultbetween0and2pi).Angle[Polygon]:Createsallanglesofapolygoninmathematicallypositiveorientation(i.

e.,counterclockwise). Note:Ifthepolygonwascreatedincounterclockwiseorientation,yougettheinteriorangles.Ifthepolygonwascreatedinclockwiseorientation,yougettheexteriorangles.

Note:Alsoseetools Angleand AnglewithGivenSize

3.3.5. PointCommands

Center

UKEnglish:CentreCenter[Conic]:Returnsthecenterofacircle,ellipse,orhyperbola.Note:Alsoseetool MidpointorCenter

Centroid

Centroid[Polygon]:Returnsthecentroidofthepolygon.

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Corner

Corner[Number n of Corner]:CreatesapointatthecorneroftheGraphicsView(n=1,2,3,4)whichisnevervisibleonscreen.

Corner[Image, Number n of Corner]:Createsapointatthecorneroftheimage(n=1,2,3,4).

Corner[Text, Number n of Corner]:Createsapointatthecornerofthetext(n=1,2,3,4).

Note:Thenumberingofthecornersiscounterclockwiseandstartsatthelowerleftcorner.

Extremum

UKEnglish:TurningPointExtremum[Polynomial]:Yieldsalllocalextremaofthepolynomialfunctionaspointson

thefunctiongraph.

Focus

Focus[Conic]:Yields(all)focioftheconicsection.

InflectionPoint

InflectionPoint[Polynomial]:Yieldsallinflectionpointsofthepolynomialaspointsonthefunctiongraph.

Intersect

Intersect[Line g, Line h]:Yieldstheintersectionpointoflinesgandh.Intersect[Line, Conic]:Yieldsallintersectionpointsofthelineandconicsection

(max.2).Intersect[Line, Conic, Number n]:Yieldsthenthintersectionpointofthelineand

theconicsection.Intersect[Conic c1, Conic c2]:Yieldsallintersectionpointsofconicsectionsc1

andc2(max.4).Intersect[Conic c1, Conic c2, Number n]:Yieldsthenthintersectionpointof

conicsectionsc1andc2.Intersect[Polynomial f1, Polynomial f2]:Yieldsallintersectionpointsof

polynomialsf1andf2.Intersect[Polynomial f1, Polynomial f2, Number n]:Yieldsthenth

intersectionpointofpolynomialsf1andf2.Intersect[Polynomial, Line]:Yieldsallintersectionpointsofthepolynomialandthe

line.Intersect[Polynomial, Line, Number n]:Yieldsthenthintersectionpointofthe

polynomialandtheline.Intersect[Function f, Function g, Point A]:Calculatestheintersectionpoint

offunctionsfandgbyusingNewton'smethodwithinitialpointA.Intersect[Function, Line, Point A]:Calculatestheintersectionpointofthe

functionandthelinebyusingNewton'smethodwithinitialpointA.

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Note:Alsoseetool IntersecttwoObjects

Midpoint

Midpoint[Point A, Point B]:ReturnsthemidpointofpointsAandB.Midpoint[Segment]:Returnsthemidpointofthesegment.Note:Alsoseetool MidpointorCenter

Point

Point[Line]:Returnsapointontheline.Point[Conic]:Returnsapointontheconicsection.Point[Function]:Returnsapointonthefunction.Point[Polygon]:Returnsapointonthepolygon.Point[Vector ]:Returnsapointonthevector.Point[Point, Vector]:Createsanewpointbyaddingthevectortothegivenpoint.Note:Alsoseetool NewPoint

Root

Root[Polynomial]:Yieldsallrootsofthepolynomialasintersectionpointsofthefunctiongraphandthexaxis.

Root[Function, Number a]:YieldsonerootofthefunctionusingtheinitialvalueaforNewton'smethod.

Root[Function, Number a, Number b]:Yieldsonerootofthefunctionintheinterval[a,b](regulafalsi).

Vertex

Vertex[Conic]:Returns(all)verticesoftheconicsection.

3.3.6. VectorCommands

CurvatureVector

CurvatureVector[Point, Function]:Yieldsthecurvaturevectorofthefunctioninthegivenpoint.

CurvatureVector[Point, Curve]:Yieldsthecurvaturevectorofthecurveinthegivenpoint.

Direction

Direction[Line]:Yieldsthedirectionvectoroftheline. Note:Alinewithequationax+by=chasthedirectionvector(b,a).

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PerpendicularVector

PerpendicularVector[Line]:Returnstheperpendicularvectoroftheline. Note:Alinewithequationax+by=chastheperpendicularvector(a,b).

PerpendicularVector[Vector v]:Returnstheperpendicularvectorofthegivenvector. Note:Avectorwithcoordinates(a,b)hastheperpendicularvector(b,a).

UnitPerpendicularVector

UnitPerpendicularVector[Line]:Returnstheperpendicularvectorwithlength1ofthegivenline.

UnitPerpendicularVector[Vector]:Returnstheperpendicularvectorwithlength1ofthegivenvector.

UnitVector

UnitVector[Line]:Yieldsthedirectionvectorwithlength1ofthegivenline.UnitVector[Vector]:Yieldsavectorwithlength1,whichhasthesamedirectionand

orientationasthegivenvector.

Vector

Vector[Point A, Point B]:CreatesavectorfrompointAtopointB.Vector[Point]:Returnsthepositionvectorofthegivenpoint.Note:Alsoseetool VectorbetweenTwoPoints

3.3.7. SegmentCommand

Segment

Segment[Point A, Point B]:CreatesasegmentbetweentwopointsAandB.Segment[Point A, Number a]:CreatesasegmentwithlengthaandstartingpointA.

Note:Theendpointofthesegmentiscreatedaswell.Note:Alsoseetools SegmentbetweenTwoPointsand SegmentwithGivenLength

fromPoint

3.3.8. RayCommand

Ray

Ray[Point A, Point B]:CreatesaraystartingatpointAthroughpointB.Ray[Point, Vector v]:Createsaraystartingatthegivenpointwhichhasthedirection

vectorv.Note:Alsoseetool RaythroughTwoPoints

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3.3.9. PolygonCommand

Polygon

Polygon[Point A, Point B, Point C,...]:ReturnsapolygondefinedbythegivenpointsA,B,C,

Polygon[Point A, Point B, Number n]:Createsaregularpolygonwithnvertices(includingpointsAandB).

Note:Alsoseetools Polygonand RegularPolygon

3.3.10. LineCommands

AngleBisector

AngleBisector[Point A, Point B, Point C]:ReturnstheanglebisectoroftheangledefinedbypointsA,B,andC. Note:PointBisapexofthisangle.

AngleBisector[Line g, Line h]:Returnsbothanglebisectorsofthelines.Note:Alsoseetool AngleBisector

Asymptote

Asymptote[Hyperbola]:Yieldsbothasymptotesofthehyperbola.

Axes

Axes[Conic]:Returnsthemajorandminoraxisofaconicsection.

ConjugateDiameter

ConjugateDiameter[Line, Conic]:Returnstheconjugatediameterofthediameterthatisparalleltotheline(relativetotheconicsection).

ConjugateDiameter[Vector, Conic]:Returnstheconjugatediameterofthediameterthatisparalleltothevector(relativetotheconicsection).

Directrix

Directrix[Parabola]:Yieldsthedirectrixoftheparabola.

Line

Line[Point A, Point B]:CreatesalinethroughtwopointsAandB.Line[Point, Parallel Line]:Createsalinethroughthegivenpointparalleltothe

givenline.Line[Point, Direction Vector v]:Createsalinethroughthegivenpointwith

directionvectorv.

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Note:Alsoseetool LinethroughTwoPoints

MajorAxis

MajorAxis[Conic]:Returnsthemajoraxisoftheconicsection.

MinorAxis

MinorAxis[Conic]:Returnstheminoraxisoftheconicsection.

PerpendicularLine

PerpendicularLine[Point, Line]:Createsalinethroughthepointperpendiculartothegivenline.

PerpendicularLine[Point, Vector]:Createsalinethroughthepointperpendiculartothegivenvector.

Note:Alsoseetool PerpendicularLine

PerpendicularBisector

PerpendicularBisector[Point A, Point B]:YieldstheperpendicularbisectorofthelinesegmentAB.

PerpendicularBisector[Segment]:Yieldstheperpendicularbisectorofthesegment.Note:Alsoseetool PerpendicularBisector

Polar

Polar[Point, Conic]:Createsthepolarlineofthegivenpointrelativetotheconicsection.

Note:Alsoseetool PolarorDiameterLine

Tangent

Tangent[Point, Conic]:Creates(all)tangentsthroughthepointtotheconicsection.Tangent[Line, Conic]:Creates(all)tangentstotheconicsectionthatareparallelto

thegivenline.Tangent[Number a, Function]:Createsthetangenttothefunctionatx=a.Tangent[Point A, Function]:Createsthetangenttothefunctionatx=x(A).

Note:x(A)isthexcoordinateofpointA.Tangent[Point, Curve]:Createsthetangenttothecurveinthegivenpoint.Note:Alsoseetool Tangents

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3.3.11. ConicSectionCommands

Circle

Circle[Point M, Number r]:YieldsacirclewithmidpointMandradiusr.Circle[Point M, Segment]:YieldsacirclewithmidpointMwhoseradiusisequalto

thelengthofthegivensegment.Circle[Point M, Point A]:YieldsacirclewithmidpointMthroughpointA.Circle[Point A, Point B, Point C]:YieldsacirclethroughthegivenpointsA,B

andC.Note:Alsoseetools Compass, CirclewithCenterthroughPoint, CirclewithCenter

andRadius,and CirclethroughThreePoints

Conic

Conic[Point A, Point B, Point C, Point D, Point E]:ReturnsaconicsectionthroughthefivegivenpointsA,B,C,D,andE. Note:Iffourofthepointslieononelinetheconicsectionisnotdefined.

Note:Alsoseetool ConicthroughFivePoints

Ellipse

Ellipse[Point F, Point G, Number a]:CreatesanellipsewithfocalpointsFandGandsemimajoraxislengtha. Note:Condition:2a>Distance[F,G]

Ellipse[Point F, Point G, Segment]:CreatesanellipsewithfocalpointsFandGwherethelengthofthesemimajoraxisequalsthelengthofthegivensegment.

Ellipse[Point F, Point G, Point A]:CreatesanellipsewithfociFandGpassingthroughpointA.

Note:Alsoseetool Ellipse

Hyperbola

Hyperbola[Point F, Point G, Number a]:CreatesahyperbolawithfocalpointsFandGandsemimajoraxislengtha. Note:Condition:0

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OsculatingCircle

OsculatingCircle[Point, Function]:Yieldstheosculatingcircleofthefunctioninthegivenpoint.

OsculatingCircle[Point, Curve]:Yieldstheosculatingcircleofthecurveinthegivenpoint.

Parabola

Parabola[Point F, Line g]:ReturnsaparabolawithfocalpointFanddirectrixg.Note:Alsoseetool Parabola

3.3.12. FunctionCommands

ConditionalFunctions

YoucanusetheBooleancommandIfinordertocreateaconditionalfunction.Note:Youcanusederivativesandintegralsofsuchfunctionsandintersectconditionalfunctionslikenormalfunctions.Examples:

f(x) = If[x < 3, sin(x), x^2]givesyouafunctionthatequalssin(x)forx

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Function

Function[Function, Number a, Number b]:Yieldsafunctiongraph,thatisequaltofontheinterval[a,b]andnotdefinedoutsideof[a,b].

Note:Thiscommandshouldbeusedonlyinordertodisplayfunctionsinacertaininterval.Example:f(x) = Function[x^2, -1, 1]givesyouthegraphoffunctionx2intheinterval[1,1].Ifyouthentypeing(x) = 2 f(x)youwillgetthefunctiong(x)=2x2,butthisfunctionisnotrestrictedtotheinterval[1,1].

Integral

Integral[Function]:Yieldstheindefiniteintegralforthegivenfunction.Note:AlsoseecommandforDefiniteintegral

Polynomial

Polynomial[Function]:Yieldstheexpandedpolynomialfunction. Example:Polynomial[(x - 3)^2]yieldsx26x+9.

Polynomial[List of n points]:Createstheinterpolationpolynomialofdegreen1throughthegivennpoints.

Simplify

Simplify[Function]:Simplifiesthetermsofthegivenfunctionifpossible. Examples:

Simplify[x + x + x]givesyouafunctionf(x)=3x. Simplify[sin(x) / cos(x)]givesyouafunctionf(x)=tan(x). Simplify[-2 sin(x) cos(x)]givesyouafunctionf(x)=sin(2x).

TaylorPolynomial

TaylorPolynomial[Function, Number a, Number n]:Createsthepowerseriesexpansionforthegivenfunctionaboutthepointx=atoordern.

3.3.13. ParametricCurveCommand

Curve

Curve[Expression e1, Expression e2, Parameter t, Number a, Number b]:YieldstheCartesianparametriccurveforthegivenxexpressione1andyexpressione2(usingparametert)withinthegiveninterval[a,b].Example:Inputofc = Curve[2 cos(t), 2 sin(t), t, 0, 2 pi]createsacirclewithradius2aroundtheoriginofthecoordinatesystem.

Note:Parametriccurvescanbeusedwithpredefinedfunctionsandarithmeticoperations.Example:Inputc(3)returnsthepointatparameterposition3oncurvec.

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Note:Usingthemouseyoucanalsoplaceapointonacurveusingtool NewPointorcommandPoint.Sincetheparametersaandbaredynamicyoucoulduseslidervariablesaswell(seetool Slider).

CommandsforParametricCurves

Curvature[Point, Curve]:Calculatesthecurvatureofthecurveinthegivenpoint.CurvatureVector[Point, Curve]:Yieldsthecurvaturevectorofthecurveinthegiven

point.Derivative[Curve]:Returnsthederivativeoftheparametriccurve.Derivative[Curve, Number n]:Returnsthe