Vectors-1.1 OVERVIEW 3D Analytic Geometry - Equations and Graphs 2017.06.05 .docx Page 1 of 16

Page1of16 TomKMadison,WITomKMadison,WI

Beforeproceedingwith3Danalyticgeometrywepointoutthatattemptingtodrawa3D(threedimensional)shapeona2D(twodimensional)sheetofpaperoracomputerscreenproducesinherentlyambiguousresultsbecauseinformationisguaranteedtobelost.EXAMPLE: Somepeoplewillthinkthatthereddotisonthefrontsurfaceofthecube,somewillthinkthatitisonthebacksurface,butinrealityitcouldbesomewhereinbetweenandstillgeneratethis2Dpicture.Doyouthinkthatthesecondpictureshowstheinsideortheoutsideofabox?Whatyourbrainisdoingmustbeveryinteresting.

Nonethelessprojecting3Dobjectsontoa2Drepresentationisusuallytractable.Photographsworklikethis1.Also,thefollowingpictureisinherentlyambiguousbutwecanhelpcomprehensionofthetotalsitutationbyrotatingthepicturearounddifferentaxes.https://www.desmos.com/calculator/zbkralj0r4

Soevenwiththeinherentambiguity,the3Dcaseusuallyisn’ttoobad.However,attemptingtorepresenta4dimensionalentityasa2Ddrawinginvolvessomuchinformationlossthatitusuallyresultsinsomethinguseless.Inthespecialcasewherethe4thdimensionrepresentstime,usinganimationistheobviousgraphicalrepresentation.Otherwiseagraphicalrepresentationisproblematic. 1Thehumanretinaisessentiallyacurved2Dsurfacesoithasthesameinformationlossproblem.However,whentwoeyesworktogether,eachonesendsaseparateimagefromadifferentpointofviewtothebrain.Thisextrainformationgreatlyenhancestheperceptionof3dimensionalobjects.Trytheexperimentoflookatsomethingwithjustoneeye,andthentheothereye,andthenboth.Thisiswhathappenswhenwatchinga3Dmovie.

Vectors-1.1 OVERVIEW 3D Analytic Geometry - Equations and Graphs 2017.06.05 .docx Page 2 of 16

Page2of16 TomKMadison,WITomKMadison,WI

AnalyticgeometryinthreedimensionsThe2Dgeometricalideasfromtheprecedingchapterscanbenaturallyextendedto3geometricaldimensions.Weaddathirdaxis( z -axis)perpendiculartothe x − y planewith x relatedto y asshownbelow.Ifthepositivez-axispointsupwardsthenyouhavea"righthanded"coordinatesystem.Ifthepositivez-axispointsdownwardsthenyouhavea"lefthanded"coordinatesystem.Byconventionwewillonlydiscusstheright-handedcoordinatesystem.https://www.desmos.com/calculator/yo5ht4gchv

Youcanlocatepointsin3-spacelikethis:(1,2,3)representsthepointwhere x = 1 and y = 2 and z = 3 (3,0,4)representsthepointwhere x = 3 and y = 0 and z = 4

However,whenyouplotsinglepointsin3Dontoaflatsurfaceit’simpossibletofigureoutexactlywheretheyactuallyare.Toomuchinformationhasbeenlost.

Vectors-1.1 OVERVIEW 3D Analytic Geometry - Equations and Graphs 2017.06.05 .docx Page 3 of 16

Page3of16 TomKMadison,WITomKMadison,WI

EquationsofSurfacesin3-Space.Asurfaceconsistsofallthepointsthatsatisfyanyoftheequationtypesbelow.Thereareavarietyofwaysthatalgebraicequationsspecifysurfacesin3-space.Thethreemaingroupsoftheseequationsare:1)Explicitfunctionrulesoftheform z = f (x, y), y = g(x, z), x = h(y, z) 2)Implictlydefinedfunctionrules f (x, y, z) = 0 3)Parametricequationsets x = f (t), y = g(t), z = h(t){ } WeuseCalcPlot3Dtodrawtheimagesbelow.Itcanbeinvokedfromthemainmenu.AppendixNgivesanoverviewofhowtouseit.ArighthandedCartesiancoordinatesystem.

Vectors-1.1 OVERVIEW 3D Analytic Geometry - Equations and Graphs 2017.06.05 .docx Page 4 of 16

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EXAMPLES:ExplicitlydefinedsurfacesFor x = 0 yandzarenotspecifiedsoallvaluesofyandzarepermittedonthesurface(aplane).For y = −1 xandzarenotspecifiedsoallvaluesofxandzarepermittedonthesurface(aplane).For z = 1 xandyarenotspecifiedsoallvaluesofxandyarepermittedonthesurface(aplane).

yisnotspecifiedsoallvaluesofyarepermittedforanycombinationofxandzonthesurface

Vectors-1.1 OVERVIEW 3D Analytic Geometry - Equations and Graphs 2017.06.05 .docx Page 5 of 16

Page5of16 TomKMadison,WITomKMadison,WI

xisnotspecifiedsoallvaluesofxarepermittedforanycombinationofyandzonthesurface.

zisnotspecifiedsoallvaluesofzarepermittedforanycombinationofxandyonthesurface.

Vectors-1.1 OVERVIEW 3D Analytic Geometry - Equations and Graphs 2017.06.05 .docx Page 6 of 16

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zisnotspecifiedsoallvaluesofzarepermittedforanycombinationofxandyonthesurfaces

.

Vectors-1.1 OVERVIEW 3D Analytic Geometry - Equations and Graphs 2017.06.05 .docx Page 7 of 16

Page7of16 TomKMadison,WITomKMadison,WI

xisnotspecifiedsoallvaluesofxarepermittedforanycombinationofyandzonthesurface(s).Youbasicallyspecifya2Dgraphonthey-zplaneandthenyouextendthatforeverinthe+and-directionsalongthex-axis

Vectors-1.1 OVERVIEW 3D Analytic Geometry - Equations and Graphs 2017.06.05 .docx Page 8 of 16

Page8of16 TomKMadison,WITomKMadison,WI

yisnotspecifiedsoallvaluesofyarepermittedforanycombinationofxandzonthesurface(s).Youbasicallyspecifya2Dgraphonthex-zplaneandthenyouextendthatforeverinthe+and–directionsalongthey-axis.

zisnotspecifiedsoallvaluesofzarepermittedforanycombinationofxandyonthesurface(s).Youbasicallyspecifya2Dgraphonthex-yplaneandthenextendeditforeverinthe+and-directionsalongthez-axis.

Vectors-1.1 OVERVIEW 3D Analytic Geometry - Equations and Graphs 2017.06.05 .docx Page 9 of 16

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Vectors-1.1 OVERVIEW 3D Analytic Geometry - Equations and Graphs 2017.06.05 .docx Page 10 of 16

Page10of16 TomKMadison,WI

Equationsthatimplicitlydefineasetoffunctions.TheimplicitequationofaplanehasthegeneralformNx x − x0( ) + Ny y − y0( ) + Nz z − z0( ) = 0 where x0, y0, z0( ) isapointontheplaneandtheNx , Ny , and Nz characterizealineperpendiculartotheplane2.Notethat

foraconstantC, C Nx x − x0( ) + Ny y − y0( ) + Nz z − z0( )( ) = 0 givesthesameresult.Itisinterestingthatyoucan’tdescribealinein3-spaceusingthiskindofequation.Itdoesnotprovideenoughconstraintsforthepermissiblepointsthatmakeuptheline.InVectors2.1weshowhowtopreciselydefinealineusingasetof3parametricequations.Quadricsurfaces(Implicitlyspecifiedby2ndorderpolynomials).

Ellipsoid x2

a2+ y2

b2+ z2

c2= 0

2Weprovidemoredetailslaterafterwedefinegeometricvectors.

Vectors-1.1 OVERVIEW 3D Analytic Geometry - Equations and Graphs 2017.06.05 .docx Page 11 of 16

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EllipticParaboloid x2

a2+ y2

b2− zc

= 0

HyperbolicParaboloid x2

a2− y2

b2− zc

= 0

Vectors-1.1 OVERVIEW 3D Analytic Geometry - Equations and Graphs 2017.06.05 .docx Page 12 of 16

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Cone x2

a2+ y2

b2− z2

c2= 0

HyperboloidofOneSheet x2

a2+ y2

b2− z2

c2− d 2 = 0

Vectors-1.1 OVERVIEW 3D Analytic Geometry - Equations and Graphs 2017.06.05 .docx Page 13 of 16

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HyperboloidofTwoSheets− x2

a2− y2

b2+ z2

c2− d 2 = 0, .

a = 1 b = 0.8 c = 1 d = 1

Aspherewithnormallinestothesurface x − a( )2 + y − b( )2 + z − c( )2 − d 2 = 0

Vectors-1.1 OVERVIEW 3D Analytic Geometry - Equations and Graphs 2017.06.05 .docx Page 14 of 16

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Somenon-quadricimplicitsurfaces.Youcangetsomeprettyweirdsurfaces.a ln x + bey + c2z2 − d 2 = 0

z = 7xy / e(x

2 + y2 )

InVectors2.1weshowhowtodefineasurfaceusingasetof3parametricequations.

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Intersectionsoftwosurfaces.Onewaytospecifyaline(curvedorstraight)in3-spaceiswithtwosimultaneousequationsofsurfaces.Graphicallytheintersectionisaline(spacecurve).Intersectionoftwoplanes. z = 1.5 z = − 0.5y + 1.5

IntersectionofHyperbolicParaboloidandaplane.

It’sveryhardtoseetheintersectionwhenwedrawthispictureona2DsurfacebutinVectors2.1weshowhowtoalgebraicallydefinealineusingasetof3parametricequations.Onceyouknowhowtodothatyoucanparameterizetheintersectionofthetwosurfacesandexplicitlydisplayitasaspacecurveasshownbelow.

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.InVectors1.1-ADV-1weshowhowtoexpressorcomputethefollowingitems.Weusetheconceptofavector,whichwedefineinthenextchapter..Equationofaplanecontainingthepoint x0, y0, z0( ) andperpendiculartoastraightline.Equationofaplanepassingthroughthreepoints.DistancefromapointtoalineDistancefromapointtoaplane.Pointintersectionofalineandaplane(unlessthelineisintheplane).Lineparalleltotheintersectionoftwoplanes.Anglebetweenplanes