Vectors-1.1 OVERVIEW 3D Analytic Geometry - Equations and Graphs 2017.06.05 .docx Page 1 of 16 Page 1 of 16 TomK Madison, WI TomK Madison, WI Before proceeding with 3D analytic geometry we point out that attempting to draw a 3D (three dimensional) shape on a 2D (two dimensional) sheet of paper or a computer screen produces inherently ambiguous results because information is guaranteed to be lost. EXAMPLE: Some people will think that the red dot is on the front surface of the cube, some will think that it is on the back surface, but in reality it could be somewhere in between and still generate this 2D picture. Do you think that the second picture shows the inside or the outside of a box? What your brain is doing must be very interesting. Nonetheless projecting 3D objects onto a 2D representation is usually tractable. Photographs work like this 1 . Also, the following picture is inherently ambiguous but we can help comprehension of the total situtation by rotating the picture around different axes. https://www.desmos.com/calculator/zbkralj0r4 So even with the inherent ambiguity, the 3D case usually isn’t too bad. However, attempting to represent a 4 dimensional entity as a 2D drawing involves so much information loss that it usually results in something useless. In the special case where the 4 th dimension represents time, using animation is the obvious graphical representation. Otherwise a graphical representation is problematic. 1 The human retina is essentially a curved 2D surface so it has the same information loss problem. However, when two eyes work together, each one sends a separate image from a different point of view to the brain. This extra information greatly enhances the perception of 3 dimensional objects. Try the experiment of look at something with just one eye, and then the other eye, and then both. This is what happens when watching a 3D movie.
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Vectors-1.1 OVERVIEW 3D Analytic Geometry - Equations and Graphs 2017.06.05 .docx Page 1 of 16
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
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.
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Page4of16 TomKMadison,WITomKMadison,WI
EXAMPLES:ExplicitlydefinedsurfacesFor x = 0 yandzarenotspecifiedsoallvaluesofyandzarepermittedonthesurface(aplane).For y = −1 xandzarenotspecifiedsoallvaluesofxandzarepermittedonthesurface(aplane).For z = 1 xandyarenotspecifiedsoallvaluesofxandyarepermittedonthesurface(aplane).
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Page9of16 TomKMadison,WITomKMadison,WI
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).
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Intersectionsoftwosurfaces.Onewaytospecifyaline(curvedorstraight)in3-spaceiswithtwosimultaneousequationsofsurfaces.Graphicallytheintersectionisaline(spacecurve).Intersectionoftwoplanes. z = 1.5 z = − 0.5y + 1.5