Defining and Plotting a Parametric Curve in 3D a(t)=(x(t),y(t),z(t)) H* clears all previous assingments so we can reuse them *L

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1 Defining and Plotting a Parametric Curve in 3D a(t)=(x(t),y(t),z(t)) Clear@"Global`*"D H* clears all previous assingments so we can reuse them *L H* Define vector functions in 3D of one variable t, i.e. a curve in 3D *L H* to typeset the alpha, beta, and sigma using esc alpha esc, ditto beta, ditto sigma *L a@t_d = 8Cos@tD, Sin@tD, t< b@t_d = 8t, 2 * t, 3 * t< s@t_d = 8Cos@tD, Sin@tD, 0< H* Plot the curves a,b, s with no plotting options defined *L H* Give the plot an assigned name and use this name in the GraphicsGrid command to plot *L aplot1 = ParametricPlot3D@a@tD, 8t, 0, 2 * Pi<D ; bplot1 = ParametricPlot3D@b@tD, 8t, 0, 2 * Pi<D ; splot1 = ParametricPlot3D@s@tD, 8t, 0, 2 * Pi<D ; H* Plot the curve with some plotting options *L aplot2 = ParametricPlot3D@a@tD, 8t, 0, 2 * Pi<, PlotStyle Ø Directive@Red, Thickness@0.015DD, AxesLabel Ø 8x, y, z<, LabelStyle Ø Directive@Black, 15DD; bplot2 = ParametricPlot3D@b@tD, 8t, 0, 2 * Pi<, PlotStyle Ø Directive@Red, Thickness@0.015D, DashedD, AxesLabel Ø 8x, y, z<, LabelStyle Ø Directive@Black, 15DD; splot2 = ParametricPlot3D@s@tD, 8t, 0, 2 * Pi<, PlotStyle Ø Directive@Red, Thickness@0.015D, DotDashedD, AxesLabel Ø 8x, y, z<, LabelStyle Ø Directive@Black, 15DD; H* Display the two plots side by side *L GraphicsGrid@

2 2 BasicPlotting.nb 88aplot1, bplot1, splot1<, 8aplot2, bplot2, splot2<< D H* Display all the plots on one set of axes *L plots1 = Show@8aplot1, bplot1, splot1<d; plots2 = Show@8aplot2, bplot2, splot2<d; H* Now plot the combined plots side by side *L GraphicsGrid@88plots1, plots2< <D 8Cos@tD, Sin@tD, t< 8t, 2 t, 3 t< 8Cos@tD, Sin@tD, 0< x y 6 4 z y z x y z x x y z 2 0

3 BasicPlotting.nb 3 Defining and Plotting Explicit Functions z=f(x,y)

4 4 BasicPlotting.nb H* clears all previous assingments so we can reuse them *L H* Define some explicit functions z=fhx,yl, z=ghx,yl, z=hhx,yl, i.e. surfaces in 3D *L x_, y_ D = x ^ 2 + y ^ 2 g@x_, y_d = Cos@xD * Sin@yD h@x_, y_d = 2 * x - 3 * y + 2 H* Plot surfaces with no plot options *L R = 5; plotf1 = Plot3D@f@x, yd, 8x, -R, R<, 8y, -R, R<D; plotg1 = Plot3D@g@x, yd, 8x, -R, R<, 8y, -R, R<D; ploth1 = Plot3D@h@x, yd, 8x, -R, R<, 8y, -R, R<D; H* Plot surfaces with some basic plot options *L R = 5; plotf2 = Plot3D@f@x, yd, 8x, -R, R<, 8y, -R, R<, PlotStyle Ø Directive@GreenDD; plotg2 = Plot3D@g@x, yd, 8x, -R, R<, 8y, -R, R<, PlotStyle Ø Directive@RedD, Mesh Ø False D; ploth2 = Plot3D@h@x, yd, 8x, -R, R<, 8y, -R, R<, PlotStyle Ø Directive@Blue, Opacity@0.6DD, Mesh -> FalseD; H* Plot surfaces with some more plot options *L Plot3D@f@x, yd, 8x, -R, R<, 8y, -R, R<, ImageSize Ø 8400, 400<, ColorFunction Ø "SunsetColors", Background Ø Black, Boxed -> False, Mesh Ø 20, PlotStyle Ø Directive@Opacity@0.8DDD H* Plot surfaces side by side *L GraphicsGrid@ 88plotf1, plotg1, ploth1<, 8plotf2, plotg2, ploth2<<d H* Group plots on one set of axes then display pairs side by side *L plot1 = Show@ 8plotf1, plotg1, ploth1< D; plot2 = Show@ 8plotf2, plotg2, ploth2< D; GraphicsGrid@ 88plot1, plot2<< D

5 BasicPlotting.nb x2 + y2 Cos@xD Sin@yD 2+2x-3y 5

6 6 BasicPlotting.nb

7 BasicPlotting.nb 7 Defining and Plotting Implicit Functions f(x,y,z)=c where c is some constant ü Recall Implicit Functions f(x,y)=c and their relation to z=f(x,y) Clear@"Global`*"D H* clears all previous assingments so we can reuse them *L H* define an expression in the variables x and y *L f@x_, y_d = x * Cos@yD; H* ContourPlot is how you plot an implicit function fhx,yl=c *L plot1 = ContourPlot@8f@x, yd ã -1, f@x, yd ã 1 ê 2<, 8x, -2, 2<, 8y, -2, 2<, AxesLabel Ø 8x, y<, ContourStyle Ø Directive@Black, Thickness@0.01DDD; H* In terms of the surface z=fhx,yl, how do we geometrically interpret fhx,yl=c? *L plot2 = Plot3D@8f@x, yd, -1, 1 ê 2<, 8x, -2, 2<, 8y, -2, 2<, PlotStyle Ø 8Directive@Green, Specularity@White, 20DD, Directive@Black, Opacity@0.6DD, Directive@Black, Opacity@0.4DD<, Lighting Ø "Neutral", Mesh Ø False, ImageSize Ø 8500, 500<, AxesLabel Ø 8x, y, z<d; H* Plot the ContourPlot and the Surface with z planes side by side *L GraphicsGrid@88 plot1, plot2 << D H* Just for fun lets make an animation for various values of the "z-slice" *L P1 = Manipulate@ContourPlot@f@x, yd ã c, 8x, -3, 3<, 8y, -3, 3<, AxesLabel Ø 8x, y<, ContourStyle Ø Directive@Black, Thickness@0.01DDD, 8c, -2, 3, 1<D; P2 = Manipulate@Plot3D@8f@x, yd, c<, 8x, -3, 3<, 8y, -3, 3<, PlotStyle Ø 8Directive@GreenD, Directive@Black, Opacity@0.6DD<, Mesh Ø False, ImageSize Ø 8500, 500<, AxesLabel Ø 8x, y, z<, ImageSize Ø 8200, 200<D, 8c, -2, 3, 1<D; GraphicsGrid@88P1, P2<<, ImageSize Ø 81200, 1200<D;

8 8 BasicPlotting.nb

9 BasicPlotting.nb 9 ü Implicit Functions f(x,y,z)=c and their relation to w=f(x,y,z) (ToDo When Get Faster Machine--ContourPlot3D) H* Define two expression each depending on three variables *L F@x_, y_, z_d = x ^ 2 + y ^ 2 + z ^ 2 G@x_, y_, z_d = Cos@xD * Sin@yD * z H* ContourPlot3D is how to plot FHx,y,zL=c *L H* ContourPlot3D@8F@x,y,zD==1,F@x,y,zD==4<, 8x,-2,2<,8y,-2,2<,8z,-2,2<,ContourStyleØ 8Directive@BlueD, Directive@Green,Opacity@0.5DD < D *L x 2 + y 2 + z 2 z Cos@xD Sin@yD

10 10 BasicPlotting.nb Curve in 2D a(t)=(x(t),y(t)), Curve b(t)=f(a(t)) on Surface z=f(x,y) in 3D

11 BasicPlotting.nb 11 H* clears all previous assingments so we can reuse them *L H* Define a functionêsurface z= fhx,yl on which we will "project" a curve *L f@x_, y_d = x ^ 2 + y ^ 2 * Sin@x * yd H* Define a 2D curve ahtl in the Hx,yL-plane *L x@t_d = Cos@tD y@t_d = Sin@tD a2@t_d = 8x@tD, y@td< H* Trick: Think of the 2D curve ahtl as a 3D curve in the Hx,y,z=0L-plane *L a3@t_d = 8x@tD, y@td, 0< H* Define a 3D curve bhtl= fhahtll which is constrained to the surface z=fhx,yl *L af@t_d = 8x@tD, y@td, f@x@td, y@tdd< H* Plots of the above *L a2plot = ParametricPlot@a2@tD, 8t, 0, 2 * Pi<, PlotStyle Ø Directive@ Black, Thickness@0.01DD, AxesLabel Ø 8x, y<, Frame Ø TrueD; a3plot = ParametricPlot3D@a3@tD, 8t, 0, 2 * Pi<, PlotStyle Ø Directive@ Black, Thickness@0.01DD, AxesLabel Ø 8x, y, z<, Boxed Ø TrueD; afplot = ParametricPlot3D@af@tD, 8t, 0, 2 * Pi<, PlotStyle Ø Directive@ Black, Thickness@0.015DD, AxesLabel Ø 8x, y, z<, Boxed Ø TrueD; fplot = Plot3D@f@x, yd, 8x, -1, 1<, 8y, -1, 1<, PlotStyle Ø Directive@Red, Opacity@0.8D, Specularity@White, 20DD, Mesh Ø False, AxesLabel Ø 8x, y, z<, Boxed Ø True, ImageSize Ø 8500, 500<D; H* Bring the 2D and 3D plots together *L plot1 = Show@8afplot, fplot, a3plot< D; GraphicsGrid@ 88a2plot, a3plot, plot1<< D

12 12 BasicPlotting.nb x 2 + y 2 Sin@x yd Cos@tD Sin@tD 8Cos@tD, Sin@tD< 8Cos@tD, Sin@tD, 0< 9Cos@tD, Sin@tD, Cos@tD 2 + Sin@tD 2 Sin@Cos@tD Sin@tDD= Bring All The Above Together: Curve of Intersection Problems ü (In Class Problem) Find the curve of intersection of the two surfaces f(x, y) = x + 2 y - 2 and g(x, y) = 2 x + y + 3. (Hint: Do the algebra work by hand) (Extra: Come up with plots to check your answer)

13 BasicPlotting.nb 13 (An Answer) : Remember that f(x,y)=x+2y-2 g(x,y)=2x+y+3 are explicit functions of x and y which we interpret to mean z=x+2y-2 and z=2x+y+3. To be an intersection curve the x, y and z's must all be equal. Our equations scream for us to focus on the zʼs first (since they are already solved for z. Clear@"Global`*"D H* clears all previous assingments so we can reuse them *L H* We want to find when the z values equal, so set them equal OR find when their difference is 0 *L Solve@x + 2 * y H2 * x + y + 3L ã 0, xd H* option 1, Solve for x in terms of y, i.e. find xhyl *L Solve@x + 2 * y H2 * x + y + 3L ã 0, yd H* option 2, Solve for y in terms of x, that is find yhxl *L H* At this point we can write at least two solutions *L a@y_d = y, y, 2 * H-5 + yl + y + 3 <; H* y is playing the role of the parameter t *L b@x_d = 8x, 5 + x, x + 2 * H5 + xl - 2< ; H* x is playing the role of the parameter t *L PossibleAnswer1 ã Simplify@a@tDD PossibleAnswer2 ã Simplify@b@tDD H* plot these curves *L SolutionCurve1 = ParametricPlot3D@a@yD, 8y, 0, 5<, PlotStyle Ø Directive@Green, Thickness@0.015DDD; SolutionCurve2 = ParametricPlot3D@b@xD, 8x, -1, 0<, PlotStyle Ø Directive@Green, Thickness@0.015DDD; H* plot the two original surfaces *L SurfacesPlot = Plot3D@8x + 2 * y - 2, 2 * x + y + 3<, 8x, -5, 5<, 8y, -5, 5<, PlotStyle Ø 8Directive@RedD, Directive@ Blue, Opacity@0.8DD<D; GraphicsGrid@ 88Show@SolutionCurve1, SurfacesPlot D, Show@SolutionCurve2, SurfacesPlot D<< D 88x Ø -5 + y<<

14 14 BasicPlotting.nb 88y Ø 5 + x<< PossibleAnswer1 ã t, t, t< PossibleAnswer2 ã 8t, 5 + t, t< ü (HomeWork 1) Find the curve of intersection of the two surfaces x + 3 y + 4 z - 2 = 0 and (x - 1) + (2 y + 2) - (z - 2) = 0 (Hint : Do the algebra work by hand) (Extra : Come up with plots to check your answer) (An Answer) : So this problem is like the In Class Problem but both surfaces are Implicit. Donʼt panic when you see an implicit form of a function. As a start, you can always try to turn it into an explicit form by solving for the easiest variable.

15 BasicPlotting.nb 15 H* clears all previous assingments so we can reuse them *L H* So the surfaces are implicit fhx,y,zl=c form, so remedy this "problem" by solving for the explicit form x=fhy,zl. *L foo = Expand@Hx - 1L + H2 * y + 2L - Hz - 2L ã 0D foo1 = Solve@foo, xd foo2 = Solve@x + 3 * y + 4 * z - 2 ã 0, xd a@y_, z_d = -3-2 * y + z; b@y_, z_d = 2-3 * y - 4 * z; H* So now we have two explicit functions for x as a function of y and z. Now repeat steps from Homework 1 *L SurfacesPlot = ContourPlot3D@8x + 3 * y + 4 * z - 2 ã 0, Hx - 1L + H2 * y + 2L - Hz - 2L ã 0<, 8x, -10, 10<, 8y, -10, 10<, 8z, -10, 10<, ContourStyle Ø 8Directive@RedD, Directive@ Blue, Opacity@0.8DD<, AxesLabel Ø 8x, y, z<d; H* Solve for y as a function of z *L Solve@a@y, zd - b@y, zd ã 0, yd H* Define a curve with parameter z and show its plot along with the two surfaces *L Solution@z_D = * H5-5 * zl + z, 5-5 * z, z < SolutionPlot = ParametricPlot3D@Solution@zD, 8z, 0, 2<, PlotStyle Ø Directive@Green, Thickness@0.015DD, AxesLabel Ø 8x, y, z<d; foo3 = GraphicsGrid@ 88Show@SolutionPlot, SurfacesPlot D<<, ImageSize Ø 8500, 500< D 3 + x + 2 y - z ã 0 88x Ø -3-2 y + z<< 88x Ø 2-3 y - 4 z<< 88y Ø -5 H-1 + zl<< H5-5 zl + z, 5-5 z, z< 3 + x + 2 y - z ã 0 88x Ø -3-2 y + z<< 88x Ø 2-3 y - 4 z<< 88y Ø -5 H-1 + zl<< H5-5 zl + z, 5-5 z, z<

16 16 BasicPlotting.nb 3 + x + 2 y - z ã 0 88x Ø -3-2 y + z<< 88x Ø 2-3 y - 4 z<< 88y Ø -5 H-1 + zl<< H5-5 zl + z, 5-5 z, z<

17 BasicPlotting.nb 17 ü (HomeWork 2) Find the curve of intersection of the two surfaces f(x,y)=-x+2y and x+(y+2)+2(z+3)=0. (Hint: Do the algebra work by hand) (Extra: Come up with plots to check your answer) (An Answer): Am I trying to trick you by giving you one implicit and one explicit function? No. Just follow the same steps as in previous examples. SurfacesPlot = Plot3D@ 8-x + 2 * y, -1 ê 2 * Hx + y L - 4<, 8x, -5, 5<, 8y, -5, 5<, PlotStyle Ø 8Directive@RedD, Directive@ Blue, Opacity@0.8DD<D; CurvePlot = ParametricPlot3D@ 8H5 * y + 8L, y, -H5 * y + 8L + 2 * y<, 8y, -5, 5<, PlotStyle Ø Directive@Green, Thickness@0.015DDD; Show@SurfacesPlot, CurvePlot D

18 18 BasicPlotting.nb ü (HomeWork, A Little More Challenging) Find the curve of intersection of the two surfaces x^2*y-z=5 and xy-z=4 (Hint: Do the algebra work by hand) (Extra: Come up with plots to check your answer) (An Answer) SurfacesPlot = ContourPlot3D@ 8x^2 * y - z ã 5, x * y - z ã 4<, 8x, -5, 5<, 8y, -5, 5<, 8z, -5, 5<, ContourStyle Ø 8Directive@RedD, Directive@ Blue, Opacity@0.8DD<D; a@x_d = 8x, 1 ê Hx^2 - xl, 1 ê Hx - 1L - 4< H* Notice from the formula for ahxl the problem values x= 0 and x=1 so take note when plotting *L CurvePlot1 = ParametricPlot3D@a@xD, 8x, -5, -0.1<, PlotStyle Ø Directive@Green, Thickness@0.015DDD; CurvePlot2 = ParametricPlot3D@a@xD, 8x, 0.1, 0.9<, PlotStyle Ø Directive@Green, Thickness@0.015DDD; CurvePlot3 = ParametricPlot3D@a@xD, 8x, 1.01, 5<, PlotStyle Ø Directive@Green, Thickness@0.015DDD; Show@SurfacesPlot, CurvePlot1, CurvePlot2, CurvePlot3D :x, 1, x + x x >

19 BasicPlotting.nb 19