H* Define 2 Points in R 3 *L P = 81, 2, 3< Q = 84, 6, 6< PQvec = Q - P. H* Plot a Single Red Point of "Size" 0.05 *L

Transcription

1 Define and plotting a point and vector H* Define 2 Points in R 3 *L P = 81, 2, 3< Q = 84, 6, 6< PQvec = Q - P H* Plot a Single Red Point of "Size" 0.05 *L Graphics3D@8PointSize@0.05D, Red, Point@PD<D H* Plot Multiple Points *L PQPlot = Graphics3D@88PointSize@0.05D, Blue, Point@PD<, 8PointSize@0.05D, Green, Point@QD<<D H* Plot a Single Black Arrow from P to Q of "Thickness" 0.02 *L Graphics3D@8Thickness@0.02D, Black, Arrow@8P, Q<D<D 81, 2, 3< 84, 6, 6< 83, 4, 3<

2 2 BasicPlottingDemo.nb

3 BasicPlottingDemo.nb 3 Defining and Plotting a Curve r(t)={x(t),y(t),z(t)}

4 4 BasicPlottingDemo.nb = 84 * Cos@tD, 4 * Sin@tD * Cos@tD, t< P1 = ParametricPlot3D@r@tD, 8 t, 0, 8 <, PlotStyle Ø Directive@Thickness@0.02D, Black D, AspectRatio Ø 1, AxesLabel Ø 8x, y, z<, Ticks Ø None, LabelStyle Ø Directive@Large, Bold, BlackD D 84 Cos@tD, 4 Cos@tD Sin@tD, t< z x y Defining and Plotting an Explicit Function z=f(x,y)

5 BasicPlottingDemo.nb 5 f@x_, y_d = 2 * Sin@xD * Cos@yD + 3 P2 = Plot3D@8f@x, yd<, 8 x, -2, 2 <, 8y, -2, 2 <, AxesLabel Ø 8x, y, z<, PlotStyle Ø Directive@Opacity@0.8DD, PlotRange Ø All, LabelStyle Ø Directive@Large, Bold, BlackDD Cos@yD Sin@xD Plotting an Implicit Function g(x,y)=c

6 6 BasicPlottingDemo.nb P3 = ContourPlot@x ^ 2 + y ^ 2 ã 1, 8x, -2, 2<, 8y, -2, 2 <, ContourStyle Ø Directive@Black, Thickness@0.02DDD 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, -4, 4<, 8y, -4, 4<, 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, -4, 4<, 8y, -4, 4<, 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

7 BasicPlottingDemo.nb x Cos@yD

8 8 BasicPlottingDemo.nb Contour Plots: The relationship between the surface g(x,y,z)=c and the 4D thing w=g(x,y,z)

9 BasicPlottingDemo.nb 9 g@x_, y_, z_d = x ^ 2 + y ^ 2 - z ^ 2 P4 = ContourPlot3D@g@x, y, zd ã 1, 8x, -2, 2<, 8y, -2, 2<, 8z, -1, 1<, AxesLabel Ø 8x, y, z<, Mesh Ø True, ContourStyle Ø Directive@FaceForm@White, GrayD, Specularity@White, 40D, Opacity@0.6DD, Ticks Ø None, LabelStyle Ø Directive@Large, Bold, BlackD, AspectRatio Ø 1D x 2 + y 2 - z 2 Curves on Surfaces:

10 10 BasicPlottingDemo.nb y_d = 2 * Sin@xD ^ 2 * Cos@yD ^ x@t_d = Cos@tD y@t_d = Sin@tD P5a = ParametricPlot3D@ 8x@tD, y@td, f@x@td, y@tdd<, 8t, 0, 2 * Pi<, PlotStyle Ø Directive@Black, Thickness@0.02DD, AxesLabel Ø 8x, y, z<, PlotRange Ø All, LabelStyle Ø Directive@Large, Black, BoldD, Ticks Ø NoneD; P5b = Plot3D@8f@x, yd<, 8 x, -2, 2 <, 8y, -2, 2 <, AxesLabel Ø 8x, y, z<, PlotStyle Ø None, PlotRange Ø AllD; P5 = Show@8P5a, P5b <D Cos@yD 2 Sin@xD 2 Cos@tD Sin@tD z x y Curves of Intersection Remember that f (x, y) = x + 2 y - 2 g (x, y) = 2 x + y + 3 are explicit functions of x and y which we interpret to mean z = x + 2 y - 2 and z = 2 x + 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 z' s first (since they are already "solved" for z.

11 BasicPlottingDemo.nb 11 Remember that f (x, y) = x + 2 y - 2 g (x, y) = 2 x + y + 3 are explicit functions of x and y which we interpret to mean z = x + 2 y - 2 and z = 2 x + 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 z' s 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,

12 12 BasicPlottingDemo.nb 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<< 88y Ø 5 + x<< PossibleAnswer1 ã t, t, t< PossibleAnswer2 ã 8t, 5 + t, t<

13 BasicPlottingDemo.nb 13