1. Textbook: Chapter 11 Part : 3D-Plotting CSE 1010

2. Demos … before we start … 1/3 3D-Plot and What Matlab Can Do … >> teapotdemo

3. Demos … before we start … 2/3 The Earth … load topo.mat [x,y,z] = sphere(50); cla reset axis square off props.AmbientStrength = 0.1; props.DiffuseStrength = 1; props.SpecularColorReflectance =.5; props.SpecularExponent = 20; props.SpecularStrength = 1; props.FaceColor= 'texture'; props.EdgeColor = 'none'; props.FaceLighting = 'phong'; props.Cdata = topo; surface(x,y,z,props); light('position',[-1 0 1]); light('position',[-1.5 0.5 -0.5], 'color', [.6.2.2]); view(3)

4. Demos … before we start … 3/3 Matlab Logo … L = 40*membrane(1,25); logoFig = figure('Color',[0 0 0]); logoax = axes('CameraPosition', [-193.4013 -265.1546 220.4819],... 'CameraTarget',[26 26 10],... 'CameraUpVector',[0 0 1],... 'CameraViewAngle',9.5,... 'DataAspectRatio', [1 1.9],... 'Position',[0 0 1 1],... 'Visible','off',... 'XLim',[1 51],... 'YLim',[1 51],... 'ZLim',[-13 40],... 'parent',logoFig); s = surface(L,... 'EdgeColor','none',... 'FaceColor',[0.9 0.2 0.2],... 'FaceLighting','phong',... 'AmbientStrength',0.3,... 'DiffuseStrength',0.6,... 'Clipping','off',... BackFaceLighting','lit',... 'SpecularStrength',1.1,... 'SpecularColorReflectance',1,... 'SpecularExponent',7,... 'Tag','TheMathWorksLogo',... 'parent',logoax); l1 = light('Position',[40 100 20],... 'Style','local',... 'Color',[0 0.8 0.8],... 'parent',logoax); l2 = light('Position',[.5 -1.4],... 'Color',[0.8 0.8 0],... 'parent',logoax);

5. Demos … before we start … 3/3

6. Contents Part 1 - 2D Plotting = 3D Plotting Part 2 - Linear 3D Plotting - Direct Linear Plotting - Parametric Linear Plotting Part 3 - 3D Mesh Plotting Part 4 - 3D Surface Plotting

7. 3D – Plotting Part 1 When 2D Potting = 3D-Plotting Linear Plotting

8. 3D – Plotting – Part 1 When 2D Plotting = 3D Plotting 1 – Create a 2D Plot 2 – Click icon “Rotate 3D”  3 – Woow!  We can extend 2-plots by adding a set of z values Syntax: Plot3 ( x, y, z, str) str is an optional parameter, It specifies color/line style The default is “solid blue” … Examples 

9. Example Plot3 ( x, y, z, str) str  is an optional parameter,  specifies color/line style defaultsolid blue … Examples 

10. 3D-PLotting Part 2 Linear 3D Plotting

11. 3D – Plotting – Part 2 Linear 3D Plotting Linear 3-D Plotting looks like … How do we do that …? 

12. Linear 3D Plotting x = 0 : 0.1 : 3.* pi;% each plot has the same set of x values y1 = zeros(size(x));% The y values for the first plot are all 0 z1 = sin(x);% z2 = sin(2.*x);% The 2 nd and 3 rd plots are sin(x) at different freq. z3 = sin(3.*x);% y3 = ones(size(x));% The y values of the 2 nd and 3 rd plots y2 = y3./2;% are all 0.5 and 1 respectively plot3(x,y1,z1, 'r', x,y2,z2, 'b', x, y3, z3, 'g') grid on% activates the grid xlabel('x-axis') ylabel('y-axis') zlabel('z-axis') NOTE:.* and./ :

14. 3D-PLooting Part 3 Parametric Plotting

15. 3D – Plotting – Part 3 Linear Parametric 3D Plot 3-D parametric plots allow the variables on each axis to be dependent on a separate, independent variable that defines a path in the plotting space  Two examples - The Spiral - 3D Motion of a particle

16. 3D – Plotting – Part 3: Linear Parametric 3D Example 1 – The Spiral The spiral as an example of a 3-D plot where two of the dimensions, x and y, are dependent on the third, independent parameter. The independent parameter is the rotation angle, θ, varying from 0 to 10 π (five complete revolutions). The x and y values are mapped as sin( θ ) and cos( θ )— the classic means of describing a circle. The spiral effect is accomplished by plotting θ on the z-axis.

17. theta = 0 : 0.1 : 10.*pi; % theta = θ plot3(sin(theta),cos(theta),theta) title( 'parametric curve based on angle'); grid on Example 1 = The Spiral

18. This is a fully parametric plot: - The values of ALL three coordinates are mappings of an independent parameter, t. - This is a plot of the 3D motion of a particle receiving random impulses in all three axes. Note the use of text anchored in x-y-z space to label points on the graph. Script …  Example 2: 3D Motion of a Particle

19. N = 20; dvx = rand(1, N) - 0.5 % random v changes dvy = rand(1, N) - 0.5 dvz = rand(1, N) - 0.5 vx = cumsum(dvx); % integrate to get v vy = cumsum(dvy); vz = cumsum(dvz); x = cumsum(vx); % integrate to get pos y = cumsum(vy); z = cumsum(vz); plot3(x,y,z) grid on title(‘All 3 axes varying with parameter t') text(0,0,0, ‘Start'); text(x(N),y(N),z(N), ‘End');

21. Part 4 Other Plot Capabilities bar3(x, y) bar3h(x, y) stem3(x, y,z) scatter3(x, y, z) pie3(y)

22. Other 3D Plot Capabilitie – Part 4 bar3(x, y) x = [ ] y = [ ] for k = -5:5 T = 1 + k + (k * 2) + (k * 3) x = [x k] % stores values for x-axis y = [y T] % stores values for y-axis end bar3(x, y) xlabel('x-axis') ylabel('y-axis') zlabel('z-axis')

23. Other 3D Plot Capabilitie – Part 4 bar3h(x, y) x = [ ] y = [ ] for k = -5:5 T = 1 + k + (k * 2) + (k * 3) x = [x k] % stores values for x-axis y = [y T] % stores values for y-axis end bar3h(x, y) % Does not work well!!! xlabel('x-axis') ylabel('y-axis') zlabel('z-axis')

24. Other 3D Plot Capabilitie – Part 4 stem3(x, y,z,’fill’) t = 0:0.2:10 x = t y = sin(t) z = t.* (1.5) stem3(x,y,z,'fill') grid on xlabel('x-axis') ylabel('y-axis') zlabel('z-axis')

25. Other 3D Plot Capabilitie Part 4 scatter3(x, y,z,’filled’) [x,y,z] = sphere(16); X = [x(:)*.5 x(:)*.75 x(:)]; Y = [y(:)*.5 y(:)*.75 y(:)]; Z = [z(:)*.5 z(:)*.75 z(:)]; S = repmat([1.75.5]*10,prod(size(x)),1); C = repmat([1 2 3],prod(size(x)),1); scatter3(X(:),Y(:),Z(:),S(:),C(:),'filled‘); view(-60,60) % view is explained later % sphere: built-in function % repmat: replicate a matrix % prod(x): product of all the % items in x % sphere  See next slide

26. sphere and peaks sphere and peaks are 2 built-in Matlab functions Their main use is to test the visualization (graphics) effects

27. sphere(n) The sphere function generates the x -, y -, and z- coordinates of a unit sphere for use with surf and mesh. sphere generates a sphere consisting of 20-by-20 faces. sphere(n) draws a surf plot of an n-by-n sphere in the current figure. [X,Y,Z] = sphere(n) returns the coordinates of a sphere in three matrices that are (n+1)-by-(n+1) in size. You draw the sphere with surf(X,Y,Z) or mesh(X,Y,Z). % Matlab script % From the Matlab website % sphere(16) optional parm sphere axis equal

28. Peaks in brief  Details peaks is a function of two variables, obtained by translating and scaling Gaussian distributions, which is useful for demonstrating mesh, surf, pcolor, contour, etc.meshsurfpcolorcontour Example of use: Z = peaks; % returns a 49-by-49 matrix % Matlab script % From the Matlab website [xx,yy,zz] = peaks(120); surf(xx,yy,zz) colormap 'default' shading faceted axis tight title('peaks') xlabel('x'),ylabel('y'),zlabel('z')

29. Matlab built-in function: peaks Function Syntax Z = peaks; Z = peaks(n); Z = peaks(V); Z = peaks(X,Y); peaks; peaks(N); peaks(V); peaks(X,Y); [X,Y,Z] = peaks; [X,Y,Z] = peaks(n); [X,Y,Z] = peaks(V); Description peaks is a function of two variables, obtained by translating and scaling Gaussian distributions, which is useful for demonstrating mesh, surf, pcolor, contour, etc. meshsurfpcolorcontour Z = peaks; returns a 49-by-49 matrix. Z = peaks(n); returns an n-by-n matrix. Z = peaks(V); returns an n-by-n matrix, where n = length(V). Z = peaks(X,Y); evaluates peaks at the given X and Y (which must be the same size) and returns a matrix the same size. peaks(...) (with no output argument) plots the peaks function with surf. > > help peaks … or go to: http://www.mathworks.com/access/helpdesk/help/techdoc/ref/peaks.html

30. Other 3D Plot Capabilitie – Part 4 pie3(y) y = [ 23 56 87 12 9 ] pie3(y)

31. Surface Plots Matlab Rendering Capabilities

32. 3D-Plot: Surface Plots 4 Basic concepts: 1 – plaid 2 – meshgrid 3 – mesh 4 - surf Plaid The “plaid” is a Matlab concept, NOT a Matlab function The underlying surface is referred to as plaid because of its conceptual similarity to a Scottish tartan pattem. To design such a pattern, one needs only to specify the color sequence of the horizontal and vertical threads. In the same way, we specify a plaid by defining vectors of the row and column data configurations. The simplest surface plots are obtained by defining a z value for each point on an x-y plaid.

33. meshgrid(x,y) … accepts the x 1xm and y 1xm vectors that bound the edges of the plaid, and replicates the rows and columns appropriately to produce xx nxm and yy 1xm, containing the x and y values (respectively) of the complete plaid This enables us in general to compute mappings for the 3-D coordinates of the figure we want to plot

34. mesh(xx,yy) mesh(xx,yy) plots the surface as white facets outlined by colored lines The line coloring uses one of many color maps (listed in Appendix A of the textbook), where the color is selected in proportion to the parameter You can turn the white facets transparent with the command “ hidden off ”.

35. surf(xx,yy,zz) plots the surface as colored facets outlined by black lines The line coloring by default is selected in proportion to the zz parameter The lines can be removed by using one of a number of shading commands listed in Appendix A of the textbook Also on the next slide

36. Shading Commands shading : shades a surface plot with one color per grid section SHADING controls the color shading of SURFACE and PATCH objects. SURFACE and PATCH objects are created by the functions SURF, MESH, PCOLOR, FILL, and FILL3 shading flat sets the shading of the current graph to flat. shading interp sets the shading to interpolated. shading faceted sets the shading to faceted, which is the default.  Read: > > >> help shading

37. More on Mesh and Surface Plots Common Rendering tools Available in Matlab Mesh mesh(x,y,z) Mesh Curtain meshz(x,y,z) Mesh Contour meshc(x,y,z) Waterfal waterfall(x,y,z) 3D Contour Plot contour3(x,y,z,n) 2D Contour Plot contour(x,y,z,n) Surf surf(x,y,z) Surface and contour surfc(x,y,z) Surface and lighting surfl(x,y,z) Details and Examples 

38. More on Mesh and Surface Plots Common Rendering tools Available in Matlab  The MESH Series In the following slides we use the same script to plot: x = -3:0.25:3; y = -3:0.25:3; [X, Y] = meshgrid(x,y) % build the plaid Z = (1.8).^ (-1.4) * sqrt((X.^2) + (Y.^2)).* cos (0.5 * Y ).* sin(X); The only line of code that is different is the Matlab function used to plot our [X,Y] values in the red box

39. Mesh mesh(x,y,z) x = -3:0.25:3; y = -3:0.25:3; [X, Y] = meshgrid(x,y) % build the plaid Z = (1.8).^(-1.4) * sqrt((X.^2) + (Y.^2)).* cos (0.5 * Y).* sin(X); mesh(X,Y,Z) xlabel('x-coordinate'); ylabel('y-coordinate'); zlabel('z-coordinate');

40. Mesh Curtain meshz(x,y,z) x = -3:0.25:3; y = -3:0.25:3; [X, Y] = meshgrid(x,y) % build the plaid Z = (1.8).^(-1.4) * sqrt((X.^2) + (Y.^2)).* cos (0.5 * Y).* sin(X); meshz(X,Y,Z) xlabel('x-coordinate'); ylabel('y-coordinate'); zlabel('z-coordinate'); %Draws a curtain around %the mesh

41. Mesh Contour meshc(x,y,z) x = -3:0.25:3; y = -3:0.25:3; [X, Y] = meshgrid(x,y) % build the plaid Z = (1.8).^(-1.4) * sqrt((X.^2) + (Y.^2)).* cos (0.5 * Y).* sin(X); meshc(X,Y,Z) xlabel('x-coordinate'); ylabel('y-coordinate'); zlabel('z-coordinate'); % Draws a contour plot % beneath the mesh

42. Mesh Waterfall waterfall(x,y,z) x = -3:0.25:3; y = -3:0.25:3; [X, Y] = meshgrid(x,y) % build the plaid Z = (1.8).^(-1.4) * sqrt((X.^2) + (Y.^2)).* cos (0.5 * Y).* sin(X); waterfall(X,Y,Z) xlabel('x-coordinate'); ylabel('y-coordinate'); zlabel('z-coordinate'); % Draws a mesh % in one direction % only

43. 3D Contour contour3(x,y,z,n) x = -3:0.25:3; y = -3:0.25:3; [X, Y] = meshgrid(x,y) % build the plaid Z = (1.8).^(-1.4) * sqrt((X.^2) + (Y.^2)).* cos (0.5 * Y).* sin(X); contour3(X,Y,Z,50) xlabel('x-coordinate'); ylabel('y-coordinate'); zlabel('z-coordinate'); % n is the number of % contour levels % optional

44. 2D Contour contour(x,y,z,n) x = -3:0.25:3; y = -3:0.25:3; [X, Y] = meshgrid(x,y) % build the plaid Z = (1.8).^(-1.4) * sqrt((X.^2) + (Y.^2)).* cos (0.5 * Y).* sin(X); contour(X,Y,Z,20) xlabel('x-coordinate'); ylabel('y-coordinate'); zlabel('z-coordinate'); % n is the number of % contour levels % optional % input is 3D % output is 2D

45. More on Mesh and Surface Plots Common Rendering tools Available in Matlab  The SURF (Surface) Series

46. Surface – Simple Version surf(x,y,z) x = -3:0.25:3; y = -3:0.25:3; [X, Y] = meshgrid(x,y) % build the plaid Z = (1.8).^(-1.4) * sqrt((X.^2) + (Y.^2)).* cos (0.5 * Y).* sin(X); surf(X,Y,Z) xlabel('x-coordinate'); ylabel('y-coordinate'); zlabel('z-coordinate'); %

47. Surface with parameters surf(x,y,z,c) % Color a sphere with the pattern of +1s and -1s % in a Hadamard matrix k = 5; n = 2^k-1; [x,y,z] = sphere(n); c = hadamard(2^k); surf(x,y,z,c); colormap ([1 1 0;0 1 1]) axis equal % Example from the % Matlab website

48. The “sinc” function (sin(r)/r) where r = radius we need to add eps to avoid inf when dividing by zero » [xx,yy]= meshgrid(-4.*pi:pi./5:4.*pi); » R=sqrt(xx.^2 + yy.^2)+eps; % radius » zz=sin(R)./R; » surf(xx,yy,zz) » axis tight eps : smallest possible difference between 2 floating point numbers inf : infinity

49. mesh(X,Y,Z) versus surf(X,Y,Z)

50. Surface and Contour Plot surfc(x,y,z) x = -3:0.25:3; y = -3:0.25:3; [X, Y] = meshgrid(x,y) % build the plaid Z = (1.8).^(-1.4) * sqrt((X.^2) + (Y.^2)).* cos (0.5 * Y).* sin(X); surfc(X,Y,Z) xlabel('x-coordinate'); ylabel('y-coordinate'); zlabel('z-coordinate'); %

51. Surface Plot with Lighting surfl(x,y,z) x = -3:0.25:3; y = -3:0.25:3; [X, Y] = meshgrid(x,y) % build the plaid Z = (1.8).^(-1.4) * sqrt((X.^2) + (Y.^2)).* cos (0.5 * Y).* sin(X); surfl(X,Y,Z) xlabel('x-coordinate'); ylabel('y-coordinate'); zlabel('z-coordinate'); %

52. Surface Plot with Triangles trisurf(tri,x,y,z) [x,y]=meshgrid(1:15,1:15); tri = delaunay(x,y); % Matlab built-in function z = peaks(15); trisurf(tri,x,y,z) % Example from the % Matlab website

53. Misc. Examples 

54. The Cube - Introduction Example 1 Uses the surf function  Listing

55. The Cube – Listing (textbook, p. 264) xx = [-1 -1 1 1 -1 % A-B-C-D-A -1 -1 1 1 -1] % D-E-F-G-D yy = [ 1 -1 -1 1 1 % A-B-C-D-A 1 -1 -1 1 1] % D-E-F-G-D zz = [ 1 1 1 1 1 % A-B-C-D-A -1 -1 -1 -1 -1] % D-E-F-G-D subplot(1, 2, 1) surf(xx, yy, zz) axis equal shading interp view(-36, 44) axis off xx = [ 0 0 0 0 0 % P-P-P-P-P -1 -1 1 1 -1 % A-B-C-D-A -1 -1 1 1 -1 % D-E-F-G-D 0 0 0 0 0] % Q-Q-Q-Q-Q yy = [ 0 0 0 0 0 % P-P-P-P-P 1 -1 -1 1 1 % A-B-C-D-A 1 -1 -1 1 1 % D-E-F-G-D 0 0 0 0 0] % Q-Q-Q-Q-Q zz = [ 1 1 1 1 1 % P-P-P-P-P 1 1 1 1 1 % A-B-C-D-A -1 -1 -1 -1 -1 % D-E-F-G-D -1 -1 -1 -1 -1] % Q-Q-Q-Q-Q subplot(1, 2, 2) surf(xx, yy, zz) axis equal shading interp view(-36, 44) % axis off

56. The Cube – 3D - Plot (p. 264)

57. Here we use the function meshgrid(x,y) to define the plaid This plot is created in three parts: 1.Develop the underlying plaid specifying the x-y location of every point on the x-y plane 2.Calculate the z values from the plaid 3.Call some function that will accept the plaid and these z values to produce the required plot meshgrid(x,y,z,C) Example: The Parabolic Dish (pp. 264-266)

58. % Parabolic dish x = -3:3; y = x; [xx,yy]= meshgrid(x,y); % generates the plaid zz = xx.^2 + yy.^2; % mapping of the z coordinates mesh(xx,yy,zz) % color is proportional to mesh height axis tight title('z = x^2 + y^2') xlabel('x'),ylabel('y'),zlabel('z') NOTE: Many more possibilities Read: >> help mesh

59. Recommended Exercises pp. 266-267: Exercise 11.2

60. Working with Colors 

61. 3D – Basic Peaks Colors (p. 267) [xx,yy,zz] = peaks(30); % Matlab peaks surf(xx,yy,zz) colormap 'default' shading interp axis tight title('peaks') xlabel('x'),ylabel('y'),zlabel('z') More on Colors 

62. Colors in Matlab (for 2D and 3D) pcolor colormap shading alpha

63. pcolor % Example n = 6;r = (0:n)'/n; theta = pi*(- n:n)/n; X = r*cos(theta); Y = r*sin(theta); C = r*cos(2*theta); pcolor(X,Y,C)axis equal tight In Matlab, pcolor - pseudocolor plot - is a rectangular array of cells with colors determined by C. MATLAB creates a pseudocolor plot using each set of four adjacent points in C to define a surface rectangle (i.e., cell). Complete description at: http://www.mathworks.com/access/helpdesk/help/techdoc/ref/pcolor.html Syntax: pcolor(C) pcolor(X,Y,C) pcolor(axes_handles,...) h = pcolor(...)

64. Explanations More about Colormap (Matlab web page): http://www.mathworks.com/access/helpdesk/help/techdoc/ref/colormap.html Built-in Colors User Defined Colors  Next Slide

65. User Defined Colors See Presentation: CSE1010_2009Fall_W-13_Ch11=ColorsInMatlab.ppt

66. Shading % From the Matlab website % Compare flat, faceted, interpolated-shaded shape subplot(3,1,1) sphere(16) axis square shading flat title('Flat Shading') subplot(3,1,2) sphere(16) axis square shading faceted title('Faceted Shading') subplot(3,1,3) sphere(16) axis equal shading interp title('Interpolated Shading')

67. alpha % Set transparency properties for objects in current axes surf(peaks); alpha(0.5);

68. More on … alpha plot(get(gcf,'Alphamap')) plot(alphamap('vup')) alphamap('increase',.4) plot(get(gcf,'Alphamap')) [x,y,z] = meshgrid(-1.25:.1:-.25,-2:.2:2,-2:.1:2); v = x.*exp(-x.^2-y.^2-z.^2); h = slice(x,y,z,v,[-1 -.75 -.5],[],[0]); alpha('color') set(h,'EdgeColor','none','FaceColor',... 'interp', 'FaceAlpha','interp') alphamap('rampdown') alphamap('increase',.1) colormap(hsv)

69. comet3: animation of a linear plot t = -10*pi:pi/250:10*pi; comet3((cos(2*t).^2).*sin(t),(sin(2*t).^2).*cos(t),t);

70. Drawnow: animation of a surface x = 0:pi/100:4*pi;y = x [X,Y]= meshgrid(x,y); z = 3*sin(X) + cos(Y); h = surf (z); axis tight; set (gca, 'nextplot', 'replacechildren'); shading interp; colormap (jet) m = 1; for k = 0 : pi/100 : 2* pi z = (sin(X) + cos (Y) ).* sin (k); set (h, 'Zdata',z); M(m) = getframe; m = m + 1; movie(M,2); drawnow End % Note: File: animation02.m

71. A better 3D Plot: Color and Lighting of the Peaks Textbook: p. 268 Let’s change the parameter to peaks to 120, and add the line light angle (60, 45) at the bottom of the script. This illuminates the surface with a light at the specified azimuth and elevation angle (in degrees).

72. [xx,yy,zz] = peaks(120); surf(xx,yy,zz) lightangle(60, 45) colormap 'default' shading interp axis tight title('peaks') xlabel('x'),ylabel('y'),zlabel('z') % Illumination of the surface with % a light at the specified azimuth % and elevation angle (degrees)

73. view(azimuth,elevation) The position of the viewer (the viewpoint) determines the orientation of the axes. You specify the viewpoint in terms of azimuth and elevation, or by a point in three-dimensional space. view(az,el) and view([az,el]) set the viewing angle for a three-dimensional plot. The default view angles are: az = 20º el = 30º

74. Surface Plotting – Parametric Surfaces Working with polar coordinates Examples: The cylinder (uses ONE angle) and the Sphere (uses TWO angles) Listing 

75. The Cylinder (p. 270) facets = 120; len = 2; radius = 1; % Constants to define the smoothness of the cylinder thr = linspace(0, 2*pi, facets); % Definition of a plaid using x and theta xr = [0 len]; [x, th] = meshgrid( xr, thr ); y = radius * cos(th); z = radius * sin(th); surf(x, y, z); shading interp axis equal,axis tight,axis off lightangle(60, 45) % Illuminate the figure alpha(0.8) % Sets the transparency of the surface so that view(-50, 35) % a portion of the hidden details can show through

76. The Sphere (textbook p. 271) facets = 120; radius = 1; thr = linspace(0, 2*pi, facets); % range of theta phir = linspace(0, pi, facets); % range of phi [th, phi] = meshgrid( thr, phir ); % builds the plaid in theta and phi x = radius * cos(phi); y = radius * sin(phi).* cos(th); z = radius * sin(phi).* sin(th); surf(x, y, z); shading interp colormap copper axis equal, axis tight, axis off lightangle(60, 45)

77. The Ellipsoid Effects: Meshing / No Meshing Shading parameters 

78. Ellipsoid – mesh view % From the Matlab website [x, y, z] = ellipsoid(0,0,0,5.9,3.25,3.25,30); surfl(x, y, z) % shading = not used colormap spring axis equal

79. Ellipsoid – faceted flat % From the Matlab website [x, y, z] = ellipsoid(0,0,0,5.9,3.25,3.25,30); surfl(x, y, z) shading flat colormap spring axis equal

80. Ellipsoid - Interpolation % From the Matlab website [x, y, z] = ellipsoid(0,0,0,5.9,3.25,3.25,30); surfl(x, y, z) shading interp colormap spring axis equal

81. Ellipsoid – mesh only % From the Matlab website clc; clear; [x, y, z] = ellipsoid(0,0,0,5.9,3.25,3.25,30); surfl(x, y, z) mesh(x, y, z) colormap spring axis equal

82. Bodies of Rotation - pp. 272 - 276 Rotating Continuous Functions Bodies of rotation are created by rotating a linear curve about a specified axis, that is, by rotating a general function v = f (u) defined over a range of u values about the x or z axes.

83. Rotating Continuous Functions

84. Rotating the v = u { 2 } about the x and z axes % set up the plaid facets = 100; x = linspace(0, 5, facets); th = linspace(0, 2*pi, facets); [xx tth] = meshgrid(x, th); % rotate about the x axis subplot(1, 2, 1) rr = xx.^2; yy = rr.* cos(tth); zz = rr.* sin(tth); surf(xx, yy, zz, xx); shading interp, axis tight xlabel('x'), ylabel('y'), zlabel('z') title('x^2 rotated about the x axis') % rotate about the z axis subplot(1, 2, 2) rr = xx; zz = rr.^2; xx = rr.* cos(tth); yy = rr.* sin(tth); surf(xx, yy, zz); shading interp, axis tight xlabel('x'), ylabel('y'), zlabel('z') title('x^2 rotated about the z axis')

86. Rotating Discrete Functions “2D profile” of a fictitious machine part and the picture created when that profile is rotated about the x-axis. The figure is generated by the code shown on the next slide 

87. x = [0 0 3 3 1.75 1.75 2 2 1.75 1.75 3 4... 5.25 5.25 5 5 5.25 5.25 3 3 6 6]; y = [0.5.5.502.502.55.55 1.75 1.75... 2.5 2.5 1.5 1.5 1.4 1.4....55.55.502.502.5.5 0]; subplot(1, 2, 1) plot(x, y) axis ([-1 7 -1 5]), axis off title('2-D profile') facets = 200; subplot(1, 2, 2) [xx tth] = meshgrid( x, linspace(0, 2*pi, facets) ); radius = meshgrid( y, 1:facets); yy = radius.* cos(tth); zz = radius.* sin(tth); surf(xx, yy, zz); shading interp axis square, axis tight, axis off colormap copper lightangle(60, 45) alpha(0.8) title('rotated object')

88. Other 3-D Surface Plot Capabilities

89. Other 3-D Surface Plot Capabilities 1/2 MATLAB can also create special-purpose plots with the following functions: alpha (x) sets the transparency of the surfaces: 0 <= x <= 1, where 0 means completely transparent and 1 is opaque.  The options available can be studied with >> help alpha contour (z) produces a contour plot of the plaid surface defined by z.  The options available can be studied with >> help bar3 [x,y, z] = cylinder(n) constructs the meshgrid for a cylinder with n facets in each direction.  For more options, see >> help cylinder [x,y,z] = ellipsoid(n) constructs the meshgrid for an ellipsoid with n facets in each direction.  For more options, see >> help ellipsoid lightangle (az, el) sets the angle of a light source (angles in degrees).  For more options, see >> help lightangle

90. Other 3-D Surface Plot Capabilities 2/2 meshc (x, y, z) makes a mesh plot with contours below.  For more options, see >> help meshc. meshz (x, y, z) makes a mesh plot with vertical line extensions  For more options, see >> help meshz. pie3 (y) makes a 3-D pie chart of the values in y.  For more options, see >> help pie3. [x,y,z] = sphere(n) constructs the meshgrid for a sphere with n facets in each direction.  For more options, see >> help sphere. surfc (x, y, z) makes a surface plot with contours below.  For more options, see >> help surfc surf z (x, y, z) makes a surface plot with vertical line extensions.  For more options, see >> help surf z. waterfall (x, y, z) makes a mesh plot with vertical line extensions only in the x direction.  For more options, see» help surf z.

92. Assembling Compound Surfaces We can assemble more complex solid bodies by constructing simple surfaces and concatenate the data before submitting it to the rendering machine. Example: A Solid Disk  (from the textbook)

93. %basic parameters facets = 200; len = 2; radius = 3; radial = [0, radius]; th = linspace(0, 2*pi, facets); along = [0 len]; % build the front face [r1, tth] = meshgrid( radial, th ); x1 = zeros(size(r1)); y1 = r1.*cos(tth); z1 = r1.*sin(tth); % build the curved surface [l, tth] = meshgrid(along, th); x2 = l; y2 = radius*cos(tth); z2 = radius*sin(tth); % build the back face x3 = len*ones(size(r1)); [r3, tth] = meshgrid(radial(end:-1:1), th); y3 = r3.*cos(tth); z3 = r3.*sin(tth); % assemble and draw the three parts x = [x1 x2 x3]; y = [y1 y2 y3]; z = [z1 z2 z3]; surf(x, y, z); shading interp colormap winter axis equal, axis tight, axis off

95. The Klein Bottle (textbook, p. 279) http://www.mathworks.com/products/matlab/demos.html?file =/products/demos/shipping/matlab/xpklein.html

96. Visualizing Geographic Data (p. 283) Example (from the textbook): You have been given two files of data: atlanta.txt, which presents the streets of Atlanta in graphical form, and ttimes.txt, which gives the travel times between Atlanta suburbs and the city center. You have been asked to present these data sets in a manner that will help to visualize and validate the data 

97. Analyzing the Data First we proceed to determine the nature of the data by opening the files in a text editor and examining their format and content. 1. Determine the file format: The first step is to open the data files in a plain text editor (the MATLAB editor would work fine). The format appears to be consistent with that of a text file delimited by tab characters. Since there are no strings in the file, it should be suitable to be read using MATLAB's built-in dlmread(...) function.

98. 2. Discern the “street map” file content: 2.The table on the next slides shows the first few lines of the file atlanta.txt simplified by omitting certain irrelevant columns. 3.The numbers in columns 3—6 are pairs, the first of the pair being a large negative number, and the second a smaller positive number. 4.Assuming that each row of this file is a street segment, these could be the x-y coordinates of the ends of a line. 2.A little thought confirms this guess when we realize that the latitude of Atlanta is —84º 42' relative to the Greenwich meridian, and its longitude is 330 65'— clearly, the values in these columns are 1,000,000 times the latitude and longitude of points within the city, probably each end of street segments. 3.Column 7 contains numbers mostly in the range 1—6, which could indicate the type of street. We could explore this idea by coloring each line according to that value

99. 3.Discern the “travel time” file content: Table 11.2 shows the first few lines of the file ttimes.txt simplified by omitting certain irrelevant columns. The same latitude/longitude values occur in columns 4 and 5, but they are not repeated, suggesting that the data in this file are in a different form. Examining the first two columns, the numbers in column 2 cycle repeatedly from 1 to 75, with column 1 counting the number of cycles up to 75. Furthermore, the values in column 5 are the same whenever column 1 is the same, and the values in column 4 are the same whenever the value in column 2 matches. This seems to be much like the plaid that results from a meshgrid(...) function call. The values in column 6 then become evident—they would be the z values of the plaid, and it seems reasonable to assume that they represent the travel time in minutes.

100. Street Map Data (part) … - 84546080.0033988480.00- 84558400.0033995480.001.00 … - 84243880.0033780010.00- 84249980.0033800840.001.00 … - 84243590.0033780060.00- 84249740.0033800840.001.00 … - 84509920.0033944340.00 - 84517200.0033958190.001.00 … - 84510420.0033944930.00- 84516490.0033957280.001.00 … - 84252840.0033895840.00- 84247360.0033899290.001.00 … - 84247360.0033899290.00- 84240250.0033903630.001.00 … - 84240250.0033903630.00- 84216090.0033911010.001.00 … - 84216090.0033911010.00 - 4203990.0033913990.001.00

101. % draw the streets raw = dlmread('atlanta.txt'); streets = raw(:,3:7); [rows, cols] = size(streets) colors = 'rgbkcmo'; for in = 1: rows x = streets(in,[1 3])/1000000; y = streets(in,[2 4])/1000000; col = streets(in,5); col(col < 1) = 7; col(col > 6) = 7; plot(x,y,colors(col)); hold on end % plot the travel times tt = dlmread('ttimes.txt'); [rows,cols] = size(tt) for in = 1:rows r = tt(in, 1); c = tt(in, 2); xc(r,c) = tt(in, 4)/1000000; yc(r,c) = tt(in, 5)/1000000; zc(r,c) = tt(in, 6); end surf(xc, yc, zc) shading interp alpha(.5) grid on axis tight xlabel('Longitude') ylabel('Latitude') zlabel('Travel Time (min)') view(-30, 45)

103. User Interaction with Plots Video at: http://www.mathworks.com/products/demos/shipping/matlab/WhatsNewR2008a_GraphicsAndGUIBuilding.html Brushing (textbook, p. 279) Linking a Plot to the Source Data … more: see video

104. Brushing Data Items Brush Tool: Script: th = linspace(1, 20*pi, 500) y = sin(th) plot(th,y)

105. Linking a Plot to the Source Data

106. Summary Basic two-dimensional line plots are accomplished by using plot(x,y) ■ Two-dimensional parametric plots are accomplished by using plot(x,y), where both x and y are dependent on another independent variable ■ Three-dimensional line and parametric plots are accomplished by using plot3(x,y,z) ■ Basic three-dimensional surface plots are accomplished by building a plaid using meshgrid(x,y), computing the zz layer as a function of xx and yy, then plotting the surface using mesh(xx, yy, zz) or surf(xx, yy, zz) ■ Parametric surface plots, like parametric line plots, are achieved by building the plaid with two independent variables and making xx, yy, and zz functions of those independent variables ■ Bodies of rotation are a special case of parametric surface plots where one of the independent variables is an angle with values between 0 and 2 (radians).