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Research Embassy

Oloyede

[MATLAB GRAPHICS]


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* * * * * * * being the TEXT presented at the workshop organized by the

Department of Industrial and Production Engineering, University of Ibadan.

Nigeria.

Facilitator: Oloyede I,

Department of Industrial and Production Engineering,

University of Ibadan. Nigeria. Contact:08053049890

All Praises, Adoration, Holiness,

Greeting are due to none but GOD

who taught MEN by PEN, who

taught MEN what he knows not. The

Omnipotent, The Omniscience.


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Contents:

Plot

Surface

Contour

Cylinder

Sphere


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GRAPHS

clear; % clear all variables from memoryclose all; % close all figure windowsh=input('Enter the step-size h - ') ;x=0:h:20; % build an array of points [0,h,2h,...,20]

f=exp(-x/6).*cos(x); % build the array [f(0),f(h),...f(20)]plot(x,f)fprintf(' Plot completed, h = %g \n',h)

clear;close all;dphi=pi/100; % set the spacing in azimuthal angleN=30; % set the number of azimuthal tripsphi=0:dphi:N*2*pi;theta=phi/N/2; % go from north to south oncer=1; % sphere of radius 1% convert spherical to Cartesianx=r*sin(theta).*cos(phi);y=r*sin(theta).*sin(phi);z=r*cos(theta);% plot the spiralplot3(x,y,z,'b-')axis equal

Surface, Contour, and Vector Field Plots

Meshgrid and NdgridMatlab displays functions of the type f(x, y), either by making a contour plot (like atopographic map) or by displaying the function as height above the xy plane like aperspective drawing. Start by defining arrays x and y that span the region that you wantto plot, then create the function f(x, y) over the plane, and finally either use contour, surf,ormesh.%begin plots

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%*********************************************************% Define the arrays x and y% Warning: dont make the step size too small or you will% kill the system%*********************************************************clear;close all;x=-1:.1:1;y=0:.1:1.5;%*********************************************************% Use meshgrid to convert these 1-d arrays into 2-d matrices of% x and y values over the plane%*********************************************************[X,Y]=meshgrid(x,y);%*********************************************************% Get f(x,y) by using f(X,Y). Note the use of .* with X and Y% rather than with x and y%*********************************************************

f=(2-cos(pi*X)).*exp(Y);%*********************************************************% Note that this function is uphill in y between x=-1 and x=1

% and has a valley at x=0%*********************************************************surf(X,Y,f);%end plots

Matlab has another command called ndgridwhich is similar to meshgridbut does theconversion to two dimensions the other way round. The plots look the same, but f(i, j)is now in the f(x, y) order:

x=-1:.1:1;y=0:.1:1.5;[X,Y]=ndgrid(x,y);f=(2-cos(pi*X)).*exp(Y);surf(X,Y,f);length(x)length(y)size(f)

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Contour Plots and Surface Plots

close all;%begin contour%*********************************************************% make a contour plot by asking Matlab to evenly space N contours

% between the minimum of f(x,y) and the maximum f(x,y) (the default)%*********************************************************N=40;contour(X,Y,f,N);title('Contour Plot')xlabel('x')ylabel('y')pause

%*********************************************************% You can also tell Matlab which contour levels you want to plot.% To do so load an array (V in this case) with the values of the

% levels you want to see. In this example they will start at% the minimum of f, end at the maximum of f, and there will% be 21 contours.%% You can even print contour labels on the plot, which is% a big help, by assigning the plot to a variable name% and using the clabel command, as shown below. Only% every other contour is labeled in this example%*********************************************************close all;top=max(max(f)); % find the max and min of fbottom=min(min(f));dv=(top-bottom)/20; % interval for 21 equally spaced contoursV=bottom:dv:top;cs=contour(X,Y,f,V);clabel(cs,V(1:2:21)) % give clabel the name of the plot and% an array of the contours to labeltitle(Contour Plot)xlabel(x)ylabel(y)pause%*********************************************************% Now make a surface plot of the function with the viewing% point rotated by AZ degrees from the x-axis and% elevated by EL degrees above the xy plane%*********************************************************close all;

surf(X,Y,f); % or you can use mesh(X,Y,f) to make a wire plotAZ=30;EL=45;view(AZ,EL);title(Surface Plot)xlabel(x)ylabel(y)pause%*********************************************************% If you want to manually change the viewing angle of% a surface plot, click on the circular arrow icon


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% on the figure window, then click and move the% pointer on the graph. Try it until you get the% hang of it.%% Heres a piece of code that lets you fly around the% surface plot by continually changing the viewing angles% and using the pause command; I think youll be impressed%*********************************************************close all;surf(X,Y,f);title(Surface Plot)xlabel(x)ylabel(y)EL=45;for m=1:100AZ=30+m/100*360;view(AZ,EL);pause(.1); % pause units are in secondsendpause

%*********************************************************% This same trick will let you make animations of% both xy and surface plots. To make this surface% oscillate up and down like a manta ray you could% do this.%*********************************************************dt=.1;for m=1:100t=m*dt;g=f*cos(t);surf(X,Y,g);AZ=30;EL=45;view(AZ,EL);title(Surface Plot)xlabel(x)ylabel(y)axis([-1 1 -1 1 min(min(f)) max(max(f))])pause(.1)end

for n=1:.2:5n10 = 10*n;x = linspace(-2,2,n10);y = x./(1+x.^2);plot(x,y,'r')title(sprintf('Graph %g. Plot based upon n = %g points.'..., (n+1)/2, n10))

axis([-2,2,-.8,.8])xlabel('x')ylabel('y')gridpause(3)end


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close all;clear;% Curve r(t) = < t*cos(t), t*sin(t), t >.t = -10*pi:pi/100:10*pi;x = t.*cos(t);

y = t.*sin(t);h = plot3(x,y,t);set(h,'LineWidth',1.25)title('Curve u(t) = < t*cos(t), t*sin(t), t >')h = get(gca,'Title');set(h,'FontSize',12)xlabel('x')h = get(gca,'xlabel');set(h,'FontSize',12)ylabel('y')h = get(gca,'ylabel');set(h,'FontSize',12)zlabel('z')h = get(gca,'zlabel');set(h,'FontSize',12)grid

% Surface plot of the hyperbolic paraboloid z = y^2 - x^2

x = -1:0.05:1;y = x;[xi, yi] = meshgrid(x,y);zi = yi.^2 - xi.^2;mesh(xi, yi, zi)axis offsurf(xi,yi,zi);

x = -1:.05:1;

y = x;

[xi,yi] = meshgrid(x,y);

zi = yi.^2 - xi.^2;

surfc(xi,yi,zi)

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colormap copper

shading interp

view([25,15,20])

grid off

title('Hyperbolic paraboloid z = y^2 x^2')

h = get(gca,'Title');

set(h,'FontSize',12)

xlabel('x')

h = get(gca,'xlabel');


ylabel('y')

h = get(gca,'ylabel');


zlabel('z')

h = get(gca,'zlabel');


pause(5)

figure

contourf(zi), hold on, shading flat

[c,h] = contour(zi,'k-'); clabel(c,h)

title('The level curves of z = y^2 - x^2.')h = get(gca,'Title');


xlabel('x')

h = get(gca,'xlabel');


ylabel('y')

h = get(gca,'ylabel');


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Hyperbolic paraboloid z = y2

x2

y

z


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Sphere and cylinder

MATLAB has some functions for generating special surfaces. We will be concerned mostly with twofunctions- sphere and cylinder.The command sphere(n) generates a unit sphere with center at the origin using (n+1)2points. If functionsphere is called without the input parameter, MATLAB uses the default value n = 20. You can translatethe center of the sphere easily. In the following example we will plot graph of the unit sphere with center

at (2, -1, 1)[x,y,z] = sphere(30);

surf(x+2, y-1, z+1)

cylinder([1 0])title('Unit cone')

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The level curves of z = y2

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t = 0:pi/100:pi;r = sin(t);plot(r,t)cylinder(r,15)shading interp

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Unit cone

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Create a Series of Polar Plots.

% An Mfile script to produce "flower petal" plotstheta = -(pi):0.01:pi;rho(1,:) = 2*sin(5*theta).^2;rho(2,:) = cos(10*theta).^3;rho(3,:) = sin(theta).^2;rho(4,:) = 5*cos(3.5*theta).^3;for i = 1:4

polar(theta,rho(i,:))pause

end

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Try entering these commands in an M-file called oloyede.m. This file is now a MATLAB

script. Typing oloyede at the MATLAB command line executes the statements in the

script.

After the script displays a plot, press Return to move to the next plot. There are no input

or output arguments; petals creates the variables it needs in the MATLAB workspace.

When execution completes, the variables (i, theta, and rho) remain in the workspace. To

see a listing of them, enter whos at the command prompt.

Inline FunctionsMatlab will let you define expressions inside a script for use as functions within thatscript only. For instance, if you wanted to use repeated references to the function

you would use the following syntax (to make both a line plot in x with y = 2 and to makea surface plot)

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clear;close all;f=inline(sin(x.*y)./(x.^2+y.^2),x,y);x=-8:.1:8;y=x;plot(x,f(x,2))[X,Y]=meshgrid(x,y);surf(X,Y,f(X,Y))

The expression that defines the function is in the first string argument to inline and theother string entries tell inline which argument goes in which input slot when you use thefunction.

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Comparing Interpolation Methods

This example compares two-dimensional interpolation methods on a 7-by-7 matrix of

data.

1 Generate the peaks function at low resolution:[x,y] = meshgrid(-3:1:3);z = peaks(x,y);surf(x,y,z)2 Generate afinermeshforinterpolation:[xi,yi] = meshgrid(-3:0.25:3);3 Interpolate usingnearestneighborinterpolation:zi1 = interp2(x,y,z,xi,yi,'nearest');4 Interpolate usingbilinearinterpolation:zi2 = interp2(x,y,z,xi,yi,'bilinear');5 Interpolate usingbicubicinterpolation:zi3 = interp2(x,y,z,xi,yi,'bicubic');

for2 x1 0, 2 x2 4, and 0 x3 2. To evaluate and plot this function:

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x1 =- 2:0.2:2;x2 = -2:0.25:2;x3 = -2:0.16:2;[X1,X2,X3] = ndgrid(x1,x2,x3);z = X2.*exp(-X1.^2 -X2.^2- X3.^2);slice(X2,X1,X3,z,[-1.28 2],2,[-2 0.2])

References

Ross L. Spencer,2000 : introduction to Matlab, Department of Physics and Astronomy

Brigham Young University

Edward Neuman Department of Mathematics Southern Illinois University at Carbondale

[email protected]

D. Hanselman and B. Littlefield, Mastering MATLAB 5. A Comprehensive Tutorial and

Reference, Prentice Hall, Upper Saddle River, NJ, 1998.

P. Marchand, Graphics and GUIs with MATLAB, Second edition, CRC Press, Boca

Raton,1999.

K. Sigmon, MATLAB Primer, Fifth edition, CRC Press, Boca Raton, 1998.

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