160 Lectures

Quibble Question

How long would it take you to walk to the moon (assuming there was a footpath)? A day? a week? a month? a year? a lifetime? more?

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Plotting

We only need simple plotting in 160. Attaway Section 2.5 is really enough.
But if you want to have fun, Matlab has super-sophisticated plotting (Attaway Chapter 10 is a start). Back in 1996 the hardcopy manual for plotting and graphics was 200 pages long.
Functionality we need for the rest of this course is all mentioned in Sect. 2.5 but it's a terse treatment.
Matlab help and examples (templates) are useful tools (too much to remember otherwise).

Getting Started

The following help request is enough to get all sorts of pointers to useful Matlab functions. More help is available for each individually.

>> help graph2d Elementary X-Y graphs. plot - Linear plot. loglog - Log-log scale plot. semilogx - Semi-log scale plot. semilogy - Semi-log scale plot. ... Axis control. axis - Control axis scaling and appearance. zoom - Zoom in and out on a 2-D plot. grid - Grid lines. ... Graph annotation. plotedit - Tools for editing and annotating plots. title - Graph title. xlabel - X-axis label. ylabel - Y-axis label. ... Hardcopy and printing. print - Print graph or Simulink system; or save graph to M-file. printopt - Printer defaults. ... See also graph3d, specgraph.

The Minimum You Need to Know

A few built-in functions:
plot, semilogx, semilogy, loglog, surf, clf, figure, hold, xlabel, ylabel, title, legend, grid.
How to control how the plot-line looks using arguments to plot.
The basic appearance and functionality of the Figure Window (where plots show up).

Basic Built-in Plot Functions

plot, surf produce 2-D line and 3-D surface plots in the figure window.
figure, clf, create and clear plot-windows.
hold allows several plots to be drawn into one figure. Calling hold alone is a 'toggle'. Clearer is hold on to start accumulating plots in the current figure and hold off to start replacing the current figure with each new plot command.
grid with various suffixes is used to control the appearance of grid lines in the plot.
xlabel, ylabel, title, and legend label your plot in various ways for inclusion into a report so your reader knows what the axes mean, what the plot is supposed to show, and what data is represented by the different lines on the plot.
Use help to discover more about these functions!!

Calling plot()

There is no substitute for help plot, (unless you want to find out even more by reading the online manual)

Plot Vanilla: plot(X,Y) Plots vector Y versus vector X. Various OK things happen if X or Y is a matrix or scalar.
Plot Usual: plot(X,Y, 'option_string') Plots vector Y versus vector X. The short (usually 2 or 3-character, 1 to 4 characters possible) option string describes the color, plot-markers, or line-types in the plot. For example, 'r-' produces a red line.
Plot Trivial but Useful Special Case: plot(Y) Plots elements of vector Y versus their index. So if Y is an N-vector, X axis is 1, 2, 3,...., N.
To produce log-log and semi-log plots, use loglog(), semilogx(), and semilogy(), which have the same syntax as plot().
There are all sorts of additional commands and arguments that allow you to change the viewpoint, the mode of display, the shading, the colors, etc., etc. Some of this can be tracked down using help. Other parts of it require knowing something about 3D visualization and graphics, and more complete reading in the manual.

Calling surf()

A lot more possible complexity, having an extra dimension to visualize. help is still some help. "Surf" is short for "surface" Here think of a function z(x,y), and think of visualizing z as a height above the x-y plane.

Surf Easy Special Case surf(A) Displays a surface corresponding to the values in the 2-D array A, where the row is considered to correspond to the y coordinate, the column to the x coordinate, and the array value to the z coordinate. With beautiful false-color height coding. And grid lines. It may be necessary to make the call view(3) to get a nice 3-D view.
Surf Usual: surf(xrange, yrange, A) This call takes a vector of x values (the xrange) a vector of y values (the yrange), and a 2-D array of z values of size(yrange) rows by size(xrange) cols, and produces a visualization of the surface.

plot() Option Strings

The options string sent as a last argument to plot (and discussed in Attaway 2.5.1.1) controls how your "plot line" looks, for example:

'bo' plot points as blue circles
'mx' plot points as magenta x's
'r-' connect data points, plot as red solid line
'g-.' connect data points, plot as green dash-dot line
'r-o' red line AND circles.

So Data = [1 3 4 5]; plot(Data, 'r-'); hold on; plot(Data, 'b*');

Yields

Attaway shows at end of 2.5.2 a little 9-line script that creates an x-range, plots sin(x) and cos(x) on same axes, and illustrates the use of legend and title commands. Check it out for yourselves.

Saving a Plot as a File

You will be featuring plots in your project writeups as a major tool in helping yourself and your readers understand ('visualize') your results.

Make sure you have informative title, legend, x and y axis labels.
Make sure your axis and data ranges show the phenomena you want to illustrate.
Pick colors, line styles, and plot symbols that maximally clarify your results.

In the resulting Figure window with your gorgeous plot,

Go to File
Go to Save As...
Type a descriptive name into the File Name box, no suffix.
Go to Files of Type drop-down and pick a type. I like JPEG image (*.jpg) -- pretty universal, but sometimes has compression artifacts; *.png is a compressionless alternative. When you pick a type the proper suffix appears on your file name
Klik on Save
New file appears in your working directory. Use it in your word processor to create final .PDF file for handing in.

Some Real-Life Situations

Suppose I generate answers y(x) for x = [1 10 100 1000 10000] and want to plot y against x, but don't want 10000-long vectors 99.95% full of zeros?
How do I put two or more graphs on same axes in one plot?
How do I generate more than one plot at once? Add legends? Colors?

This page shows about 2% of Matlab's plotting power but it's all we really need.

Plot Demos
% Set up some data xdata = zeros(1,10); ydata = zeros(1,10); for elt= 1:10 xdata(elt) = elt; ydata(elt) = elt*elt; end plot(ydata); % defaults to plot(ydata, 'b-') title('1. line graph of i versus i*i');

figure; % create a new plot window... %we'll wind up with 10 of em plot(ydata, 'rd'); title('2. i versus i*i, with red diamonds');

figure; plot(ydata, 'c'); % cyan hold on; plot(ydata+13, 'm'); % magenta title('3. hold on to superimpose 2 plots'); hold off % else the rest will be superimposed

Spaced-out equally-spaced X data points:
xdata = [20 40 60 80]; ydata = zeros(1,4); yindex = 1; for n = 1:4 ydata(n) = xdata(n) * xdata(n); end figure plot(ydata, '-*'); % plot line and * title('4. x versus x*x for sparse x, squeezed'); figure plot(xdata,ydata,'-d'); % due to autoscale % and equally-spaced X % values, graph looks same title('5. n versus n*n, sparse n, unsqueezed');

Spaced-out UNequally-spaced X data points:
Plotting power functions like x = y², Volume = 1/Pressure.
The log-log plot.

xdata = [2 4 8 16 32 64 ]; ydata = zeros(1,6); for n = 1:length(xdata) ydata(n) = xdata(n) ^ 2; % power law end figure plot(ydata, '-o'); % plot with line and circle, title('6. y = x*x, unequally spaced xs, squeezed'); figure plot(xdata,ydata,'r-o'); % now graphs different title('7. y = x*x, unequally spaced xs unsqueezed');

figure % log-log plot for power relationship % yields straight line no matter how x values spaced loglog(xdata, ydata,'g-d'); title('8. y = x*x, log-log plot');

Plotting Exponential relationships like
y =e^x, DecayingVal = InitVal^(-.1*t)
The semi-log plot.
xdata = linspace(1,10,10); ydata = zeros(1,10); for n = 1:10 ydata(n) = 100 ^ (-.1 * xdata(n)); % exponential end; figure; plot(xdata, ydata, 'm-'); title('9. exponential function');

figure; semilogy(xdata, ydata, 'g-*'); % semi-log plot title('10. exponential function, semi-log plot');

Fun with Surface Plots

Making 2-D plots using surf can be neat. It can get complicated, but there are some defaults that can be used to generate some pretty cool graphics without a lot of trouble. For example.

a = -1 : .05 : 1; % a 41 element vector from -1 to 1 b = a .* a; % a^2 pointwise c = linspace(1, 1, 41); % a 41 element vector of 1s x2 = c' * b; % array representing z(x,y) = x^2 y2 = b' * c; % array representing z(x,y) = y^2 view(3); % default 3-D view figure surf(x2); % plot the x2 array as a surface figure surf(y2); % plot the y2 array as a surface

Produces the following figure windows (note the toolbar!). These were produced by a window-grab, not by "Save As..."
(Note: The window produced by surf may need to be closed before another surface plot can be made)

We can add x^2 and y^2 to get a paraboloid (illustrating the squared Euclidean distance function / 2),
x2_y2 = (x2 + y2) / 2; % z(x,y) = (x^2 + y^2) / 2 figure surf(x2_y2);

or take the outer product of a^2 with itself to get a particular 4th order function
x2y2 = 1 - b' * b; % z(x,y) = 1 - x^2 * y^2 figure surf(x2y2);

Which produces

For even more fun we can combine the last two,
z4 = x2y2 + x2_y2; figure surf(z4);

or make our own function using for loops.
z = zeros(41,41); for m = 1:41 for n = 1:41 y = (m - 21) / 20; x = (n - 21) / 20; z(m,n) = (x*x + y*y) * ... sin(1.7*pi*x) * cos(2.7*pi*y); end end figure surf(z);

Plotting Fundamentals: (Attaway Ch 2.5, (10))