*This twelfth article of the mathematical journey through open source, shows the mathematical visualization in octave.*

Mathematics is incomplete without visualization, without drawing the results, and without plotting the graphs. *octave* uses the powerful *gnuplot* as the backend of its plotting functionality. And in the frontend provides various plotting functions. Let’s look at the most beautiful ones.

## Basic 2D Plotting

The most basic plotting is using the *plot()* function, which takes the Cartesian x & y values. Additionally, you may pass, as how to plot, i.e. as points or lines, their style, their colour, label, etc. Supported point styles are: +, *, o, x, ^, and lines are represented by -. Supported colours are: k (black), r (red), g (green), b (blue), y (yellow), m (magenta), c (cyan), w (white). With this background, here is how you plot a sine curve, and Figure 12 shows the plot.

```
$ octave -qf
octave:1> x = 0:0.1:2*pi;
octave:2> y = sin(x);
octave:3> plot(x, y, "^b");
octave:4> xlabel("x ->");
octave:5> ylabel("y = sin(x) ->");
octave:6> title("Basic plot");
octave:7>
```

*xlabel()*, and *ylabel()* adds the corresponding labels, and *title()* adds the title. Multiple plots can be done on the same axis as follows, and Figure 13 shows the plots. Note the usage of *legend()* to mark the multiple plots.

```
$ octave -qf
octave:1> x = 0:0.1:2*pi;
octave:2> plot(x, sin(x), "*", x, 1 + sin(x), "-", x, cos(x), "o");
octave:3> xlabel("x ->");
octave:4> ylabel("y ->");
octave:5> legend("sine", "1 + sine", "cosine");
octave:6> title("Multiple plots");
octave:7>
```

## Advanced 2D Figures

Now, if we want to have the multiple graphs on the same sheet but with different axes as shown in Figure 14, here is how to do that:

```
octave:1> t = 0:0.1:6*pi;
octave:2> subplot(2, 1, 1);
octave:3> plot(t, 325 * sin(t));
octave:4> xlabel("t (sec)");
octave:5> ylabel("V_{ac} (V)");
octave:6> title("AC voltage curve");
octave:7> grid("on");
octave:8> subplot(2, 1, 2);
octave:9> plot(t, 5 * cos(t));
octave:10> xlabel("t (sec)");
octave:11> ylabel("I_{ac} (A)");
octave:12> title("AC current curve");
octave:13> grid("on");
octave:14> print("-dpng", "multiple_plots_on_a_sheet.png");
octave:15>
```

Note the usage of *subplot()*, taking the matrix dimensions (row, column) and the plot number to create the matrix of plots. In the example above, it created a 2×1 matrix of plots. As add-ons, we have used the *grid(“on”)* to show up the dotted grid lines, and *print()* to save the generated figure as a .png file.

It is not always easy to plot everything in Cartesian co-ordinates, or rather many things are easier to plot in polar co-ordinates, e.g. a spiral, circle, heart, etc. The following code & Figure 15 shows a few such examples. Shown along with is a technique of modifying the figure properties, after drawing the figure using the *set()* function. Here it modifies the line thickness.

```
octave:1> th = 0:0.1:2*pi;
octave:2> r1 = 1.1 .^ th;
octave:3> r2 = 7 * cos(th);
octave:4> r3 = 5 * (1 - cos(th));
octave:5> r = [r1; r2; r3];
octave:6> ph = polar(th, r, "-");
octave:7> set(ph, "LineWidth", 4);
octave:8> legend("spiral", "circle", "heart");
octave:9>
```

There are many other possible ways of drawing various interesting 2-D figures for all kind of mathematical & scientific requirements. So, before closing on 2-D plotting, let’s look into just one more often needed drawing – plotting with log axis, and more over with two y-axes on a single plot. The function for that is *plotyy()*. Note the *plotyy()* calling the corresponding function pointers *@plot*, *@semilogy* passed to it, in the following code segment. Figure 16 shows the output.

```
octave:1> x = 0:0.1:2*pi;
octave:2> y1 = sin(x);
octave:3> y2 = exp(exp(x));
octave:4> ax = plotyy(x, y1, x, y2, @plot, @semilogy);
octave:5> xlabel("x ->");
octave:6> ylabel(ax(1), "sine ->");
octave:7> ylabel(ax(2), "e^{e^x} ->");
octave:8> title("Mixed plots");
octave:9>
```

## 3D Visualization

And finally, let’s do some 3-D plotting. *plot3()* is the simplest *octave* function to do a simple 3-D drawing, taking the set of (x, y, z) points. Here’s the sample code to draw the dwindling sinusoidal curve shown in Figure 17:

```
octave:1> x = -10:0.1:10;
octave:2> y = 10:-0.1:-10;
octave:3> z = x .* sin(x - y);
octave:4> plot3(x, y, z, "-", "LineWidth", 4);
octave:5> xlabel("x ->");
octave:6> ylabel("y ->");
octave:7> zlabel("z ->");
octave:8> title("Dwindling sinusoidal");
octave:9> grid("on");
octave:10>
```

In case, we want to plot the values of a 2-D matrix against its indices (x, y), it could be done using *mesh()*, one of the many other 3-D plotting functions. Figure 18 shows the same, created using the following code:

```
octave:1> x = 0:0.1:2*pi;
octave:2> y = 0:0.1:2*pi;
octave:3> z = sin(x)' * sin(y);
octave:4> mesh(x, y, z);
octave:5> xlabel("x ->");
octave:6> ylabel("y ->");
octave:7> zlabel("z ->");
octave:8> title("3-D waves");
octave:9>
```

## What’s next?

Hope you enjoyed the colours of drawing. Next, we would look into *octave* from a statisticians perspective.

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David NelsonThank you, Anil,

I would like to eliminate the gridlines on the plot and see–and save–just color information.

From advice elsewhere, I tried separately:

surface(x,y,z,’linestyle’,’none’), shading(‘interp’), colormap(gray);

surface(x,y,z,’edgecolor’,’none’), shading(‘interp’), colormap(gray);

(because I really only want intensities) and both do result in plots which visually do not show

visible lines between the values of the mesh. (I understand that the ‘interp’ shading interpolates missing values).

However, when I “Save-as” the result as a ‘png’, the image always contains residual evidence of the lines.

Also, if I may, is there a way to lock the aspect ratio of resulting plots? Despite the fact that my x,y values are symmetric, the plots are always over a rectangular grid, instead of square.

Anil Kumar PugaliaPost authorDavid, I am getting it perfectly fine. Just try “graphics_toolkit(“gnuplot”);”, the first thing when you start octave, and then do your drawings, prints, etc. Let us know, if it helps.

Regarding the aspect ratio, you may try pbaspect() for plot box and daspect() for data. Without parameter, they return the current corresponding aspect ratios. With a 3-D vector, they change the corresponding aspect ratios.

lakshmikruthigasir, I have following code, The values of displ in stage1 and stage2 are plotted in displ vs time graph. Thus a single curve is obtained.

displ=zeros(100,2)

tim=zeros(100,1)

for i=2:100

v=5;

dt=1*10^-4;

function [displ_f]=stage1(displ_i)

dt=1*10^-4;

v=5;

displ_f=displ_i+v*dt;

endfunction

function [displ_f]=stage2(displ_i)

dt=1*10^-4;

v=5;

displ_f=displ_i+v*dt;

endfunction

if (displ(i-1,1)=0)

disp(“IN stage 1”);

[displ(i,1)]=stage1(displ(i-1,1));

displ(i,2)=v;

tim(i,1)=tim(i-1)+dt;

elseif (displ(i-1,1)>=0.020 && displ(i-1,1)==0)

disp(“IN stage 2″);

[displ(i,1)]=stage2(displ(i-1,1));

displ(i,2)=displ(i-1,2);

tim(i,1)=tim(i-1)+dt;

end

end

disp(” displ vs tim”)

plot(tim(:,1),displ(:,1));

How will I get stage1 curve in orange color and stage2 curve in blue color in the same plot to know which stage has which displ values? kindly help me.

Anil Kumar PugaliaPost authorYou may have to draw them as two separate curves.

BTW, I am not getting any curve from your above program.

John McGeorgeHello.I am new to Octave.Would like to plot the rear suspension in 3D.not sure how to do that.I am a 61 year old. Please need help.Thanks

Anil Kumar PugaliaPost authorDo you have the mathematical equation of the same? If yes, plug it in, as per the above examples.