Plotting graphs is indispensable in science and engineering as a tool for communicating information and results. MATLAB provides an extensive set of commands that facilitate the creation of high-quality plots.

In this activity we will explore the basic commands to create and manipulate surface plots.

Use the MATLAB Live Editor to edit and run this Live Script in your browser or your desktop.

- Read each section carefully.
- Run the code within a section before starting to read the next.
- To run the code from each section, position the cursor on the code with the mouse and then click on the Run Section button (from the toolstrip) or click on the blue stripe on left side of that section as shown below:

Remark: Run the code of each section from top to bottom, otherwise you may get an error.

- The end of a section is indicated with a thin line, like the next one -

Let , a function of two variables, with domain given by the region , in the xy-plane.

To plot this surface first compute for many points , plot those points in 3D, then join them up to make it look like a surface.

In MATLAB, this is easy. First, we have to make grids of the x and y values where we want to calculate the values of . In MATLAB we can do this using the command:

[X, Y] = meshgrid(1:4);

The meshgrid command takes the vector (1,2,3,4) and produces two matrices called X and Y whose values correspond to the integer valued x and y coordinates of the domain, that is, , of .

Now, we can produce a new matrix called Z that holds the values at each point of the integer grid as follows:

Z = X.^2- Y.^2;

Remark: Here we are using row vectors (also known as arrays) which means that the operations multiplication, division and exponentiation must have the period "." before each operator: *, /, ^. Learn more about it here: Arithmetic operations.

surf(X,Y,Z)

To see how this works, run this section.

[X, Y] = meshgrid(1:4);

Z = X.^2- Y.^2;

surf(X,Y,Z)

The first thing you may notice is that the surface is not smooth. To create a smoother version just replace 1:4 by 1:0.25:4, in the previous code. Re-run the section to appreciate the result. Learn more about the colon operator here: colon operator.

In MATLAB we can specify the colours for a surface plot by including a fourth matrix input, C. The surface plot uses Z for height and C for colour.

Specify the colours using a colormap, which uses single numbers to stand for colours on a spectrum. When you use a colormap, C is the same size as Z. Add a colour bar to the graph to show how the data values in C correspond to the colours in the colormap.

For example, consider the function on the region , .

To include the colormap to this surface we use the following code. Run this section to see the output:

[X,Y] = meshgrid(-2:0.2:2);

Z = X.*exp(-X.^2-Y.^2);

C = Z;

surf(X,Y,Z,C)

colorbar

In MATLAB we can create a semitransparent surface by specifying the FaceAlpha name-value pair with 0.5 as the value. To allow further modifications, assign the surface object to the variable s.

For example:

[X,Y] = meshgrid(-5:.5:5);

Z = Y.*sin(X) - X.*cos(Y);

s = surf(X,Y,Z,'FaceAlpha',0.5)

Use the variable s to access and modify properties of the surface object after it is created. For example, hide the edges by setting the EdgeColor property. To do this just add the following line

s.EdgeColor = 'none';

at the end of the previous code. Re-run this section to see the output.

Note: There are a wide variety of properties that can be modified. For more details see: Surface properties.

Plot the following surfaces in the indicated region.

- for and .
- for and .

Write your code here:

% Plot f(x,y)

% Plot g(x,y)

Define two new X,Y grids using the code:

[X1, Y1] = meshgrid(0:0.5:5);

[X2, Y2] = meshgrid(0:5, -1:6);

Plot the function

over the grids given by (X1,Y1) and (X2,Y2).

Write your code here:

What do you notice? What do you wonder? Write in the code above, as comment, at least two differences you notice about the two surface plots.