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  Plotting Functions of two variables

Plotting Functions of Two Variables:

Functions on Rectangular Grids:

Presume you wish to plot a function f(x, y) on the rectangle a ≤ x ≤ b and c ≤ y ≤ d. The graph of the function of two variables is of course a three dimensional object. Visualizing the graph is habitually very useful.

For instance presume you have a formula:

f(x, y) = x sin(xy)

And you are involved in the function on the region 0 ≤ x ≤ 5, π ≤ y ≤ 2π. A manner to plot this function in Mat lab would be the following sequence of commands

> f = inline(’x.*sin(x.*y)’,’x’,’y’)
> [X,Y] = meshgrid(0:.1:5,pi:.01*pi:2*pi);
> Z = f(X,Y)
>mesh(X,Y,Z)

This will produce a 3-D plot that you are able to rotate by clicking on the rotate icon and then dragging with the mouse.

Instead of the command mesh you could utilize the command:

>surf(X,Y,Z)

The key command in this series is [X Y] = meshgrid(a:h:b,c:k:d) which generates matrices of x and y values in X and Y. Enter

>size(X)
>size(Y)
>size(Z)

to see that each one of these variables is a 101 × 51 matrix. To observe the first few entries of X enter

>X(1:6,1:6)

And to see the first several values of Y type

>Y(1:6,1:6)

You must observe that the x values in X begin at 0 on the left column and increase from left to right. The y values on the other have begun at π at the top and increase from top to bottom. Note down that this arrangement is flipped from the typical arrangement in the x-y plane.

In the command [X Y] = meshgrid(a:h:b,c:k:d) h is the increase in the x direction and k is the increment in the y direction.

Habitually we will calculate:

h = (b – a) m and k = (d − c) n

Where m is the number of intervals in the x direction as well as n is the number of intervals in the y direction. To acquire a good plot it is best if m and n can be set between 10 and 100.

For another illustration of how meshgrid works try the following and look at the output in X and
Y.

> [X,Y] = meshgrid(0:.5:4,1:.2:2);

Scattered Data and Triangulation:

Habitually we are interested in objects whose bases are not rectangular. For instance, data does not typically come arranged in a nice rectangular grid rather measurements are occupied where convenient.

In Mat lab we are able to produce triangles for a region by recording the coordinates of the vertices and recording which vertices belong to each triangle. The subsequent script program produces such a set of triangles:

% mytriangles
% Program to produce a triangulation.
% V contains vertices, which are (x,y) pairs
V = [ 1/2 1/2 ; 1 1 ; 3/2 1/2 ; .5 1.5 ; 0 0
1 0 ; 2 0 ; 2 1 ; 1.5 1.5 ; 1 2
0 2 ; 0 1]
% x, y are row vectors containing coordinates of vertices
x = V(:,1)’;
y = V(:,2)’;
% Assign the triangles
T = delaunay(x,y)

You are able to plot the triangles using the following command

>trimesh(T,x,y)

You can as well prescribe values (heights) at each vertex directly (say from a survey):

> z1 = [ 2 3 2.5 2 1 1 .5 1.5 1.6 1.7 .9 .5 ];

Or by means of a function:

> f = inline(’abs(sin(x.*y)).^(3/2)’,’x’,’y’);
> z2 = f(x,y);

The resulting profiles are able to be plotted:

>trimesh(T,x,y,z1)
>trisurf(T,x,y,z2)

Every row of the matrix T corresponds to a triangle so T(i) gives triangle number i. The 3 corners of triangle number i are at indices T(i,1), T(i,2), and T(i,3). Therefore for example to get the y-coordinate of the second point of triangle number 5 enter

>y(T(5,2))

To see other illustration of regions defined by triangle get mywedge.m and mywasher.m from the class web site as well as run them. Every of these programs defines vectors x and y of x and y values of vertices and a matrix T. As prior to T is a list of sets of three integers. Every triple of integers indicates which vertices to connect in a triangle.

To plot a function say f(x, y) = x2 − y2 on the washer figure try

>mywasher
> z = x.^2 - y.^2
>trisurf(T,x,y,z)

Note again that this plot is able to be rotated using the icon and mouse.

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