gibson:teaching:fall-2014:math445:lecture16-diary

Matlab vocabulary

- meshgrid
- pcolor
- contour, contourf
- surf, surfc
- quiver
- shading (flat, faceted, or interp)
- colorbar
- axis (equal, tight)

The `meshgrid`

function is essential for Matlab's 3D graphics. Meshgrid creates 2D arrays of x,y data covering the x,y, plane, over which a function can be evaluated and graphed. Example:

>> x = linspace(-2,2,5) x = -2 -1 0 1 2 >> y = linspace(-3,3,7) y = -3 -2 -1 0 1 2 3 >> [X,Y] = meshgrid(x,y) X = -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 Y = -3 -3 -3 -3 -3 -2 -2 -2 -2 -2 -1 -1 -1 -1 -1 0 0 0 0 0 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3

It is conventional to use capital letters (X,Y) for the matrix output of meshgrid from small letter (x,y) vector inputs. Observe how the output matrices X varies left to right, along the x axis, and Y varies up and down, along the y axis. Together they provide x,y, coordinates for a grid of points on the x,y plane in the region , .

We can then evaluate a function over that 2D array via elementwise matrix operations. For example, this Matlab code would evaluate

Z = X.^2 + Y.^2

The `pcolor`

function produces a pseudocolor or checkerboard plot of Z as a function of x,y.

pcolor(X,Y,Z) colorbar axis equal axis tight xlabel('x'); ylabel('y'); title('z = x^2 + y^2')

You can modify the appearance of the pseudocolor with the `shading`

command. Try
`shading flat`

, `shading interp`

, and `shading faceted`

.

The `contour`

function plots contours or level curves of z=f(x,y). That is, it plots curves
on which f(x,y) is constant.

[X,Y] = meshgrid(x,y); Z = X.^2 + Y.^2; contour(X,Y,Z) xlabel('x'); ylabel('y'); title('z= x^2 + y^2') colorbar axis equal axis tight

As you can see, the level curves of are circles .

`contourf`

is the same as `contour`

, except that the regions between contour lines are filled with color.

The previous plots were all looking straight down at the (x,y) plane, with the value of z = f(x,y) encoded as a color. The `surf`

function will plot z = f(x,y) in 3D, as a surface of height z over the (x,y) plane.

surf(X,Y,Z) % draw z=f(x,y) as a surface over x,y xlabel('x'); ylabel('y'); zlabel('z') axis equal; axis tight

It's also possible to draw more complicated surfaces (surfaces that are not simple graphs of the form ). Here's an example of how to draw a sphere by parameterizing its surface in terms of angles and .

% make mesh over theta, phi theta = linspace(0,2*pi,50); % angle between x and y phi = linspace(0,pi,25); % angle down from z axis [Theta, Phi] = meshgrid(theta, phi); % form 2D mesh in theta, phi % Parameterize surface of sphere in terms of theta, phi % (note that x^2 + y^2 + z^2 = 1) X = cos(Theta).*sin(Phi); Y = sin(Theta).*sin(Phi); Z = cos(Phi); % Draw parametrized surface of sphere with surf surf(X,Y,Z); axis equal xlabel('x'); ylabel('y'); zlabel('z') axis equal

gibson/teaching/fall-2014/math445/lecture16-diary.txt · Last modified: 2014/11/18 07:25 by gibson