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

Math 445: 3D graphics

Matlab vocabulary

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

meshgrid:

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

pcolor: pseudocolor plot

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.

contour, contourf

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.

surf, surfc

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

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