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--- gibson:teaching:fall-2014:math445:lecture16-diary [2014/11/18 06:53]
gibson created
+++ gibson:teaching:fall-2014:math445:lecture16-diary [2014/11/18 07:25] (current)
gibson [surf, surfc]
@@ Line 11: / Line 11: @@
   * axis (equal, tight)
-====== meshgrid: ======
+===== 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:
 <code matlab>
 >> x = linspace(-2,2,5)
 x =
@@ Line 46: / Line 46: @@
      3     3     3     3
 </code>
-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.
+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 $-2 \leq x \leq 2$, $-3 \leq y \leq 3$.
-===== pcolor: pseudocolor plot =====
+We can then evaluate a function $z = f(x,y)$ over that 2D array via elementwise matrix operations. For example, this Matlab code would evaluate $z = f(x,y) = x^2 + y^2$
-% 3D graphics: plot z=f(x,y) versus x,y
+<code matlab>
+Z = X.^2 + Y.^2
+</code>
-% need to evaluate f over aplane x,y
+===== pcolor: pseudocolor plot =====
-% define simple function f(x,y) = 1/2 x^2 + y^2
+The ''pcolor'' function produces a pseudocolor or checkerboard plot of Z as a function of x,y.
+<code matlab>
+pcolor(X,Y,Z)
+colorbar
+axis equal
+axis tight
+xlabel('x'); ylabel('y'); title('z = x^2 + y^2')
+</code>
+You can modify the appearance of the pseudocolor with the ''shading'' command. Try
+''shading flat'', ''shading interp'', and ''shading faceted''.
-x = linspace(-2,2,5)
+===== contour, contourf =====
-x =
+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.
-    -2    -1     0     1     2
-y = linspace(-3,3,6)
-y =
-   -3.0000   -1.8000   -0.6000    0.6000    1.8000    3.0000
-[X,Y] = meshgrid(x,y);
-X
-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
-Y
-Y =
-   -3.0000   -3.0000   -3.0000   -3.0000   -3.0000
-   -1.8000   -1.8000   -1.8000   -1.8000   -1.8000
-   -0.6000   -0.6000   -0.6000   -0.6000   -0.6000
-.6000    0.6000    0.6000    0.6000    0.6000
-.8000    1.8000    1.8000    1.8000    1.8000
-.0000    3.0000    3.0000    3.0000    3.0000
-X
-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
-Y
-Y =
-   -3.0000   -3.0000   -3.0000   -3.0000   -3.0000
-   -1.8000   -1.8000   -1.8000   -1.8000   -1.8000
-   -0.6000   -0.6000   -0.6000   -0.6000   -0.6000
-.6000    0.6000    0.6000    0.6000    0.6000
-.8000    1.8000    1.8000    1.8000    1.8000
-.0000    3.0000    3.0000    3.0000    3.0000
-% Now use componentwise arithmetic to generate z=f(x,y)
-Z = 0.5*X.^2 + Y.^2
-Z =
-.0000    9.5000    9.0000    9.5000   11.0000
-.2400    3.7400    3.2400    3.7400    5.2400
-.3600    0.8600    0.3600    0.8600    2.3600
-.3600    0.8600    0.3600    0.8600    2.3600
-.2400    3.7400    3.2400    3.7400    5.2400
-.0000    9.5000    9.0000    9.5000   11.0000
-% simplest 3D plot is a contour plot
-contour(X,Y,Z)
-% contour plot shows levels curves, Z = constant
-y = linspace(-3,3,20)
-y =
-  Columns 1 through 7
-   -3.0000   -2.6842   -2.3684   -2.0526   -1.7368   -1.4211   -1.1053
-  Columns 8 through 14
-   -0.7895   -0.4737   -0.1579    0.1579    0.4737    0.7895    1.1053
-  Columns 15 through 20
-.4211    1.7368    2.0526    2.3684    2.6842    3.0000
-x = linspace(-2,2,20)
-x =
-  Columns 1 through 7
-   -2.0000   -1.7895   -1.5789   -1.3684   -1.1579   -0.9474   -0.7368
-  Columns 8 through 14
-   -0.5263   -0.3158   -0.1053    0.1053    0.3158    0.5263    0.7368
-  Columns 15 through 20
-.9474    1.1579    1.3684    1.5789    1.7895    2.0000
+<code matlab>
 [X,Y] = meshgrid(x,y);
 Z = X.^2 + Y.^2;
 contour(X,Y,Z)
-xlabel('x')
+xlabel('x'); ylabel('y'); title('z= x^2 + y^2')
-ylabel('y')
 colorbar
-title('z= x^2 + y^2')
 axis equal
 axis tight
-contourf(X,Y,Z) % filled contour plot
+</code>
-axis equal
+As you can see, the level curves of $z = f(x,y) = x^2 + y^2$ are circles $x^2 + y^2 = c$.
-axis tight
-colorbar
+''contourf'' is the same as ''contour'', except that the regions between contour lines are filled with color.
+===== surf, surfc =====
-pcolor(X,Y,Z)
+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.
-pcolor(X,Y,Z) % draw each matrix elem as a color
-colorbar
-shading flat % takes away grid lines
-shading interp % interpolate shading
+<code matlab>
-% 3D graphics
-clf()
 surf(X,Y,Z) % draw z=f(x,y) as a surface over x,y
-xlabel('x')
+xlabel('x'); ylabel('y'); zlabel('z')
-ylabel('y')
+axis equal; axis tight
-zlabel('z')
+</code>
-shading flat
-shading interp
-% Parametric plot of sphere
+It's also possible to draw more complicated surfaces (surfaces that are not simple graphs of the form
+$z = f(x,y)$). Here's an example of how to draw a sphere by parameterizing its surface in terms of angles $\phi$ and $\theta$.
+<code matlab>
 % make mesh over theta, phi
-theta = linspace(0,2*pi,50);
+theta = linspace(0,2*pi,50); % angle between x and y
-phi = linspace(0,pi,25);
+phi   = linspace(0,pi,25);   % angle down from z axis
-% those are vectors, transform to matrix mesh
-[Theta, Phi] = meshgrid[theta, phi];
- [Theta, Phi] = meshgrid[theta, phi];
-                        |
-Error: Unbalanced or unexpected parenthesis or bracket.
-[Theta, Phi] = meshgrid(theta, phi);
+[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')
-% use extra variable C for color coding
-C = X;
-surf(X,Y,Z,C); axis equal
-xlabel('x'); ylabel('y'); zlabel('z')
-C = sin(X);
-surf(X,Y,Z,C); axis equal
-% surfc : surfaceplot with contour on x,y
-surfc(X,Y,Z,C); axis equal
-% quiver plot
-% used to show vector fields
-load y.asc
-load x.asc
-load vx.asc
-load vy.asc
-clf()
-quiver(x,y,vx,vy);
 axis equal
-axis tight
+</code>
-xlabel('x'); ylabel('y');
-title('vector field vx,vy over plane x,y')
-% subplots
-% make an array of plots within a figure window
-clf()
-subplot(2,2,1) % 1st subplot in 2 x 2 array
-quiver(x,y,vx,vy);
-axis equal
-axis tight
-subplot(2,2,2) % 2nd subplot in 2 x 2 array
-surfc(X,Y,Z,C); axis equal
-subplot(2,2,3) % 2nd subplot in 2 x 2 array
-surf(X,Y,Z,Z); axis equal
-subplot(2,2,4) % 4th
-x = linspace(0,pi,20);

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