gibson:teaching:spring-2016:math445:lecture:graphics3d
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gibson:teaching:spring-2016:math445:lecture:graphics3d [2016/03/29 08:45] gibson |
gibson:teaching:spring-2016:math445:lecture:graphics3d [2016/03/29 09:02] (current) gibson [Math 445 lecture 16: 3D graphics] |
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| - | ====== Math 445 lecture 16: 3D graphics ====== | + | ====== Math 445 lecture 16: 2D, 3D graphics ====== |
| + | Some examples of plotting 2d functions $f(x,y)$ in Matlab. | ||
| + | |||
| + | ---- | ||
| ==== 1d graphs ==== | ==== 1d graphs ==== | ||
| Line 23: | Line 26: | ||
| produces | produces | ||
| {{ :gibson:teaching:spring-2016:math445:onedplot.png?direct&400 |}} | {{ :gibson:teaching:spring-2016:math445:onedplot.png?direct&400 |}} | ||
| + | |||
| + | ---- | ||
| ==== 2d graphs with meshgrid ==== | ==== 2d graphs with meshgrid ==== | ||
| - | Grpahing a 2d function $f(x,y)$ is very similar. For example, to make a graph of the function | + | Graphing a 2d function $f(x,y)$ is very similar. For example, to make a graph of the function |
| - | $f(x,y) = 2*(x-2)^2 + (y-1)^2$ on $-4 \leq x \leq 4, -4 \leq y \leq 4$ | + | $f(x,y) = 2*(x-1)^2 + (y-1)^2$ on $-4 \leq x \leq 4, -4 \leq y \leq 4$ |
| - Create vectors ''x'' and ''y'' of gridpoints on $x$ and $y$ axes with ''linspace'' | - Create vectors ''x'' and ''y'' of gridpoints on $x$ and $y$ axes with ''linspace'' | ||
| Line 36: | Line 41: | ||
| For example, | For example, | ||
| <file matlab plot2d.m> | <file matlab plot2d.m> | ||
| - | % Plot f(x,y) = 2*(x-2)^2 + (y-1)^2 on -4 < x < 4, -4 < y < 4 | + | % Plot f(x,y) = 2*(x-1)^2 + (y-1)^2 on -4 < x < 4, -4 < y < 4 |
| x = linspace(-4,4,30); % create 1d gridpoints on x and y axes | x = linspace(-4,4,30); % create 1d gridpoints on x and y axes | ||
| Line 43: | Line 48: | ||
| [X,Y] = meshgrid(x,y); % create 2d mesh of gridpoints over x,y | [X,Y] = meshgrid(x,y); % create 2d mesh of gridpoints over x,y | ||
| - | F = 2*(X-2).^2 + (Y-1).^2; % evaluate the function f(x,y) over mesh | + | F = 2*(X-1).^2 + (Y-1).^2; % evaluate the function f(x,y) over mesh |
| pcolor(x,y,F); % or contour(X,Y,F), surf(x,y,F), etc. | pcolor(x,y,F); % or contour(X,Y,F), surf(x,y,F), etc. | ||
| Line 50: | Line 55: | ||
| xlabel('x'); | xlabel('x'); | ||
| ylabel('y'); | ylabel('y'); | ||
| - | title('f(x,y) = 2(x-2)^2 + (y-1)^2') | + | title('f(x,y) = 2(x-1)^2 + (y-1)^2') |
| shading interp % or faceted, flat | shading interp % or faceted, flat | ||
| </file> | </file> | ||
| + | {{ :gibson:teaching:spring-2016:math445:lecture:twodplot.png?direct&400 |}} | ||
| + | |||
| + | ---- | ||
| + | |||
| + | ==== quiver plots and the gradient=== | ||
| + | |||
| + | The function $f(x,y) = x e^{-x^2 - y^2}$ has gradient | ||
| + | |||
| + | \begin{eqnarray*} | ||
| + | \nabla f = \left( \begin{array}{c} \partial f/\partial x \\ \partial f/\partial y \end{array} \right) = \left( \begin{array}{c} (1-x^2) e^{-x^2 - y^2} \\ -2xy e^{-x^2 - y^2} \end{array} \right) | ||
| + | \end{eqnarray*} | ||
| + | |||
| + | Here's the Matlab code to make a contour plot of $f(x,y)$ with its gradient superimposed as a quiver plot. | ||
| + | |||
| + | <file matlab plotgradient.m> | ||
| + | % Plot f(x,y) = x e^(-x^2 - y^2) with contours and its gradient as arrows. | ||
| + | x = linspace(-2, 2, 30); | ||
| + | y = linspace(-2, 2, 30); | ||
| + | |||
| + | [X,Y] = meshgrid(x,y); | ||
| + | |||
| + | F = X .* exp(-X.^2 - Y.^2); % evaluate f(x,y) on mesh | ||
| + | contour(x,y,F); % draw contour plot of f | ||
| + | |||
| + | % let dfdy = y component of grad F = df/dy | ||
| + | |||
| + | dfdx = (1-2*X.^2) .* exp(-X.^2 - Y.^2); % evaluate dfdx on mesh | ||
| + | dfdy = -2*X.*Y .* exp(-X.^2 - Y.^2); % evaluate dfdy on mesh | ||
| + | |||
| + | hold on | ||
| + | quiver(x,y,dfdx,dfdy); % draw quiver plot | ||
| + | |||
| + | xlabel('x') | ||
| + | ylabel('y') | ||
| + | title('f(x,y) = x e^{-x^2 - y^2} with \nabla f arrow plot') | ||
| + | colorbar() | ||
| + | </file> | ||
| + | {{ :gibson:teaching:spring-2016:math445:lecture:quiver.png?direct&500 |}} | ||
gibson/teaching/spring-2016/math445/lecture/graphics3d.1459266307.txt.gz · Last modified: 2016/03/29 08:45 by gibson
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