Graphic Objects

Graphic Objects

Graphics objects are the basic drawing elements used by MATLAB to display data.

Each instance of an object has a unique identifier called a handle.

Using this handle, you can manipulate the object properties of an existing graphics object.

set changes the property of the object.

get returns the property of the object. (Try.)

figure creates figure graphics objects using default property values. (Try.)

Graphic Object Hierarchy

1See p. 293, Lent.

Example: Set Figure Size

1 screen size = get(0, 'ScreenSize') % 0: the screen

2 figure('Position', [1 screen size(4)/2 ...

screen size(3)/2 screen size(4)/2]);

Seeaccessing-object-handles.

The current figure, denoted by gcf, is the window designated to receive graphics output.

The current axes, denoted by gca, are the targets for commands that create axes children.

The current object, denoted by gco, is the graphics object created or clicked on by the mouse.

Example

1 clear all;

2 clc

3 % main

4 x = -0.5*pi:0.01*pi:0.5*pi;

5 h = plot(x,sin(1./x)); % plot a sin curve

6 set(h,'marker','o'); % set marker to 'o'

7 set(h,'markerSize',5); % set marker size to 5

8 set(h,'lineWidth',1.5); % set line width to 1.5

9 set(h,'lineStyle',':'); % set line style to dot

10 set(h,'markerEdgeColor','r'); % set marker edge ...

color to green

11 set(h,'markerFaceColor','g'); % set marker face ...

color to yellow

You can use theplot toolin the figures to change the properties.

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Axis (1/3)

plot automatically decides the ranges of axis.

axis([xmin, xmax , ymin, ymax ]) sets the ranges of the current axes.

1 clear; clc;

2 % main

3 x = 0:0.1:4*pi;

4 y = sin(x);

5 plot(x,y);

6 grid on;

7 axis([-inf, inf, -0.5, 2]); % [xmin xmax ymin ymax]

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Axis (2/3)

axis autois the default setting.

axis tightmakes the axes shape to conform to the plotted data.

axis equalmakes the x- and y-data units equal.

axis squaremakes the x- and y-axes equal in length.

Axis (3/3)

Change the location of the tick marks on the plot by setting xtick and ytick of the axes.

Use gca to set the properties for the current axes.

1 clear all;

2 clc

3 % main

4 x = 0:0.1:4*pi;

5 plot(x,sin(x)+sin(3*x));

6 axis tight;

7 set(gca,'ytick',[-1 -0.3 0.1 1]);

8 set(gca,'yticklabel',{'Min','Threshold','Critical ...

Point','Max'});

9 grid on;

Try ’xtick’ and ’xticklabel’.

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Threshold Critical Point Max

subplot

Multiple plots can be placed in the same figure.

subplot(m, n, p) divides the current figure into an m-by-n grid and creates an axes in the grid position specified by p.

For example,

1 clear; clc;

2 % main

3 x = 0:0.1:4*pi;

4 subplot(2,2,1);plot(x,sin(x));title('Subplot 1');

5 subplot(2,2,2);plot(x,cos(x));title('Subplot 2');

6 subplot(2,2,3);plot(x,sin(x).*exp(-x/5));

7 title('Subplot 3');

8 subplot(2,2,4);plot(x,x.ˆ2);title('Subplot 4');

9 grid on;

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Summary: Line Plots

plot 2-D line plot

plotyy 2-D line plots with y-axes on both left and right side loglog Log-log scale plot

semilogx Semilogarithmic plot semilogy Semilogarithmic plot errorbar Plot error bars along curve

fplot Plot function between specified limits ezplot Easy-to-use function plotter

Summary: More Plots

bar Bar graph

barh Plot bar graph horizontally hist Histogram plot

rose Angle histogram plot pareto Pareto chart

area Filled area 2-D plot pie Pie chart

stem Plot discrete sequence data stairs Stairstep graph

scatter Scatter plot

contour Contour plot of matrix

bar

bar(x , y , ’style’) draws bars for each column in y at locations specified in x , and ’style’ specifies the style of the bars.

For example,

1 Clear all; clc;

2 x = 1900:10:2000;

3 y = [75.995,91.972,105.711,123.203,131.669,...

4 150.697,179.323,203.212,226.505,249.633,281.422];

5 bar(x,y);

2Seebar.

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 0

50 100 150 200 250 300

Example: Available Styles

1 clear; clc;

2

3 y = round(rand(5,3)*10);

4 figure;

5 subplot(2,2,1);

6 title('Group');

7 bar(y,'grouped');

8

9 subplot(2,2,2);

10 title('Stack');

11 bar(y,'stacked');

12

13 subplot(2,2,3);

14 title('Histc');

15 bar(y,'histc');

16

17 subplot(2,2,4);

18 title('Baseline');

19 h = bar(y, 'LineWidth',1 ,'EdgeColor', 'red', ...

'LineStyle', ':');

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Histc

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Hist

plotyy

plotyy(x 1, y 1, x 2, y 2) plots x 1 versus y 1 with y-axis labeling on the left and plots x 2 versus y 2 with y-axis labeling on the right.

For example,

1 clear all; clc;

2 x=0:0.01:20;

3 y1=200*exp(-0.05*x).*sin(x);

4 y2=0.8*exp(-0.5*x).*sin(10*x);

5 plotyy(x,y1,x,y2);

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errorbar

errorbar(x , y , e) plots y versus x with symmetric error bars 2 × e(i ).

1 clear; clc;

2 % main

3 x = linspace(0,2*pi,30);

4 y = sin(x);

5 e = y*0.2;

6 errorbar(x,y,e);

7 grid on;

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fplot

fplot(fun, limits) plots a function³ between specified limits.

1 clear; clc;

2 % main

3 fplot('sin(1/x)', [0.02 0.2]);

4 grid on;

Try ezplot.

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polar

polar(θ, r ) creates a polar coordinate plot of the angle θ versus the radius r .

1 clear; clc;

2 % main

3 theta = linspace(0,2*pi);

4 r = cos(4*theta);

5 polar(theta,r,'-r');

6 grid on;

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hist

hist(y ) creates a histogram bar plot of y .

Elements in y are sorted into 10 equally spaced bins along the x-axis between the minimum and maximum values of y .

1 clear; clc;

2 % main

3 n = 1e5;

4 nbins = 1e2; the number of bins

5 subplot(2,1,1); hist(rand(1,n),nbins);grid on;

6 title('Uniform distribution');

7 subplot(2,1,2); hist(randn(1,n),nbins);grid on;

8 title('Normal distribution');

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3D-Plotting by mesh

mesh(X , Y , Z ) plots the colored mesh defined by 3 matrix arguments.

1 clear; clc;

2 % main

3 x = 3:6; % the range along the x-axis

4 y = 5:9; % the range along the y-axis

5 z = zeros(length(y),length(x));

6 for i = 1:size(z,1)

7 for j = 1:size(z,2)

8 z(i,j) = y(i)*x(j); % z = f(x,y)

9 end

10 end

11 mesh(x,y,z);

12 xlabel('X');ylabel('Y');zlabel('Z');axis tight;

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Y X

Z

3D-Plotting by surf

surf(X , Y , Z ) plots the colored surface defined by 3 matrix arguments.

[X , Y ] =meshgrid(x , y ) transforms the domain specified by vectors x and y into arrays X and Y that can be used for the evaluation of functions of two variables and 3-D surface plots.

You may try meshz, meshc, surfc, and waterfall.

Example: z = xe

1 clear; clc;

2 % main

3 x = linspace(-2,2,25);

4 y = linspace(-2,2,25);

5 [xx,yy] = meshgrid(x,y); % form all x-y pairs

6 zz = xx.*exp(-xx.ˆ2-yy.ˆ2);

7 surf(xx,yy,zz);

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Exercise: sinc(x ) = sin(x ) x

There is a singularity point at x = 0.

In order to avoid this infinity, one can add an eps in the equation.

eps = 2.22044604925031e − 016

1 [X,Y]=meshgrid(-8:.5:8);

2 R=sqrt(X.ˆ2+Y.ˆ2)+eps;

3 Z=sin(R)./R;

4 surf(Z);

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plot3

plot3(x , y , z), where x , y and z are three vectorsof the same length, plots a curve in 3-dimensional coordinate through the points.

For example,

1 clear; clc;

2 % main

3 t = 0:0.1:10*pi;

4 plot3(t.*sin(t),t.*cos(t),t);

5 hold on;

6 plot3(t.*sin(t),t.*cos(t),-t,'r');

7 axis equal;

8 grid on;

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Example: z = xe

(Revisited)

We can plot z(x , y ) by plot3.

1 clear; clc;

2 % main

3 [x,y] = meshgrid(-2:0.1:2);

4 z = x.*exp(-x.ˆ2-y.ˆ2);

5 plot3(x,y,z); grid on;

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griddata

In practice, the data points are collected from the measurement in experiments.

Oneposterior analysis on these data points is to find the possible curves or surfaces to fitthe observed data points.

griddata interpolates the surface at the query points specified by (xq, yq) and returns the interpolated values.

griddata(x , y , v , xq, yq) returns a fitted surface of the form v = f (x , y ) to the scattered data in the vectors (x , y , v ).

peaks

peaks(x , y ) returns a matrix of the function values, given by z = 3(1−x )²e^{−x2−(y +1)2}−10(x

5−x³−y⁵)e^−x2−y2−1

3e^{−(x+1)2−y2}. One can simply execute peaks to obtain a figure. (Try.)

Example

1 clear; clc;

2 % main

3 x = 6*rand(100,1)-3;

4 y = 6*rand(100,1)-3;

5 z = peaks(x,y); % 100 sample points in total

6 [xq,yq] = meshgrid(-3:0.1:3);

7 zq = griddata(x,y,z,xq,yq,'cubic');

8 % Choose the linear, cubic, or nearest method to ...

interpolate.

9 mesh(xq,yq,zq); hold on;

10 plot3(x,y,z,'.','markersize',16); axis tight;

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3-D Graph Viewpoint Specification

view(AZ , EL) sets the angle of the view from which an observer sees the current 3-D plot.

AZ is the azimuth or horizontal rotation.

EL is the vertical elevation.

Note that AZ and ZL are both in degrees.

You can use the Rotate 3D button in the figure.

1 clear; clc;

2 % main

3 peaks;

4 view([60,-15]); % degree

5 colorbar % Appends a colorbar to the current axes

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Save Figures

In the menu of figure, you can save as any file type of picture file.

Exporting to Files

print(gcf,’-fileType’,’fileName’) saves the contents of the current figure with a specified file type and file name.

For example,

1 clear all;

2 clc;

3 % main

4 surf(peaks);

5 print(gcf,'-djpeg','peaks.jpg');

You can find more optional argumentshere.

Exercise: Save to Multiple Files

1 clear all;

2 clc;

3 filename set={'file1', 'file2', 'file3'};

4 for i=1:length(filename set)

5 surf(sphere(i*10));

6 print('-djpeg',fnames{i});

7 end

Solution

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A Love Tale: Cardioid

傳聞，法國著名數學家笛卡爾(Descartes)曾經流落到瑞典，

邂逅瑞典公主Kristina，並成為了公主的數學老師。兩人萌

生愛意。國王知道後，強行拆散他們，並且沒收了之後笛卡 爾寫給公主的所有信件。

後來，笛卡爾染上黑死病，臨死前給公主寄去了最後一封 信，信中只寫著一行字：

r = a(1 − sin(θ))

國王和大臣們都看不懂這是什麼意思，只好交還給公主。

公主在紙上建立了極坐標系，用筆在上面描下方程的點，看 到了方程所表示的心臟線(Cardioid)，理解了笛卡爾對自己 的深深愛意。

1 clear; clc;

2 % main: by polar

3 a = 2;

4 theta = 0:0.05:2*pi;

5 rho = a*(1 - sin(theta));

6 polar(theta,rho,'r*'); grid on;

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3 4

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3D Heart

1 clear; clc;

2 % main

3 [X Y Z] = meshgrid(-3:0.1:3, -3:0.1:3, -3:0.1:3);

4 F = ((-(X.ˆ2).*(Z.ˆ3)-(9/80).*(Y.ˆ2).*(Z.ˆ3))...

5 +((X.ˆ2)+(9/4).*(Y.ˆ2)+(Z.ˆ2)-1).ˆ3);

6 p = patch(isosurface(X,Y,Z,F,0));

7 set(p,'facecolor','w','EdgeColor','r'); % graphic ...

handling

8 daspect([1 1 1]);

9 axis tight; axis equal;

10 view(3); % set the default 3-D view, AZ = -37.5, ...

EL = 30.

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Animation

Animation is the process of creating motion and shape change illusion by means of the rapid display of a sequence of static images that minimally differ from each other.

Simply put, images are displayed in a rapid succession, usually 24, 25, 30, or 60 frames per second.

movie Play recorded movie frames comet 2-D comet plot

getframe Capture movie frame

im2frame Convert image to movie frame

frame2im Return image data associated with movie frame

Example

1 function starpoly(n,step)

2 y = zeros(1,n);

3 for k = 0:n

4 y(k+1) = ...

cos(2*pi/n*(step*k))+i*sin(2*pi/n*(step*k));

5 end

6 plot(y,'bo-'); axis equal; hold off;

7 end

1 clear; clc;

2 % main

3 t = 0:0.01:2*pi;

4 for k = 1:1:50*pi

5 plot(cos(t) + i*sin(t),'r--'); hold on;

6 starpoly(79,k);

8 frame = getframe(1);

9 im = frame2im(frame);

10 [imind,cm] = rgb2ind(im,256);

11 if k == 1

12 imwrite(imind, cm, 'go.gif', 'gif', ...

'Loopcount', inf);

13 else

14 imwrite(imind, cm, 'go.gif', 'gif', ...

'writemode', 'append');

15 end

16 end

1 >> Lecture 5

2 >>

3 >> -- Matrix Computation

4 >>

Vectors

We denote the vector space of all m-by-1 column vector x by R^m×1:

x ∈ R^m×1↔ x = (x_i) =





 x₁

... x_m





. (1)

Similarly, the row vector ~y is in form of y ∈ R^1×n ↔ y = (y_i) =

y₁· · · y_n  . (2)

Matrices

Let R be the set of all real numbers.

We denote the vector space of all m-by-n real matrices A by R^m×n:

A ∈ R^m×n ↔ A = (a_ij) =







a11 · · · a1n

... . .. ... a_m1 · · · a_mn





.

Recall the matrix indexing⁴. By subscripts: A(i , j )

By linear addressing: A(k) for some positive integer k.

The matrix definition is similar if we consider complex matrices.

Transposition

transpose(A) is the transpose of A.

1 A=magic(4)

2 transpose(A)

You can also transpose A by

1 A'

Note that if A ∈ C^m×n, then A’ is the transposition and complex conjugate⁵ of A.

5z = a + bi has a complex conjugate ¯z = a − bi .

Arithmetic Operations

Addition and subtraction are the same both for vectors and matrices.

Let a_ij, b_ij be the elements of the matrices A, B ∈ R^m×n for 1 ≤ i ≤ m and 1 ≤ j ≤ n.

Then C = A ± B can be calculated by cij = aij± bij. (Try.) For vectors,

inner product⁶ is a binary operation which takes two vectors and returns a scalar quantity.

cross product⁷ is a binary operation which takes two vectors and returns another vector perpendicular to both two vectors.

Inner Product

Let A, B ∈ R^m×1 be two column vectors. Then the inner product of A · B =

a1· · · a_m 





 b₁

... bm





.

8Seeinner product

Example

1 clear all;

2 clc

3 % main

4 x=[1;2;3];

5 y=[4;5;6];

6 z=0;

7 for i=1:3

8 z=z+x(i)*y(i);

9 end

10 z % by definition using a for loop

11 x'*y % equivalent formula

12 dot(x,y) % using built-in function

Exercise

Write a program that returns the angle between vectors x and y .

Input: vectors x , y Output: θ in degree

Solution

1 function theta = myAngle(x,y)

2 % input: two vectors x,y

3 % output: angle in degree

4 len x = sqrt(sum(x.ˆ2));

5 len y = sqrt(sum(y.ˆ2));

6 theta = acos(x'*y/len x/len y)/pi*180;

1 >> x = [1/sqrt(2);0;1/sqrt(2)];

2 >> y = [1/sqrt(2);1/sqrt(2);0];

3 >> myAngle(x,y)

4

5 ans =

6

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Cross Product

cross(x , y ) returns the cross product of the vectors x and y . Note that x and y must be 3-dimensional vectors.⁹

For example,

1 >> x=[1;0;0];

2 >> y=[0;1;0];

3 >> z=cross(x,y) % built-in

4

5 z =

6

7 0

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9 1

9Actually, only in 1-, 3-, and 7-dimensional Euclidean spaces.