E5 MatLab - 2nd class demo

Contents

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some special matrices

The identity matrix - note variable name at the left

I=eye(3)

I =
     1     0     0
     0     1     0
     0     0     1

A matrix of zeros (by default, square)

Z=zeros(3)

Z =
     0     0     0
     0     0     0
     0     0     0

A non-square matrix of zeros

Z=zeros(2,5)

Z =
     0     0     0     0     0
     0     0     0     0     0

Can also create a matrix of ones...

O=ones(3)

O =
     1     1     1
     1     1     1
     1     1     1

We can make the matrix the same size as another array

O=ones(size(Z))

O =
     1     1     1     1     1
     1     1     1     1     1

Define a matrix of random numbers

R=rand(2,5)

R =
    0.7948    0.3786    0.5328    0.9390    0.5502
    0.6443    0.8116    0.3507    0.8759    0.6225

Working with arrays and subsets of arrays

A=[1 -2 3 -4; -2 3 -4 5; -2 -4 -6 -8]

A =
     1    -2     3    -4
    -2     3    -4     5
    -2    -4    -6    -8

find size

size(A)

ans =     3     4

find number of rows (the first dimension of the A matrix)

size(A,1)

ans =     3

what is the 2nd row?

A(2,:)

ans =    -2     3    -4     5

what is the 3rd column?

A(:,3)

ans =
     3
    -4
    -6

Define an array of zeros same size as a row of A (1 row, same number of columns)

z=zeros(1, size(A,2))

z =     0     0     0     0

now change the last row

A(3,:)=z

A =
     1    -2     3    -4
    -2     3    -4     5
     0     0     0     0

Higher dimensional arrays

X=rand(2,4,3)

X(:,:,1) =
    0.2305    0.1948    0.1707    0.4357
    0.8443    0.2259    0.2277    0.3111

X(:,:,2) =
    0.9234    0.1848    0.9797    0.1111
    0.4302    0.9049    0.4389    0.2581

X(:,:,3) =
    0.4087    0.2622    0.7112    0.1174
    0.5949    0.6028    0.2217    0.2967

Take "top" array

X(:,:,1)

ans =
    0.2305    0.1948    0.1707    0.4357
    0.8443    0.2259    0.2277    0.3111

Take 3rd column of "middle" array

X(:,3,2)

ans =
    0.9797
    0.4389

Take first two elements of 2nd row of "bottom" array

X(2,1:2,3)

ans =    0.5949    0.6028

Using a function

p=3  % Define a variable "p" for no real reason.

p =     3

Define x and y values of a complicated shape (a smile)

x1Smile=linspace(-0.5,0.5);            x2Smile=linspace(0.5,-0.5);
y1Smile=0.8*x1Smile.^2+0.1;            y2Smile=01.2*x2Smile.^2;
xSmile=[x1Smile x2Smile];              ySmile=[y1Smile y2Smile];
smile=patch(xSmile,ySmile,[1 0 0]);    axis([-2 2 -2 2],'square')

Find the perimeter

smilePerimeter=perim(xSmile,ySmile)

smilePerimeter =    2.3026

Note that the variable p is unchanged

p

p =     3

Graphical transformations

Rotation

Define a point in 2-d space as a column vector (x,y)=(1,-2)

x=1; y=-2;

Draw line from the origin with a blue dotted line.

plot([0 x],[0 y],'b:');

add a blue circle at end, without erasing.

hold on;
plot(x, y, 'bo');

Now add labels, scale axes and add grid.

xlabel('x'); ylabel('y');
title('The point (x,y) = (1,-2)');
axis([-3 3 -3 3], 'square'); grid on

Define transformation matrix for a -90 degree rotation

theta = -90 * pi/180;
T=[cos(theta) sin(-theta) 0;
   sin(theta) cos(theta) 0;
   0 0 1]

T =
    0.0000    1.0000         0
   -1.0000    0.0000         0
         0         0    1.0000

Perform rotation and plot Note, for the transformations we add a third row that is always 1

xy=[x; y; 1]

xy =
     1
    -2
     1

xyp=T*xy

xyp =
   -2.0000
   -1.0000
    1.0000

xp=xyp(1)
yp=xyp(2)

xp =
    -2

yp =
   -1.0000

Plot

plot([0 xp],[0 yp],'b-','Linewidth',2);
plot(xp,yp,'bo','MarkerFaceColor','b');
title('The point (1,-2) rotated by -90^\circ')

hold off;  %Next plot will clear the screen

Translation

Now for a translation, tx=1, ty=4

tx=1;  ty=4;
T=[1 0 tx; 0 1 ty; 0 0 1]

T =
     1     0     1
     0     1     4
     0     0     1

do transformation

xyp=T*xy
xp=xyp(1);  yp=xyp(2);

xyp =
     2
     2
     1

Plot both points

plot([0 x],[0 y],'b:');  hold on;  plot(x, y, 'bo');
xlabel('x'); ylabel('y');
plot([0 xp],[0 yp],'b-','Linewidth',2);
plot(xp,yp,'bo','MarkerFaceColor','b');
title('The point (1,-2) translated by t_x=1.0, t_y=4');
axis([-3 3 -3 3],'square');  grid on

hold off;  %Next plot will clear the screen

Combined rotation and translation

theta=-90*pi/180;  tx=1;  ty=4;
T=[cos(theta) sin(-theta) tx;
   sin(theta) cos(theta) ty;
   0 0 1]

T =
    0.0000    1.0000    1.0000
   -1.0000    0.0000    4.0000
         0         0    1.0000

do transformation

xyp=T*xy
xp=xyp(1);  yp=xyp(2);

xyp =
    -1
     3
     1

Plot both points

plot([0 x],[0 y],'b:');  hold on;  plot(x, y, 'bo');
xlabel('x'); ylabel('y');
plot([0 xp],[0 yp],'b-','Linewidth',2);
plot(xp,yp,'bo','MarkerFaceColor','b');
title('(1,-2) Rotation: -90^\circ, Translation: t_x=1, t_y=4');
axis([-3 3 -3 3], 'square');  grid on;

hold off;  %Next plot will clear the screen

Manipulating graphical objects (patches)

Clear the figure, set axis limits, grid

clf;  axis([-3 3 -3 3], 'square'); grid on;

 

Define a patch by its vertices

x=[0 2 2];  y=[0 1 0];
myTriangle=patch(x,y,'r');

Rotate by 45 degrees and translate by 1, -2

xy=[x; y; ones(size(x))]

xy =
     0     2     2
     0     1     0
     1     1     1

theta=45*pi/180;  tx=1;  ty=-2;
T=[cos(theta) sin(-theta) tx;
   sin(theta) cos(theta) ty;
   0 0 1]

T =
    0.7071   -0.7071    1.0000
    0.7071    0.7071   -2.0000
         0         0    1.0000

do transformation

xyp=T*xy

xyp =
    1.0000    1.7071    2.4142
   -2.0000    0.1213   -0.5858
    1.0000    1.0000    1.0000

Pull out new x and y values

xp=xyp(1,:)
yp=xyp(2,:)

xp =    1.0000    1.7071    2.4142

yp =   -2.0000    0.1213   -0.5858

Update shape

set(myTriangle,'Xdata',xp,'Ydata',yp);

Published with MATLAB® 7.4