Question

MATLAB codes please 3. Adapt the procedure developed in Example 6 to rotate the square counterclockwise by incre ments of /10 about the origin. Stop when the square is in its original location and then rotate it in the clockwise direction until the square is in its original location again. You may want to rescale the axis by using axis((-2,2-2,21). Include the M-lile. Do not include the figure Hint: Since you are performing a computation several times, you will want to use two for loops: one loop for the counterclockwise rotation and another one for the clockwise rotation. Think about how many times you will need to go through the loops, keeping in mind that you are rotating the square counterclockwise for a full circle by increments of /10 and then rotating the square clockwise back again. EXAMPLE 6 The goal of this example is to draw the square in its original size, then cause it to disappear, and finally to redraw it as it looks after it is dilated by a factor of 9/8. If this operation is repeated ten times in succession, the square will appear to be expanding. To do this we need to first plot the square and store its handle in p: p = plot(S(1,:),(2,:)) clf After we transform the matrix by multiplying by the appropriate transformation matrix, we can erase the original figure and draw the new figure with the command set(p, 'xdata',s(1,:), 'ydata',S(2.:)); Since we are repeating the same operation ten times, we will use a for loop. After we have gone through this loop, we create another one that contracts the matrix by a factor of 8/9, thereby returning it to its original size. The following are the commands that produce the effects. Enter them in MATLAB (you need not enter the comments that follow the % signs). The pause (0.1) command is used to adjust the speed of the plots so that we can see the graphs before they are erased. You might need to change the 0.1 to suit the speed of your machine. % clear all settings for the plot S-[0,1,1,0,0,0,0,1,1,0); D1 - 9/8* eye (2); % dilation matrix P- plot (S(1, :), S (2, :)); % plot the square axis ([-1,4,-1,4]) % set size of the graph axis square, grid on % make the display square % hold the current graph for i = 1:10 S - D1.S; % dilate the square set (p. 'xdata'.S (1, :), 'ydata',s(2,:)); % erase original figure and plot % the transformed figure pause (0.1) % adjust this pause rate to suit your computer. end D2 = 8/9*eye (2): % contraction matrix for i = 1:10 S = D2S; % contract the square set(p,'xdata',s(1, :), 'ydata',s(2,:)); % erase original figure and plot % the transformed figure pause (0.1) % adjust this pause rate to suit your computer. end hold on hold off

Answer 1

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Question 3:

clf
% If you wish to verify the direction of rotation, you can
% create a spike in square by uncommenting these lines
% S=[-1,-1,1,1,-1.5,-1;
% -1,1,1,-1,-1.5,-1];

S=[-1,-1,1,1,-1; % square matrix
-1,1,1,-1,-1];
p=plot(S(1,:),S(2,:)); % plot the square
axis([-2 2, -2,2]); % adjust the axis limits
axis square, grid on
hold on

thetad = pi/10; % angle of rotations
t=ceil(2*pi/thetad); % no. of rotations required to complete a circle
D1 = [cos(thetad) sin(thetad); -sin(thetad) cos(thetad)]; % rotation matrix for counter-clockwise direction
for i=1:1:t
S=D1*S; % rotate the square
set(p,'xdata',S(1,:),'ydata',S(2,:)); % plot
pause(0.1);
end

D2 = [cos(thetad) -sin(thetad); sin(thetad) cos(thetad)]; % rotation matrix for clockwise direction
for i=1:t
S=D2*S; % rotate the square
set(p,'xdata',S(1,:),'ydata',S(2,:)); % plot
pause(0.1);
end
hold off

---------------------- SCREENSHOT OF CODE

roate_square.mx rotate_and_expand.mx + 1 - cit| 2 If you wish to verify the direction of rotation, you can 3 create a spike in square by uncommenting these lines 4 $ S=(-1,-1,1,1,-1.5, -1; 5 몸 -1,1,1,-1,-1.5,-1]; 6 7 S=(-1,-1,1,1,-1; square matrix 8 -1,1,1,-1,-1]; 9 - p=plot (S(1,:),(2,:)); plot the square 10 axis ([-2 2, -2,2]); adjust the axis limits 11 - axis square, grid on 12 - hold on 13 14 - thetad = pi/10; * angle of rotations 15 - t=ceil(2*pi/thetad); * no. of rotations required to complete a circle 16 - D1 = [cos (thetad) sin(thetad); -sin(thetad) cos (thetad) ); rotation matrix for counter-clockwise direction 17 - for i=1:1:t 18 - S-D1 S; rotate the square 19 set (P, 'xdata',S(1, :), 'ydata', 5(2,:)); plot 20 - pause (0.1); 21 - end 22 23 D2 = [cos (thetad) -3in (thetad); sin(thetad) cos (thetad) }; 3 rotation matrix for clockwise direction 24 for i=1:t 25 S-D2S; rotate the square 26 set (p, 'xdata',S(1, :), 'ydata',5 (2,:)); $plot 27 - pause (0.1); 28 end 29 - hold off

----------------------------- SAMPLE OUTPUT

1.5 0.5 OH -0.5 -1.5 -1.5 - 1 -0.5 0 0.5 1 1.5 2