MAT 343 Laboratory 4 Plotting and computer animation in MATLAB In this laboratory session we will learn: 1. How to plot in MATLAB 2. The geometric properties of special types of matrices (rotations, dilations, reflections) 3. How to perform simple computer animations in MATLAB Plotting in MATLAB If x and y are vectors of the same size, the command plot(x,y) will produce a plot of all the (xi , yi ) pairs, and each point will be connected to the next by a line segment. If the x-coordinates are taken close enough together, the graph should resemble a smooth curve. The command plot(x,y,’o’) will mark the ordered pairs with o’s, but will not connect the points. ex/10 sin x For example, to plot the function f (x) = on the interval [0, 10], set x+1 x = 0:0.2:10; y = exp(x/10).*sin(x)./(x+1); (note the . before the * and /, as these operations must be done component-wise). The command plot(x,y) will generate the graph of the function. To compare the graph with that of sin x, we could set z = sin(x); and use the command plot(x,y,x,z) to plot both curves at the same time. It is also possible to do more sophisticated types of plots in MATLAB, including polar coordinates, contour plots, and three-dimensional surfaces. Given the points (xi , yi ), rather than creating the vectors x and y, an equivalent approach, which often is more convenient, is to create a 2 × n matrix of data:   x1 x2 . . . xn A= y1 y2 . . . yn then typing plot(A(1,:),A(2,:)) In other words, the first row of A gives the x’s and the second row the y’s. EXAMPLE 1 Consider the square whose vertices are (0, 0), (1, 0), (1, 1), (0, 1). (a) Enter the x and y coordinates of these points as MATLAB vectors by setting x = [0, 1, 1, 0, 0] y = [0, 0, 1, 1, 0] THIS CONTENT IS PROTECTED AND MAY NOT BE SHARED, UPLOADED, SOLD OR DISTRIBUTED 2019 v1 Copyright@ School of Mathematical and Statistical Sciences, Arizona State University 1 The coordinates of the first points are repeated in the fourth column of x and y because we want to connect the fourth point back to the first. The command plot(x,y) draws this square in the graph window. The command axis([-2,2,-2,2]) will rescale the axes so that the square does not take up the entire figure. (b) Now let’s store the entire figure in a matrix S by storing the x-coordinate in the first row and the y-coordinate in the second row: S = [0, 1, 1, 0, 0; 0, 0, 1, 1, 0] (if x and y have already been entered we can simply type S = [x; y]). The square can then be plotted entering plot(S(1,:),S(2,:)) Again, set axis([-2,2,-2,2]), and add a grid by entering grid on. Rotation Matrices in R2 Any 2-dimensional matrix of the form  Q= cos θ − sin θ sin θ cos θ  is a rotation matrix. If v is a vector in R2 , the product Qv is the vector v rotated θ radians in the counterclockwise direction around the origin (0, 0). To rotate the vector in the clockwise direction we simply replace θ with −θ, thus   cos(−θ) − sin(−θ) −1 . Q = sin(−θ) cos(−θ) Since cos(−θ) = cos θ and sin(−θ) = − sin(θ), we have   cos θ sin θ −1 = QT . Q = − sin θ cos θ EXAMPLE 2 We can rotate the square S from Example 1 by an angle of π/4 in the counterclockwise direction by multiplying the matrix S by   cos(π/4) − sin(π/4) Q= sin(π/4) cos(π/4) The following code implements the rotation and the resulting picture is displayed in Figure 1. clear all ; % clear all variables clf ; % clear all settings for the plot S =[0 ,1 ,1 ,0 ,0;0 ,0 ,1 ,1 ,0]; plot ( S (1 ,:) , S (2 ,:) , ' linewidth ' ,2); % plot the square hold on ; theta = pi /4; % define the angle theta Q =[ cos ( theta ) , - sin ( theta ); sin ( theta ) , cos ( theta )]; % rotation matrix QS = Q * S ; % rotate the square plot ( QS (1 ,:) , QS (2 ,:) , ' -r ' , ' linewidth ' ,2); % plot the rotated square title ( ' Example of Rotation ' ); % add a title legend ( ' original square ' , ' rotated square ') % add a legend axis equal ; axis ([ -1 ,2 , -1 ,2]); % set the window grid on ; % add a grid hold off THIS CONTENT IS PROTECTED AND MAY NOT BE SHARED, UPLOADED, SOLD OR DISTRIBUTED 2019 v1 Copyright@ School of Mathematical and Statistical Sciences, Arizona State University 2 Figure 1: Original square and rotated square Remarks: • The hold on command concatenates all subsequent plot commands on the same graph. Type hold off to start a new plot. • The ’-r’ plots the figure in red. • grid on adds a grid to the axis • axis equal sets the same scale on the x and y axes. • ’linewidth’, 2 sets the width of the line to 2 points = 2/72 inch (the default is 0.5 points). Dilation and Contraction Consider the matrix   αx 0 D= . 0 αy  T  T If we multiply this matrix by any vector v = x y we obtain the vector αx x αy y which scales the vector by a factor αx in the x-direction and by a factor αy in the y-direction. The scalings are a dilation or contraction, depending on whether αx and αy are greater or smaller than one. EXAMPLE 3 Consider the case where the dilation factors are αx = 2 and αy = 3. We can plot the original square S together with the dilated version by executing the following commands. The output is displayed in Figure 2. clf S =[0 ,1 ,1 ,0 ,0;0 ,0 ,1 ,1 ,0]; plot ( S (1 ,:) , S (2 ,:) , ' linewidth ' ,2) hold on D = [2 , 0; 0 , 3]; % dilation matrix DS = D * S ; plot ( DS (1 ,:) , DS (2 ,:) , ' -r ' , ' linewidth ' ,2) title ( ' Example of Dilation ') legend ( ' original square ' , ' dilated square ' , ' location ' , ' southeast ') grid on axis equal , axis ([ -1 ,4 , -1 ,4]) % adjust the axis and the window hold off THIS CONTENT IS PROTECTED AND MAY NOT BE SHARED, UPLOADED, SOLD OR DISTRIBUTED 2019 v1 Copyright@ School of Mathematical and Statistical Sciences, Arizona State University 3 Figure 2: Original square and dilated square Shear Matrices   1 tx A matrix of the form T = is a horizontal shear transformation. When we apply this 0 1 transformation, the y coordinates are unaffected, but the x coordinates are translated linearly with y.   1 0 is a vertical shear transformation: the x coordinates are Similarly, a matrix of the form T = ty 1 unaffected, but the y coordinates are translated linearly with x. EXAMPLE 4 We will  a horizontal shear transformation of 2 units. The matrix that accomplishes that is  apply 1 2 . We can plot the original square together with the sheared version by executing the T = 0 1 following commands. The output is displayed in Figure 3. clf S =[0 ,1 ,1 ,0 ,0;0 ,0 ,1 ,1 ,0]; plot ( S (1 ,:) , S (2 ,:) , ' linewidth ' ,2) hold on T =[1 ,2;0 ,1]; % shear transformation matrix TS = T * S ; plot ( TS (1 ,:) , TS (2 ,:) , ' -r ' , ' linewidth ' ,2); title ( ' Example of horizontal shear ') legend ( ' original square ' , ' sheared square ' , ' location ' , ' southeast ') axis equal , axis ([ -1 ,4 , -1 ,4]); grid on % adjust the axis and the window hold off Composition of Transformations One can perform a composition of two (or more) transformations by multiplying by the corresponding matrices in the appropriate order. EXAMPLE 5 We can see the effect of first rotating the square 45 degrees counterclockwise and then applying the shear from EXAMPLE 3, by plotting the product T*Q*S . Note the order of the matrices; here we first apply the rotation (Q is the matrix that multiplies S first) and then we apply the shear. The THIS CONTENT IS PROTECTED AND MAY NOT BE SHARED, UPLOADED, SOLD OR DISTRIBUTED 2019 v1 Copyright@ School of Mathematical and Statistical Sciences, Arizona State University 4 Figure 3: Original square and sheared square output of the following code is displayed in Figure 4. clf S =[0 ,1 ,1 ,0 ,0;0 ,0 ,1 ,1 ,0]; plot ( S (1 ,:) , S (2 ,:) , ' linewidth ' ,2) hold on theta = pi /4; % define the angle Q =[ cos ( theta ) , - sin ( theta ); sin ( theta ) , cos ( theta )]; T =[1 ,2;0 ,1]; % shear transformation matrix TQS = T * Q * S ; plot ( TQS (1 ,:) , TQS (2 ,:) , ' -r ' , ' linewidth ' ,2); title ( ' Example of rotation and shear ') legend ( ' original square ' , ' modified square ' , ' location ' , ' southeast ') axis equal , axis ([ -1 ,4 , -1 ,4]); grid on % adjust the axis and the window hold off Figure 4: Original square and square first rotated and then sheared Animation 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,:),S(2,:)) THIS CONTENT IS PROTECTED AND MAY NOT BE SHARED, UPLOADED, SOLD OR DISTRIBUTED 2019 v1 Copyright@ School of Mathematical and Statistical Sciences, Arizona State University 5 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. clf % 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 on % 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 = D2 * S ; % 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 off EXERCISES Instructions: For each of the following exercises, create an M-file to store the MATLAB commands. Copy and paste the M-file into a text document. For problems 1 and 2, include in the text document the pictures produced by MATLAB. Resize and crop the pictures so that they do not take up too much space. If the question requires written answers, include them in the text file in the appropriate location. Make sure you clearly label and separate all the Exercises. For problems 3 and 4 you do not need to include the picture. 1. Determine the shearing transformation matrix that shears 1 units in the vertical direction. Plot the original square together with the sheared square. Use axis([-1,3,-1,3]);. Add a grid, a legend and a title (similarly to EXAMPLE 4). Include the M-file as well as the figure. 2. Consider the original square S. First apply the shear transformation from EXAMPLE 4 and then rotate the square 45◦ counterclockwise. Plot the resulting figure (together with the original square) and compare with the plot in Example 5. Are the results the same? Does the order of the transformations matter? Include the M-file as well as the figure and don’t forget to answer all questions. Hint: Here you want to apply the shear first, then rotate, whereas in Example 5 the square is first rotated and then the shear is applied. To do this, TQS = T*Q*S, where Q is the rotation THIS CONTENT IS PROTECTED AND MAY NOT BE SHARED, UPLOADED, SOLD OR DISTRIBUTED 2019 v1 Copyright@ School of Mathematical and Statistical Sciences, Arizona State University 6 matrix and T is the shearing transformation matrix. So, if you want to first shear and then rotate, in what order will you multiply the matrices? 3. Adapt the procedure developed in Example 6 to rotate the square counterclockwise by increments of π/7 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,-2,2]). Include the M-file. 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 π/7 and then rotating the square clockwise back again. 4. Adapt the procedure developed in Example 6 to show the square rotating in a counterclockwise direction about the origin by increments of π/7 for a total angle of 2π and expanding at the same time by a factor of 8/7, then stopping and rotating in the clockwise direction as it shrinks to its original size (with a contraction factor of 7/8). At the end of the program, the figure should have returned to its original size and original location. You may want to rescale the axis to axis([-7,7,-7,7]); Include the M-file. Do not include the figure. Hint: Similarly to Exercise 3, you might want to use two for loops: one loop for the counterclockwise rotation + dilation, and another one for the clockwise rotation + contraction. Think about how many times you will need to go through the loops. Homogeneous coordinates Translations are not linear transformations and consequently cannot be accomplished via matrix multiplication using a 2 × 2 matrix. To circumvent this problem we use homogeneous coordinates. The homogeneous coordinates for a point (x1 , x2 )T are (x1 , x2 , 1)T . However, when the point represented by the homogeneous coordinates (x1 , x2 , 1)T is plotted, only the x1 and x2 are taken into consideration. The 1 is ignored. This allows us to produce the translated coordinates for a figure by matrix multiplication. Suppose we want to translate the vector c1 units horizontally and c2 units vertically. In homogeneous coordinates the translated vector is (x1 + c1 , x2 + c2 , 1)T . We need a matrix M such that M (x1 , x2 , 1)T = (x1 + c1 , x2 + c2 , 1)T It is easy to verify that the matrix   1 0 c1 M =  0 1 c2  0 0 1 has the desired property. EXAMPLE 7 We will apply a horizontal translation of 2units and a vertical translation of -3 units. The matrix  1 0 2 that accomplishes that is M = 0 1 −3. To apply the transformation to the square S, we will 0 0 1 need to convert the square to homogeneous coordinates. This is done by adding a row of ones to the THIS CONTENT IS PROTECTED AND MAY NOT BE SHARED, UPLOADED, SOLD OR DISTRIBUTED 2019 v1 Copyright@ School of Mathematical and Statistical Sciences, Arizona State University 7  0  matrix S so that the new matrix is S = 0 1 with the translated version by executing the 5. 1 1 0 0 1 1 1 1 1 following  0 0. We can plot the original square together 1 commands. The output is displayed in Figure clf S =[0 ,1 ,1 ,0 ,0;0 ,0 ,1 ,1 ,0;1 ,1 ,1 ,1 ,1]; % square in homogeneous coordinates M =[1 ,0 ,2;0 ,1 , -3;0 ,0 ,1]; % translation matrix MS = M * S ; % apply the translation to the square plot ( S (1 ,:) , S (2 ,:) , 'k ' , ' linewidth ' ,2); % plot the original square in black hold on plot ( MS (1 ,:) , MS (2 ,:) , 'r ' , ' linewidth ' ,2); % plot the translated square in red legend ( ' original square ' , ' translated square ' , ' location ' , ' southwest ' ); axis equal , axis ([ -1 ,4 , -4 ,4]) , grid on % adjust the axis hold off Figure 5: Original square and translated square EXAMPLE 8 Consider the matrix S representing the original square in homogeneous coordinates. In the following M-file we translate the square horizontally using c1 = 0.2 and c2 = 0 for 40 times. We then translate the square vertically using c1 = 0 and c2 = 0.2 and 40 iterations. clf S =[0 ,1 ,1 ,0 ,0;0 ,0 ,1 ,1 ,0;1 ,1 ,1 ,1 ,1]; % square in homogeneous coordinates M1 = [1 ,0 ,0.2;0 ,1 ,0;0 ,0 ,1]; % first translation matrix M2 = [1 ,0 ,0;0 ,1 ,0.2;0 ,0 ,1]; % the second translation matrix p = plot ( S (1 ,:) , S (2 ,:)); % plot the original square axis square , axis ([ -1 ,10 , -1 ,10]) , grid on for i = 1:40 S = M1 * S ; % compute the translated square set (p , ' xdata ' ,S (1 ,:) , ' ydata ' ,S (2 ,:)); % plot the translated square pause (0.1) end for i = 1:40 S = M2 * S ; % compute the translated square set (p , ' xdata ' ,S (1 ,:) , ' ydata ' ,S (2 ,:)); % plot the translated square pause (0.1) end THIS CONTENT IS PROTECTED AND MAY NOT BE SHARED, UPLOADED, SOLD OR DISTRIBUTED 2019 v1 Copyright@ School of Mathematical and Statistical Sciences, Arizona State University 8 Reflection Matrices The reflection matrix about the line L with direction the unit vector u is given by R = 2uuT − I, where I is the identity matrix.  √ √ T For instance, to obtain the reflection matrix about the line at 45 degrees we can take u = 22 , 22 .     1/2 1/2 0 1 T Then uu = and R = . (Note that this answer makes perfect sense, since the 1/2 1/2 1 0 reflection of the vector x = (x1 , x2 )T about the line at 45 degrees is the vector (x2 , x1 )T ). In homogeneous coordinates, the reflection matrix becomes:   0 1 0 R = 1 0 0 0 0 1 The other types of basic transformations (rotations, dilations) can all be adapted to homogeneous coordinates by augmenting the matrix by the row (0, 0, 1) and the column (0, 0, 1)T . For instance, in homogeneous coordinates, the dilation matrix that dilates by a factor α becomes   α 0 0 D =  0 α 0 . 0 0 1 EXERCISES 5. Consider the translated square MS in EXAMPLE 7. Apply to this square the reflection matrix that reflects about the line at 45 degrees. Plot the translated square in black and the reflected square in red. Use axis([-4,4,-4,4]); and plot the line y = x using the command plot([-4,4],[-4,4]); (this command plots the line connecting the points (−4, −4) and (4, 4)). Add a legend and a grid. You should reproduce Figure 6. Include the M-file and the figure. Figure 6: Output of Exercise 5 6. Modify the M-file in EXAMPLE 8 adding translations that bring the square to its original position using 40 iterations and a single additional for loop. Include the M-file (for a total of three loops) You do not need to include the figure. THIS CONTENT IS PROTECTED AND MAY NOT BE SHARED, UPLOADED, SOLD OR DISTRIBUTED 2019 v1 Copyright@ School of Mathematical and Statistical Sciences, Arizona State University 9 Rotations about a point different than the origin   cos θ − sin θ performs a rotation about the origin, but what if we want sin θ cos θ to rotate about a point P = (Px , Py ) different than the origin? To accomplish this we need to The rotation matrix Q = • translate the object so that P will coincide with the origin, i.e. translate −Px units horizontally and −Py units vertically • rotate the object • translate the object back, i.e. translate Px units horizonally and Py units vertically To compose the three transformations above, we multiply the    1 0 Px cos θ − sin θ 0 1 0 1 Py   sin θ cos θ 0 0 0 0 1 0 0 1 0 appropriate matrices:  0 −Px 1 −Py  0 1 Note that the rotation matrix is written in homogeneous coordinates. EXAMPLE 9 In this example we first translate the square horizontally 8 units using increments of 0.2. We then rotate the resulting square clockwise π/2 radians around the vertex (9, 0) using increments of π/8 radians. clf S =[0 ,1 ,1 ,0 ,0;0 ,0 ,1 ,1 ,0;1 ,1 ,1 ,1 ,1]; % square in homogeneous coordinates M1 = [1 ,0 ,0.2;0 ,1 ,0;0 ,0 ,1]; % first translation matrix theta = pi /8; % define the angle theta Q =[ cos ( theta ) , - sin ( theta ) ,0; sin ( theta ) , cos ( theta ) ,0;0 ,0 ,1]; % rotation matrix about (0 ,0) QP =[1 ,0 ,9;0 ,1 ,0;0 ,0 ,1]* Q '*[1 ,0 , -9;0 ,1 ,0;0 ,0 ,1]; % rotation matrix about (9 ,0) p = plot ( S (1 ,:) , S (2 ,:)); % plot the original square axis equal , axis ([ -0.5 ,11 , -2 ,5]) , grid on for i = 1:40 S = M1 * S ; % compute the translated square set (p , ' xdata ' ,S (1 ,:) , ' ydata ' ,S (2 ,:)); % plot the translated square pause (0.1) end for i = 1:4 S = QP * S ; % compute the rotated square set (p , ' xdata ' ,S (1 ,:) , ' ydata ' ,S (2 ,:)); % plot the rotated square pause (0.1) end EXERCISES 7. Consider the square in EXAMPLE 9. The goal of this exercise is to bring back the square to its original position by first translating it horizontally to the left 8 units using 40 iterations, and then rotating it counterclockwise π/2 radians around the point (1, 0) using 4 iterations. This can be done by modifying the code in EXAMPLE 9 by adding two for loops. The first loop should translate the square while the second should rotate it around the point (1, 0). Note that the rotation is counterclockwise, while in EXAMPLE 9 it was clockwise. Include the M-file. You do not need to include the figure. THIS CONTENT IS PROTECTED AND MAY NOT BE SHARED, UPLOADED, SOLD OR DISTRIBUTED 2019 v1 Copyright@ School of Mathematical and Statistical Sciences, Arizona State University 10
Purchase answer to see full attachment