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Book4_Ch20_Python_Codes/Bk4_Ch20_01.py

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+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+# Bk4_Ch20_01.py
+
+import numpy as np
+import matplotlib.pyplot as plt
+
+alphas = np.linspace(0, 2*np.pi, 100)
+
+# unit circle
+r = np.sqrt(1.0)
+
+z1 = r*np.cos(alphas)
+z2 = r*np.sin(alphas)
+
+Z = np.array([z1, z2]).T # data of unit circle
+
+# scale
+S = np.array([[2, 0],
+              [0, 0.5]])
+
+thetas = np.array([0, 30, 45, 60, 90, 120])
+
+for theta in thetas:
+
+    # rotate
+    print('==== Rotate ====')
+    print(theta)
+    theta = theta/180*np.pi
+    R = np.array([[np.cos(theta), -np.sin(theta)],
+                  [np.sin(theta),  np.cos(theta)]])
+    
+    # translate
+    c = np.array([2, 1])
+    X = Z@S@R.T + c;
+    
+    Q = R@np.linalg.inv(S)@np.linalg.inv(S)@R.T
+    print('==== Q ====')
+    print(Q)
+    LAMBDA, V = np.linalg.eig(Q)
+    print('==== LAMBDA ====')
+    print(LAMBDA)
+    print('==== V ====')
+    print(V)
+    
+    x1 = X[:,0]
+    x2 = X[:,1]
+    
+    fig, ax = plt.subplots(1)
+    ax.plot(z1, z2, 'b') # plot the unit circle
+    ax.plot(x1, x2, 'r') # plot the transformed shape
+    ax.plot(c[0],c[1],'xk') # plot the center
+    
+    ax.quiver(0,0,1,0,color = 'b',angles='xy', scale_units='xy',scale=1)
+    ax.quiver(0,0,0,1,color = 'b',angles='xy', scale_units='xy',scale=1)
+    ax.quiver(0,0,-1,0,color = 'b',angles='xy', scale_units='xy',scale=1)
+    ax.quiver(0,0,0,-1,color = 'b',angles='xy', scale_units='xy',scale=1)
+    
+    ax.quiver(0,0,c[0],c[1],color = 'k',angles='xy', scale_units='xy',scale=1)
+    
+    ax.quiver(c[0],c[1],
+              V[0,0]/np.sqrt(LAMBDA[0]),
+              V[1,0]/np.sqrt(LAMBDA[0]),color = 'r',
+              angles='xy', scale_units='xy',scale=1)
+    
+    ax.quiver(c[0],c[1],
+              V[0,1]/np.sqrt(LAMBDA[1]),
+              V[1,1]/np.sqrt(LAMBDA[1]),color = 'r',
+              angles='xy', scale_units='xy',scale=1)
+
+    ax.quiver(c[0],c[1],
+              -V[0,0]/np.sqrt(LAMBDA[0]),
+              -V[1,0]/np.sqrt(LAMBDA[0]),color = 'r',
+              angles='xy', scale_units='xy',scale=1)
+    
+    ax.quiver(c[0],c[1],
+              -V[0,1]/np.sqrt(LAMBDA[1]),
+              -V[1,1]/np.sqrt(LAMBDA[1]),color = 'r',
+              angles='xy', scale_units='xy',scale=1)
+    
+    plt.axvline(x=0, color= 'k', zorder=0)
+    plt.axhline(y=0, color= 'k', zorder=0)
+    
+    
+    ax.set_aspect(1)
+    plt.xlim(-2,4)
+    plt.ylim(-2,4)
+    plt.grid(linestyle='--')
+    plt.show()
+    plt.xlabel('$x_1$')
+    plt.ylabel('$x_2$')

Book4_Ch20_Python_Codes/Bk4_Ch20_02.py

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+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+# Bk4_Ch20_02.py
+
+import numpy as np
+import matplotlib.pyplot as plt
+
+alphas = np.linspace(0, 2*np.pi, 100)
+
+# unit circle
+r = np.sqrt(1.0)
+
+z1 = r*1/np.cos(alphas)
+z2 = r*np.tan(alphas)
+
+Z = np.array([z1, z2]).T # data of unit circle
+
+# scale
+S = np.array([[1, 0],
+              [0, 1]])
+
+thetas = np.array([0, 30, 45, 60, 90, 120])
+
+for theta in thetas:
+
+    # rotate
+    print('==== Rotate ====')
+    print(theta)
+    theta = theta/180*np.pi
+    R = np.array([[np.cos(theta), -np.sin(theta)],
+                  [np.sin(theta),  np.cos(theta)]])
+
+    X = Z@S@R.T;
+    
+    x1 = X[:,0]
+    x2 = X[:,1]
+    
+    fig, ax = plt.subplots(1)
+    ax.plot(z1, z2, 'b') # plot the unit circle
+    ax.plot(x1, x2, 'r') # plot the transformed shape
+
+    plt.axvline(x=0, color= 'k', zorder=0)
+    plt.axhline(y=0, color= 'k', zorder=0)
+    ax.set_aspect(1)
+    plt.xlim(-3,3)
+    plt.ylim(-3,3)
+    plt.grid(linestyle='--')
+    plt.show()
+    plt.xlabel('$x_1$')
+    plt.ylabel('$x_2$')

Book4_Ch20_Python_Codes/Bk4_Ch20_03.py

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+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+# Bk4_Ch20_03.py
+
+import numpy as np
+import matplotlib.pyplot as plt
+
+a = 1.5
+b = 1
+
+x1 = np.linspace(-3,3,200)
+x2 = np.linspace(-3,3,200)
+xx1,xx2 = np.meshgrid(x1,x2)
+
+fig, ax = plt.subplots()
+
+theta_array = np.linspace(0,2*np.pi,100)
+
+plt.plot(a*np.cos(b*np.sin(theta)),b*np.sin(b*np.sin(theta)),color = 'k')
+
+colors = plt.cm.RdYlBu(np.linspace(0,1,len(theta_array)))
+
+for i in range(len(theta_array)):
+    
+    theta = theta_array[i]
+    
+    p1 = a*np.cos(theta)
+    p2 = b*np.sin(theta)
+    
+    tangent = p1*xx1/a**2 + p2*xx2/b**2 - p1**2/a**2 - p2**2/b**2
+    
+    colors_i = colors[int(i),:]
+    
+    ax.contour(xx1,xx2,tangent, levels = [0], colors = [colors_i])
+
+plt.axis('scaled')
+ax.set_xlim(-3,3)
+ax.set_ylim(-3,3)
+ax.spines['top'].set_visible(False)
+ax.spines['right'].set_visible(False)
+ax.spines['bottom'].set_visible(False)
+ax.spines['left'].set_visible(False)
+ax.axvline(x=0,color = 'k')
+ax.axhline(y=0,color = 'k')