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GitHub Repository: weijie-chen/Linear-Algebra-With-Python
 Path: blob/master/notebooks/linear_algebra_visulization.py
Views: 449
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import matplotlib.pyplot as plt

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import numpy as np

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import numpy

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def linearCombo(a, b, c):

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    '''This function is for visualizing linear combination of standard basis in 3D.

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    Function syntax: linearCombo(a, b, c), where a, b, c are the scalar multiplier, 

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    also the elements of the vector.

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    '''

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    fig = plt.figure(figsize = (10,10))

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    ax = fig.add_subplot(projection='3d')

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    ######################## Standard basis and Scalar Multiplid Vectors#########################

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    vec = np.array([[[0, 0, 0, 1, 0, 0]], # e1

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                    [[0, 0, 0, 0, 1, 0]], # e2

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                    [[0, 0, 0, 0, 0, 1]], # e3

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                    [[0, 0, 0, a, 0, 0]], # a* e1

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                    [[0, 0, 0, 0, b, 0]], # b* e2

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                    [[0, 0, 0, 0, 0, c]], # c* e3

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                    [[0, 0, 0, a, b, c]]]) # ae1 + be2 + ce3

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    colors = ['b','b','b','r','r','r','g']

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    for i in range(vec.shape[0]): 

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        X, Y, Z, U, V, W = zip(*vec[i,:,:])

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        ax.quiver(X, Y, Z, U, V, W, length=1, normalize=False, 

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                  color = colors[i] ,arrow_length_ratio = .08, pivot = 'tail',

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                  linestyles = 'solid',linewidths = 3, alpha =.6)

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    #################################Plot Rectangle Boxes##############################

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    dlines = np.array([[[a, 0, 0],[a, b, 0]],

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                      [[0, b, 0],[a, b, 0]],

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                      [[0, 0, c],[a, b, c]],

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                      [[0, 0, c],[a, 0, c]],

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                      [[a, 0, c],[a, b, c]],

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                      [[0, 0, c],[0, b, c]],

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                      [[0, b, c],[a, b, c]],

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                      [[a, 0, 0],[a, 0, c]],

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                      [[0, b, 0],[0, b, c]],

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                      [[a, b, 0],[a, b, c]]])

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    colors = ['k','k','g','k','k','k','k','k','k']

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    for i in range(dlines.shape[0]):

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        ax.plot(dlines[i,:,0], dlines[i,:,1], dlines[i,:,2], lw =3, ls = '--', color = 'black', alpha=0.5)

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    #################################Annotation########################################

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    ax.text(x = a, y = b, z = c, s= ' $(%0.d, %0.d, %.0d)$'% (a, b, c), size = 18)

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    ax.text(x = a, y = 0, z = 0, s= ' $%0.d e_1 = (%0.d, 0, 0)$'% (a, a), size = 15)

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    ax.text(x = 0, y = b, z = 0, s= ' $%0.d e_2 = (0, %0.d, 0)$'% (b, b), size = 15)

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    ax.text(x = 0, y = 0, z = c, s= ' $%0.d e_3 = (0, 0, %0.d)$' %(c, c), size = 15)

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    #################################Axis Setting######################################

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    ax.grid()

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    ax.set_xlim([0, a+1])

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    ax.set_ylim([0, b+1])

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    ax.set_zlim([0, c+1])

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    ax.set_xlabel('x-axis', size = 18)

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    ax.set_ylabel('y-axis', size = 18)

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    ax.set_zlabel('z-axis', size = 18)

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    ax.set_title('Vector $(%0.d, %0.d, %.0d)$ Visualization' %(a, b, c), size = 20)

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    ax.view_init(elev=20., azim=15)

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if __name__ == '__main__':

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    a = 7

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    b = 4

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    c = 9

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    linearCombo(a, b, c)

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def linearComboNonStd(a, b, c, vec1, vec2, vec3):

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    '''This function is for visualizing linear combination of non-standard basis in 3D.

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    Function syntax: linearCombo(a, b, c, vec1, vec2, vec3), where a, b, c are the scalar multiplier, 

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    ve1, vec2 and vec3 are the basis.

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    '''

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    fig = plt.figure(figsize = (10,10))

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    ax = fig.add_subplot(projection='3d')

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    ########################Plot basis##############################

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    vec1 = np.array([[0, 0, 0, vec1[0], vec1[1], vec1[2]]])

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    X, Y, Z, U, V, W = zip(*vec1)

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    ax.quiver(X, Y, Z, U, V, W, length=1, normalize=False, color = 'blue',arrow_length_ratio = .08, pivot = 'tail',

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          linestyles = 'solid',linewidths = 3)

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    vec2 = np.array([[0, 0, 0, vec2[0], vec2[1], vec2[2]]])

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    X, Y, Z, U, V, W = zip(*vec2)

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    ax.quiver(X, Y, Z, U, V, W, length=1, normalize=False, color = 'blue',arrow_length_ratio = .08, pivot = 'tail',

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          linestyles = 'solid',linewidths = 3)

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    vec3 = np.array([[0, 0, 0, vec3[0], vec3[1], vec3[2]]])

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    X, Y, Z, U, V, W = zip(*vec3)

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    ax.quiver(X, Y, Z, U, V, W, length=1, normalize=False, color = 'blue',arrow_length_ratio = .08, pivot = 'tail',

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          linestyles = 'solid',linewidths = 3)

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    ###########################Plot Scalar Muliplied Vectors####################

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    avec1 = a * vec1 

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    X, Y, Z, U, V, W = zip(*avec1)

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    ax.quiver(X, Y, Z, U, V, W, length=1, normalize=False, color = 'red', alpha = .6,arrow_length_ratio = a/100, pivot = 'tail',

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          linestyles = 'solid',linewidths = 3)

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    bvec2 = b * vec2

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    X, Y, Z, U, V, W = zip(*bvec2)

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    ax.quiver(X, Y, Z, U, V, W, length=1, normalize=False, color = 'red', alpha = .6,arrow_length_ratio = b/100, pivot = 'tail',

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          linestyles = 'solid',linewidths = 3)

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    cvec3 = c * vec3

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    X, Y, Z, U, V, W = zip(*cvec3)

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    ax.quiver(X, Y, Z, U, V, W, length=1, normalize=False, color = 'red', alpha = .6,arrow_length_ratio = c/100, pivot = 'tail',

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          linestyles = 'solid',linewidths = 3)

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    combo = avec1 + bvec2 + cvec3

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    X, Y, Z, U, V, W = zip(*combo)

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    ax.quiver(X, Y, Z, U, V, W, length=1, normalize=False, color = 'green', alpha = .7,arrow_length_ratio = np.linalg.norm(combo)/300, pivot = 'tail',

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          linestyles = 'solid',linewidths = 3)

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    #################################Plot Rectangle Boxes##############################

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    point1 = [avec1[0, 3], avec1[0, 4], avec1[0, 5]]

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    point2 = [avec1[0, 3]+bvec2[0, 3], avec1[0, 4]+bvec2[0, 4], avec1[0, 5]+bvec2[0, 5]]

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    line1 = np.array([point1, point2])

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    ax.plot(line1[:,0], line1[:,1], line1[:,2], lw =3, ls = '--', color = 'black', alpha=0.5)

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    point1 = [bvec2[0, 3], bvec2[0, 4], bvec2[0, 5]]

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    point2 = [avec1[0, 3]+bvec2[0, 3], avec1[0, 4]+bvec2[0, 4], avec1[0, 5]+bvec2[0, 5]]

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    line1 = np.array([point1, point2])

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    ax.plot(line1[:,0], line1[:,1], line1[:,2], lw =3, ls = '--', color = 'black', alpha=0.5)

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    point1 = [bvec2[0, 3], bvec2[0, 4], bvec2[0, 5]]

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    point2 = [cvec3[0, 3]+bvec2[0, 3], cvec3[0, 4]+bvec2[0, 4], cvec3[0, 5]+bvec2[0, 5]]

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    line1 = np.array([point1, point2])

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    ax.plot(line1[:,0], line1[:,1], line1[:,2], lw =3, ls = '--', color = 'black', alpha=0.5)

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    point1 = [cvec3[0, 3], cvec3[0, 4], cvec3[0, 5]]

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    point2 = [cvec3[0, 3]+bvec2[0, 3], cvec3[0, 4]+bvec2[0, 4], cvec3[0, 5]+bvec2[0, 5]]

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    line1 = np.array([point1, point2])

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    ax.plot(line1[:,0], line1[:,1], line1[:,2], lw =3, ls = '--', color = 'black', alpha=0.5)

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    point1 = [cvec3[0, 3], cvec3[0, 4], cvec3[0, 5]]

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    point2 = [cvec3[0, 3]+avec1[0, 3], cvec3[0, 4]+avec1[0, 4], cvec3[0, 5]+avec1[0, 5]]

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    line1 = np.array([point1, point2])

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    ax.plot(line1[:,0], line1[:,1], line1[:,2], lw =3, ls = '--', color = 'black', alpha=0.5)

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    point1 = [avec1[0, 3], avec1[0, 4], avec1[0, 5]]

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    point2 = [cvec3[0, 3]+avec1[0, 3], cvec3[0, 4]+avec1[0, 4], cvec3[0, 5]+avec1[0, 5]]

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    line1 = np.array([point1, point2])

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    ax.plot(line1[:,0], line1[:,1], line1[:,2], lw =3, ls = '--', color = 'black', alpha=0.5)

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    ##

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    point1 = [avec1[0, 3]+bvec2[0, 3]+cvec3[0, 3], 

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              avec1[0, 4]+bvec2[0, 4]+cvec3[0, 4], 

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              avec1[0, 5]+bvec2[0, 5]+cvec3[0, 5]]

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    point2 = [cvec3[0, 3]+avec1[0, 3], 

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              cvec3[0, 4]+avec1[0, 4], 

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              cvec3[0, 5]+avec1[0, 5]]

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    line1 = np.array([point1, point2])

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    ax.plot(line1[:,0], line1[:,1], line1[:,2], lw =3, ls = '--', color = 'black', alpha=0.5)

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    ##

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    point1 = [avec1[0, 3]+bvec2[0, 3]+cvec3[0, 3],

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              avec1[0, 4]+bvec2[0, 4]+cvec3[0, 4], 

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              avec1[0, 5]+bvec2[0, 5]+cvec3[0, 5]]

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    point2 = [cvec3[0, 3]+bvec2[0, 3], 

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              cvec3[0, 4]+bvec2[0, 4], 

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              cvec3[0, 5]+bvec2[0, 5]]

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    line1 = np.array([point1, point2])

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    ax.plot(line1[:,0], line1[:,1], line1[:,2], lw =3, ls = '--', color = 'black', alpha=0.5)

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    ##

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    point1 = [avec1[0, 3]+bvec2[0, 3]+cvec3[0, 3], 

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              avec1[0, 4]+bvec2[0, 4]+cvec3[0, 4], 

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              avec1[0, 5]+bvec2[0, 5]+cvec3[0, 5]]

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    point2 = [bvec2[0, 3]+avec1[0, 3], 

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              bvec2[0, 4]+avec1[0, 4], 

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              bvec2[0, 5]+avec1[0, 5]]

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    line1 = np.array([point1, point2])

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    ax.plot(line1[:,0], line1[:,1], line1[:,2], lw =3, ls = '--', color = 'black', alpha=0.5)

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    #################################Annotation########################################

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    ax.text(x = vec1[0,3], y = vec1[0,4], z = vec1[0,5], s= ' $v_1 =(%0.d, %0.d, %.0d)$'% (vec1[0,3], vec1[0,4], vec1[0,4]), size = 8)

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    ax.text(x = vec2[0,3], y = vec2[0,4], z = vec2[0,5], s= ' $v_2 =(%0.d, %0.d, %.0d)$'% (vec2[0,3], vec2[0,4], vec2[0,4]), size = 8)

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    ax.text(x = vec3[0,3], y = vec3[0,4], z = vec3[0,5], s= ' $v_3= (%0.d, %0.d, %.0d)$'% (vec3[0,3], vec3[0,4], vec3[0,4]), size = 8)

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    ax.text(x = avec1[0,3], y = avec1[0,4], z = avec1[0,5], s= ' $%.0d v_1 =(%0.d, %0.d, %.0d)$'% (a, avec1[0,3], avec1[0,4], avec1[0,4]), size = 8)

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    ax.text(x = bvec2[0,3], y = bvec2[0,4], z = bvec2[0,5], s= ' $%.0d v_2 =(%0.d, %0.d, %.0d)$'% (b, bvec2[0,3], bvec2[0,4], bvec2[0,4]), size = 8)

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    ax.text(x = cvec3[0,3], y = cvec3[0,4], z = cvec3[0,5], s= ' $%.0d v_3= (%0.d, %0.d, %.0d)$'% (c, cvec3[0,3], cvec3[0,4], cvec3[0,4]), size = 8)

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#     ax.text(x = 0, y = b, z = 0, s= ' $%0.d e_2 = (0, %0.d, 0)$'% (b, b), size = 15)

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#     ax.text(x = 0, y = 0, z = c, s= ' $%0.d e_3 = (0, 0, %0.d)$' %(c, c), size = 15)

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    #################################Axis Setting######################################

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    ax.grid()

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    ax.set_xlim([0, 15])

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    ax.set_ylim([0, 15])

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    ax.set_zlim([0, 15])

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    ax.set_xlabel('x-axis', size = 18)

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    ax.set_ylabel('y-axis', size = 18)

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    ax.set_zlabel('z-axis', size = 18)

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    #ax.set_title('Vector $(%0.d, %0.d, %.0d)$ Visualization' %(a, b, c), size = 20)

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    ax.view_init(elev=20., azim=15)

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if __name__ == '__main__':

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    a = 2

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    b = 3

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    c = 4

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    vec1 = np.array([2,1,0])

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    vec2 = np.array([0,3,1])

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    vec3 = np.array([1,2,3])

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    linearComboNonStd(a, b, c, vec1,vec2,vec3)

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