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‎linAlgVis.py

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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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