1 | #!/usr/bin/env python |
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2 | """ |
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3 | Jitter Explorer |
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4 | =============== |
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5 | |
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6 | Application to explore orientation angle and angular dispersity. |
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7 | """ |
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8 | from __future__ import division, print_function |
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9 | |
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10 | import argparse |
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11 | |
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12 | try: # CRUFT: travis-ci does not support mpl_toolkits.mplot3d |
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13 | import mpl_toolkits.mplot3d # Adds projection='3d' option to subplot |
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14 | except ImportError: |
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15 | pass |
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16 | |
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17 | import matplotlib.pyplot as plt |
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18 | from matplotlib.widgets import Slider |
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19 | import numpy as np |
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20 | from numpy import pi, cos, sin, sqrt, exp, degrees, radians |
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21 | |
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22 | def draw_beam(axes, view=(0, 0)): |
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23 | """ |
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24 | Draw the beam going from source at (0, 0, 1) to detector at (0, 0, -1) |
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25 | """ |
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26 | #axes.plot([0,0],[0,0],[1,-1]) |
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27 | #axes.scatter([0]*100,[0]*100,np.linspace(1, -1, 100), alpha=0.8) |
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28 | |
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29 | steps = 25 |
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30 | u = np.linspace(0, 2 * np.pi, steps) |
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31 | v = np.linspace(-1, 1, steps) |
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32 | |
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33 | r = 0.02 |
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34 | x = r*np.outer(np.cos(u), np.ones_like(v)) |
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35 | y = r*np.outer(np.sin(u), np.ones_like(v)) |
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36 | z = 1.3*np.outer(np.ones_like(u), v) |
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37 | |
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38 | theta, phi = view |
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39 | shape = x.shape |
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40 | points = np.matrix([x.flatten(), y.flatten(), z.flatten()]) |
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41 | points = Rz(phi)*Ry(theta)*points |
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42 | x, y, z = [v.reshape(shape) for v in points] |
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43 | |
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44 | axes.plot_surface(x, y, z, rstride=4, cstride=4, color='y', alpha=0.5) |
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45 | |
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46 | def draw_ellipsoid(axes, size, view, jitter, steps=25, alpha=1): |
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47 | """Draw an ellipsoid.""" |
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48 | a, b, c = size |
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49 | u = np.linspace(0, 2 * np.pi, steps) |
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50 | v = np.linspace(0, np.pi, steps) |
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51 | x = a*np.outer(np.cos(u), np.sin(v)) |
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52 | y = b*np.outer(np.sin(u), np.sin(v)) |
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53 | z = c*np.outer(np.ones_like(u), np.cos(v)) |
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54 | x, y, z = transform_xyz(view, jitter, x, y, z) |
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55 | |
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56 | axes.plot_surface(x, y, z, rstride=4, cstride=4, color='w', alpha=alpha) |
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57 | |
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58 | draw_labels(axes, view, jitter, [ |
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59 | ('c+', [+0, +0, +c], [+1, +0, +0]), |
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60 | ('c-', [+0, +0, -c], [+0, +0, -1]), |
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61 | ('a+', [+a, +0, +0], [+0, +0, +1]), |
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62 | ('a-', [-a, +0, +0], [+0, +0, -1]), |
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63 | ('b+', [+0, +b, +0], [-1, +0, +0]), |
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64 | ('b-', [+0, -b, +0], [-1, +0, +0]), |
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65 | ]) |
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66 | |
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67 | def draw_sc(axes, size, view, jitter, steps=None, alpha=1): |
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68 | """Draw points for simple cubic paracrystal""" |
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69 | atoms = _build_sc() |
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70 | _draw_crystal(axes, size, view, jitter, atoms=atoms) |
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71 | |
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72 | def draw_fcc(axes, size, view, jitter, steps=None, alpha=1): |
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73 | """Draw points for face-centered cubic paracrystal""" |
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74 | # Build the simple cubic crystal |
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75 | atoms = _build_sc() |
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76 | # Define the centers for each face |
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77 | # x planes at -1, 0, 1 have four centers per plane, at +/- 0.5 in y and z |
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78 | x, y, z = ( |
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79 | [-1]*4 + [0]*4 + [1]*4, |
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80 | ([-0.5]*2 + [0.5]*2)*3, |
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81 | [-0.5, 0.5]*12, |
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82 | ) |
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83 | # y and z planes can be generated by substituting x for y and z respectively |
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84 | atoms.extend(zip(x+y+z, y+z+x, z+x+y)) |
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85 | _draw_crystal(axes, size, view, jitter, atoms=atoms) |
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86 | |
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87 | def draw_bcc(axes, size, view, jitter, steps=None, alpha=1): |
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88 | """Draw points for body-centered cubic paracrystal""" |
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89 | # Build the simple cubic crystal |
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90 | atoms = _build_sc() |
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91 | # Define the centers for each octant |
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92 | # x plane at +/- 0.5 have four centers per plane at +/- 0.5 in y and z |
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93 | x, y, z = ( |
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94 | [-0.5]*4 + [0.5]*4, |
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95 | ([-0.5]*2 + [0.5]*2)*2, |
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96 | [-0.5, 0.5]*8, |
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97 | ) |
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98 | atoms.extend(zip(x, y, z)) |
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99 | _draw_crystal(axes, size, view, jitter, atoms=atoms) |
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100 | |
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101 | def _draw_crystal(axes, size, view, jitter, atoms=None): |
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102 | atoms, size = np.asarray(atoms, 'd').T, np.asarray(size, 'd') |
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103 | x, y, z = atoms*size[:, None] |
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104 | x, y, z = transform_xyz(view, jitter, x, y, z) |
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105 | axes.scatter([x[0]], [y[0]], [z[0]], c='yellow', marker='o') |
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106 | axes.scatter(x[1:], y[1:], z[1:], c='r', marker='o') |
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107 | |
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108 | def _build_sc(): |
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109 | # three planes of 9 dots for x at -1, 0 and 1 |
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110 | x, y, z = ( |
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111 | [-1]*9 + [0]*9 + [1]*9, |
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112 | ([-1]*3 + [0]*3 + [1]*3)*3, |
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113 | [-1, 0, 1]*9, |
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114 | ) |
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115 | atoms = list(zip(x, y, z)) |
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116 | #print(list(enumerate(atoms))) |
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117 | # Pull the dot at (0, 0, 1) to the front of the list |
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118 | # It will be highlighted in the view |
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119 | index = 14 |
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120 | highlight = atoms[index] |
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121 | del atoms[index] |
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122 | atoms.insert(0, highlight) |
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123 | return atoms |
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124 | |
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125 | def draw_parallelepiped(axes, size, view, jitter, steps=None, alpha=1): |
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126 | """Draw a parallelepiped.""" |
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127 | a, b, c = size |
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128 | x = a*np.array([+1, -1, +1, -1, +1, -1, +1, -1]) |
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129 | y = b*np.array([+1, +1, -1, -1, +1, +1, -1, -1]) |
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130 | z = c*np.array([+1, +1, +1, +1, -1, -1, -1, -1]) |
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131 | tri = np.array([ |
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132 | # counter clockwise triangles |
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133 | # z: up/down, x: right/left, y: front/back |
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134 | [0, 1, 2], [3, 2, 1], # top face |
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135 | [6, 5, 4], [5, 6, 7], # bottom face |
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136 | [0, 2, 6], [6, 4, 0], # right face |
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137 | [1, 5, 7], [7, 3, 1], # left face |
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138 | [2, 3, 6], [7, 6, 3], # front face |
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139 | [4, 1, 0], [5, 1, 4], # back face |
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140 | ]) |
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141 | |
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142 | x, y, z = transform_xyz(view, jitter, x, y, z) |
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143 | axes.plot_trisurf(x, y, triangles=tri, Z=z, color='w', alpha=alpha) |
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144 | |
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145 | # Draw pink face on box. |
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146 | # Since I can't control face color, instead draw a thin box situated just |
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147 | # in front of the "c+" face. Use the c face so that rotations about psi |
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148 | # rotate that face. |
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149 | if 1: |
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150 | x = a*np.array([+1, -1, +1, -1, +1, -1, +1, -1]) |
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151 | y = b*np.array([+1, +1, -1, -1, +1, +1, -1, -1]) |
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152 | z = c*np.array([+1, +1, +1, +1, -1, -1, -1, -1]) |
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153 | x, y, z = transform_xyz(view, jitter, x, y, abs(z)+0.001) |
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154 | axes.plot_trisurf(x, y, triangles=tri, Z=z, color=[1, 0.6, 0.6], alpha=alpha) |
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155 | |
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156 | draw_labels(axes, view, jitter, [ |
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157 | ('c+', [+0, +0, +c], [+1, +0, +0]), |
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158 | ('c-', [+0, +0, -c], [+0, +0, -1]), |
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159 | ('a+', [+a, +0, +0], [+0, +0, +1]), |
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160 | ('a-', [-a, +0, +0], [+0, +0, -1]), |
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161 | ('b+', [+0, +b, +0], [-1, +0, +0]), |
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162 | ('b-', [+0, -b, +0], [-1, +0, +0]), |
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163 | ]) |
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164 | |
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165 | def draw_sphere(axes, radius=10., steps=100): |
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166 | """Draw a sphere""" |
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167 | u = np.linspace(0, 2 * np.pi, steps) |
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168 | v = np.linspace(0, np.pi, steps) |
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169 | |
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170 | x = radius * np.outer(np.cos(u), np.sin(v)) |
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171 | y = radius * np.outer(np.sin(u), np.sin(v)) |
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172 | z = radius * np.outer(np.ones(np.size(u)), np.cos(v)) |
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173 | axes.plot_surface(x, y, z, rstride=4, cstride=4, color='w') |
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174 | |
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175 | def draw_jitter(axes, view, jitter, dist='gaussian', size=(0.1, 0.4, 1.0), |
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176 | draw_shape=draw_parallelepiped): |
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177 | """ |
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178 | Represent jitter as a set of shapes at different orientations. |
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179 | """ |
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180 | # set max diagonal to 0.95 |
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181 | scale = 0.95/sqrt(sum(v**2 for v in size)) |
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182 | size = tuple(scale*v for v in size) |
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183 | |
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184 | #np.random.seed(10) |
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185 | #cloud = np.random.randn(10,3) |
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186 | cloud = [ |
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187 | [-1, -1, -1], |
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188 | [-1, -1, +0], |
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189 | [-1, -1, +1], |
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190 | [-1, +0, -1], |
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191 | [-1, +0, +0], |
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192 | [-1, +0, +1], |
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193 | [-1, +1, -1], |
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194 | [-1, +1, +0], |
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195 | [-1, +1, +1], |
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196 | [+0, -1, -1], |
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197 | [+0, -1, +0], |
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198 | [+0, -1, +1], |
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199 | [+0, +0, -1], |
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200 | [+0, +0, +0], |
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201 | [+0, +0, +1], |
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202 | [+0, +1, -1], |
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203 | [+0, +1, +0], |
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204 | [+0, +1, +1], |
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205 | [+1, -1, -1], |
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206 | [+1, -1, +0], |
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207 | [+1, -1, +1], |
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208 | [+1, +0, -1], |
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209 | [+1, +0, +0], |
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210 | [+1, +0, +1], |
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211 | [+1, +1, -1], |
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212 | [+1, +1, +0], |
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213 | [+1, +1, +1], |
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214 | ] |
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215 | dtheta, dphi, dpsi = jitter |
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216 | if dtheta == 0: |
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217 | cloud = [v for v in cloud if v[0] == 0] |
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218 | if dphi == 0: |
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219 | cloud = [v for v in cloud if v[1] == 0] |
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220 | if dpsi == 0: |
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221 | cloud = [v for v in cloud if v[2] == 0] |
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222 | draw_shape(axes, size, view, [0, 0, 0], steps=100, alpha=0.8) |
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223 | scale = {'gaussian':1, 'rectangle':1/sqrt(3), 'uniform':1/3}[dist] |
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224 | for point in cloud: |
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225 | delta = [scale*dtheta*point[0], scale*dphi*point[1], scale*dpsi*point[2]] |
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226 | draw_shape(axes, size, view, delta, alpha=0.8) |
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227 | for v in 'xyz': |
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228 | a, b, c = size |
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229 | lim = np.sqrt(a**2 + b**2 + c**2) |
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230 | getattr(axes, 'set_'+v+'lim')([-lim, lim]) |
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231 | getattr(axes, v+'axis').label.set_text(v) |
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232 | |
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233 | PROJECTIONS = [ |
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234 | # in order of PROJECTION number; do not change without updating the |
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235 | # constants in kernel_iq.c |
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236 | 'equirectangular', 'sinusoidal', 'guyou', 'azimuthal_equidistance', |
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237 | 'azimuthal_equal_area', |
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238 | ] |
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239 | def draw_mesh(axes, view, jitter, radius=1.2, n=11, dist='gaussian', |
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240 | projection='equirectangular'): |
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241 | """ |
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242 | Draw the dispersion mesh showing the theta-phi orientations at which |
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243 | the model will be evaluated. |
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244 | |
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245 | jitter projections |
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246 | <https://en.wikipedia.org/wiki/List_of_map_projections> |
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247 | |
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248 | equirectangular (standard latitude-longitude mesh) |
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249 | <https://en.wikipedia.org/wiki/Equirectangular_projection> |
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250 | Allows free movement in phi (around the equator), but theta is |
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251 | limited to +/- 90, and points are cos-weighted. Jitter in phi is |
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252 | uniform in weight along a line of latitude. With small theta and |
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253 | phi ranging over +/- 180 this forms a wobbling disk. With small |
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254 | phi and theta ranging over +/- 90 this forms a wedge like a slice |
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255 | of an orange. |
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256 | azimuthal_equidistance (Postel) |
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257 | <https://en.wikipedia.org/wiki/Azimuthal_equidistant_projection> |
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258 | Preserves distance from center, and so is an excellent map for |
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259 | representing a bivariate gaussian on the surface. Theta and phi |
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260 | operate identically, cutting wegdes from the antipode of the viewing |
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261 | angle. This unfortunately does not allow free movement in either |
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262 | theta or phi since the orthogonal wobble decreases to 0 as the body |
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263 | rotates through 180 degrees. |
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264 | sinusoidal (Sanson-Flamsteed, Mercator equal-area) |
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265 | <https://en.wikipedia.org/wiki/Sinusoidal_projection> |
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266 | Preserves arc length with latitude, giving bad behaviour at |
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267 | theta near +/- 90. Theta and phi operate somewhat differently, |
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268 | so a system with a-b-c dtheta-dphi-dpsi will not give the same |
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269 | value as one with b-a-c dphi-dtheta-dpsi, as would be the case |
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270 | for azimuthal equidistance. Free movement using theta or phi |
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271 | uniform over +/- 180 will work, but not as well as equirectangular |
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272 | phi, with theta being slightly worse. Computationally it is much |
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273 | cheaper for wide theta-phi meshes since it excludes points which |
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274 | lie outside the sinusoid near theta +/- 90 rather than packing |
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275 | them close together as in equirectangle. Note that the poles |
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276 | will be slightly overweighted for theta > 90 with the circle |
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277 | from theta at 90+dt winding backwards around the pole, overlapping |
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278 | the circle from theta at 90-dt. |
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279 | Guyou (hemisphere-in-a-square) **not weighted** |
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280 | <https://en.wikipedia.org/wiki/Guyou_hemisphere-in-a-square_projection> |
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281 | With tiling, allows rotation in phi or theta through +/- 180, with |
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282 | uniform spacing. Both theta and phi allow free rotation, with wobble |
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283 | in the orthogonal direction reasonably well behaved (though not as |
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284 | good as equirectangular phi). The forward/reverse transformations |
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285 | relies on elliptic integrals that are somewhat expensive, so the |
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286 | behaviour has to be very good to justify the cost and complexity. |
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287 | The weighting function for each point has not yet been computed. |
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288 | Note: run the module *guyou.py* directly and it will show the forward |
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289 | and reverse mappings. |
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290 | azimuthal_equal_area **incomplete** |
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291 | <https://en.wikipedia.org/wiki/Lambert_azimuthal_equal-area_projection> |
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292 | Preserves the relative density of the surface patches. Not that |
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293 | useful and not completely implemented |
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294 | Gauss-Kreuger **not implemented** |
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295 | <https://en.wikipedia.org/wiki/Transverse_Mercator_projection#Ellipsoidal_transverse_Mercator> |
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296 | Should allow free movement in theta, but phi is distorted. |
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297 | """ |
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298 | # TODO: try Kent distribution instead of a gaussian warped by projection |
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299 | |
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300 | dist_x = np.linspace(-1, 1, n) |
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301 | weights = np.ones_like(dist_x) |
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302 | if dist == 'gaussian': |
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303 | dist_x *= 3 |
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304 | weights = exp(-0.5*dist_x**2) |
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305 | elif dist == 'rectangle': |
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306 | # Note: uses sasmodels ridiculous definition of rectangle width |
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307 | dist_x *= sqrt(3) |
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308 | elif dist == 'uniform': |
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309 | pass |
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310 | else: |
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311 | raise ValueError("expected dist to be gaussian, rectangle or uniform") |
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312 | |
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313 | if projection == 'equirectangular': #define PROJECTION 1 |
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314 | def _rotate(theta_i, phi_j): |
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315 | return Rx(phi_j)*Ry(theta_i) |
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316 | def _weight(theta_i, phi_j, w_i, w_j): |
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317 | return w_i*w_j*abs(cos(radians(theta_i))) |
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318 | elif projection == 'sinusoidal': #define PROJECTION 2 |
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319 | def _rotate(theta_i, phi_j): |
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320 | latitude = theta_i |
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321 | scale = cos(radians(latitude)) |
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322 | longitude = phi_j/scale if abs(phi_j) < abs(scale)*180 else 0 |
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323 | #print("(%+7.2f, %+7.2f) => (%+7.2f, %+7.2f)"%(theta_i, phi_j, latitude, longitude)) |
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324 | return Rx(longitude)*Ry(latitude) |
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325 | def _weight(theta_i, phi_j, w_i, w_j): |
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326 | latitude = theta_i |
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327 | scale = cos(radians(latitude)) |
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328 | active = 1 if abs(phi_j) < abs(scale)*180 else 0 |
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329 | return active*w_i*w_j |
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330 | elif projection == 'guyou': #define PROJECTION 3 (eventually?) |
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331 | def _rotate(theta_i, phi_j): |
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332 | from .guyou import guyou_invert |
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333 | #latitude, longitude = guyou_invert([theta_i], [phi_j]) |
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334 | longitude, latitude = guyou_invert([phi_j], [theta_i]) |
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335 | return Rx(longitude[0])*Ry(latitude[0]) |
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336 | def _weight(theta_i, phi_j, w_i, w_j): |
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337 | return w_i*w_j |
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338 | elif projection == 'azimuthal_equidistance': # Note: Rz Ry, not Rx Ry |
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339 | def _rotate(theta_i, phi_j): |
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340 | latitude = sqrt(theta_i**2 + phi_j**2) |
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341 | longitude = degrees(np.arctan2(phi_j, theta_i)) |
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342 | #print("(%+7.2f, %+7.2f) => (%+7.2f, %+7.2f)"%(theta_i, phi_j, latitude, longitude)) |
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343 | return Rz(longitude)*Ry(latitude) |
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344 | def _weight(theta_i, phi_j, w_i, w_j): |
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345 | # Weighting for each point comes from the integral: |
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346 | # \int\int I(q, lat, log) sin(lat) dlat dlog |
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347 | # We are doing a conformal mapping from disk to sphere, so we need |
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348 | # a change of variables g(theta, phi) -> (lat, long): |
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349 | # lat, long = sqrt(theta^2 + phi^2), arctan(phi/theta) |
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350 | # giving: |
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351 | # dtheta dphi = det(J) dlat dlong |
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352 | # where J is the jacobian from the partials of g. Using |
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353 | # R = sqrt(theta^2 + phi^2), |
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354 | # then |
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355 | # J = [[x/R, Y/R], -y/R^2, x/R^2]] |
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356 | # and |
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357 | # det(J) = 1/R |
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358 | # with the final integral being: |
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359 | # \int\int I(q, theta, phi) sin(R)/R dtheta dphi |
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360 | # |
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361 | # This does approximately the right thing, decreasing the weight |
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362 | # of each point as you go farther out on the disk, but it hasn't |
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363 | # yet been checked against the 1D integral results. Prior |
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364 | # to declaring this "good enough" and checking that integrals |
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365 | # work in practice, we will examine alternative mappings. |
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366 | # |
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367 | # The issue is that the mapping does not support the case of free |
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368 | # rotation about a single axis correctly, with a small deviation |
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369 | # in the orthogonal axis independent of the first axis. Like the |
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370 | # usual polar coordiates integration, the integrated sections |
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371 | # form wedges, though at least in this case the wedge cuts through |
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372 | # the entire sphere, and treats theta and phi identically. |
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373 | latitude = sqrt(theta_i**2 + phi_j**2) |
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374 | weight = sin(radians(latitude))/latitude if latitude != 0 else 1 |
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375 | return weight*w_i*w_j if latitude < 180 else 0 |
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376 | elif projection == 'azimuthal_equal_area': |
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377 | def _rotate(theta_i, phi_j): |
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378 | radius = min(1, sqrt(theta_i**2 + phi_j**2)/180) |
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379 | latitude = 180-degrees(2*np.arccos(radius)) |
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380 | longitude = degrees(np.arctan2(phi_j, theta_i)) |
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381 | #print("(%+7.2f, %+7.2f) => (%+7.2f, %+7.2f)"%(theta_i, phi_j, latitude, longitude)) |
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382 | return Rz(longitude)*Ry(latitude) |
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383 | def _weight(theta_i, phi_j, w_i, w_j): |
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384 | latitude = sqrt(theta_i**2 + phi_j**2) |
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385 | weight = sin(radians(latitude))/latitude if latitude != 0 else 1 |
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386 | return weight*w_i*w_j if latitude < 180 else 0 |
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387 | else: |
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388 | raise ValueError("unknown projection %r"%projection) |
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389 | |
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390 | # mesh in theta, phi formed by rotating z |
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391 | dtheta, dphi, dpsi = jitter |
---|
392 | z = np.matrix([[0], [0], [radius]]) |
---|
393 | points = np.hstack([_rotate(theta_i, phi_j)*z |
---|
394 | for theta_i in dtheta*dist_x |
---|
395 | for phi_j in dphi*dist_x]) |
---|
396 | dist_w = np.array([_weight(theta_i, phi_j, w_i, w_j) |
---|
397 | for w_i, theta_i in zip(weights, dtheta*dist_x) |
---|
398 | for w_j, phi_j in zip(weights, dphi*dist_x)]) |
---|
399 | #print(max(dist_w), min(dist_w), min(dist_w[dist_w > 0])) |
---|
400 | points = points[:, dist_w > 0] |
---|
401 | dist_w = dist_w[dist_w > 0] |
---|
402 | dist_w /= max(dist_w) |
---|
403 | |
---|
404 | # rotate relative to beam |
---|
405 | points = orient_relative_to_beam(view, points) |
---|
406 | |
---|
407 | x, y, z = [np.array(v).flatten() for v in points] |
---|
408 | #plt.figure(2); plt.clf(); plt.hist(z, bins=np.linspace(-1, 1, 51)) |
---|
409 | axes.scatter(x, y, z, c=dist_w, marker='o', vmin=0., vmax=1.) |
---|
410 | |
---|
411 | def draw_labels(axes, view, jitter, text): |
---|
412 | """ |
---|
413 | Draw text at a particular location. |
---|
414 | """ |
---|
415 | labels, locations, orientations = zip(*text) |
---|
416 | px, py, pz = zip(*locations) |
---|
417 | dx, dy, dz = zip(*orientations) |
---|
418 | |
---|
419 | px, py, pz = transform_xyz(view, jitter, px, py, pz) |
---|
420 | dx, dy, dz = transform_xyz(view, jitter, dx, dy, dz) |
---|
421 | |
---|
422 | # TODO: zdir for labels is broken, and labels aren't appearing. |
---|
423 | for label, p, zdir in zip(labels, zip(px, py, pz), zip(dx, dy, dz)): |
---|
424 | zdir = np.asarray(zdir).flatten() |
---|
425 | axes.text(p[0], p[1], p[2], label, zdir=zdir) |
---|
426 | |
---|
427 | # Definition of rotation matrices comes from wikipedia: |
---|
428 | # https://en.wikipedia.org/wiki/Rotation_matrix#Basic_rotations |
---|
429 | def Rx(angle): |
---|
430 | """Construct a matrix to rotate points about *x* by *angle* degrees.""" |
---|
431 | angle = radians(angle) |
---|
432 | rot = [[1, 0, 0], |
---|
433 | [0, +cos(angle), -sin(angle)], |
---|
434 | [0, +sin(angle), +cos(angle)]] |
---|
435 | return np.matrix(rot) |
---|
436 | |
---|
437 | def Ry(angle): |
---|
438 | """Construct a matrix to rotate points about *y* by *angle* degrees.""" |
---|
439 | angle = radians(angle) |
---|
440 | rot = [[+cos(angle), 0, +sin(angle)], |
---|
441 | [0, 1, 0], |
---|
442 | [-sin(angle), 0, +cos(angle)]] |
---|
443 | return np.matrix(rot) |
---|
444 | |
---|
445 | def Rz(angle): |
---|
446 | """Construct a matrix to rotate points about *z* by *angle* degrees.""" |
---|
447 | angle = radians(angle) |
---|
448 | rot = [[+cos(angle), -sin(angle), 0], |
---|
449 | [+sin(angle), +cos(angle), 0], |
---|
450 | [0, 0, 1]] |
---|
451 | return np.matrix(rot) |
---|
452 | |
---|
453 | def transform_xyz(view, jitter, x, y, z): |
---|
454 | """ |
---|
455 | Send a set of (x,y,z) points through the jitter and view transforms. |
---|
456 | """ |
---|
457 | x, y, z = [np.asarray(v) for v in (x, y, z)] |
---|
458 | shape = x.shape |
---|
459 | points = np.matrix([x.flatten(), y.flatten(), z.flatten()]) |
---|
460 | points = apply_jitter(jitter, points) |
---|
461 | points = orient_relative_to_beam(view, points) |
---|
462 | x, y, z = [np.array(v).reshape(shape) for v in points] |
---|
463 | return x, y, z |
---|
464 | |
---|
465 | def apply_jitter(jitter, points): |
---|
466 | """ |
---|
467 | Apply the jitter transform to a set of points. |
---|
468 | |
---|
469 | Points are stored in a 3 x n numpy matrix, not a numpy array or tuple. |
---|
470 | """ |
---|
471 | dtheta, dphi, dpsi = jitter |
---|
472 | points = Rx(dphi)*Ry(dtheta)*Rz(dpsi)*points |
---|
473 | return points |
---|
474 | |
---|
475 | def orient_relative_to_beam(view, points): |
---|
476 | """ |
---|
477 | Apply the view transform to a set of points. |
---|
478 | |
---|
479 | Points are stored in a 3 x n numpy matrix, not a numpy array or tuple. |
---|
480 | """ |
---|
481 | theta, phi, psi = view |
---|
482 | points = Rz(phi)*Ry(theta)*Rz(psi)*points |
---|
483 | return points |
---|
484 | |
---|
485 | # translate between number of dimension of dispersity and the number of |
---|
486 | # points along each dimension. |
---|
487 | PD_N_TABLE = { |
---|
488 | (0, 0, 0): (0, 0, 0), # 0 |
---|
489 | (1, 0, 0): (100, 0, 0), # 100 |
---|
490 | (0, 1, 0): (0, 100, 0), |
---|
491 | (0, 0, 1): (0, 0, 100), |
---|
492 | (1, 1, 0): (30, 30, 0), # 900 |
---|
493 | (1, 0, 1): (30, 0, 30), |
---|
494 | (0, 1, 1): (0, 30, 30), |
---|
495 | (1, 1, 1): (15, 15, 15), # 3375 |
---|
496 | } |
---|
497 | |
---|
498 | def clipped_range(data, portion=1.0, mode='central'): |
---|
499 | """ |
---|
500 | Determine range from data. |
---|
501 | |
---|
502 | If *portion* is 1, use full range, otherwise use the center of the range |
---|
503 | or the top of the range, depending on whether *mode* is 'central' or 'top'. |
---|
504 | """ |
---|
505 | if portion == 1.0: |
---|
506 | return data.min(), data.max() |
---|
507 | elif mode == 'central': |
---|
508 | data = np.sort(data.flatten()) |
---|
509 | offset = int(portion*len(data)/2 + 0.5) |
---|
510 | return data[offset], data[-offset] |
---|
511 | elif mode == 'top': |
---|
512 | data = np.sort(data.flatten()) |
---|
513 | offset = int(portion*len(data) + 0.5) |
---|
514 | return data[offset], data[-1] |
---|
515 | |
---|
516 | def draw_scattering(calculator, axes, view, jitter, dist='gaussian'): |
---|
517 | """ |
---|
518 | Plot the scattering for the particular view. |
---|
519 | |
---|
520 | *calculator* is returned from :func:`build_model`. *axes* are the 3D axes |
---|
521 | on which the data will be plotted. *view* and *jitter* are the current |
---|
522 | orientation and orientation dispersity. *dist* is one of the sasmodels |
---|
523 | weight distributions. |
---|
524 | """ |
---|
525 | if dist == 'uniform': # uniform is not yet in this branch |
---|
526 | dist, scale = 'rectangle', 1/sqrt(3) |
---|
527 | else: |
---|
528 | scale = 1 |
---|
529 | |
---|
530 | # add the orientation parameters to the model parameters |
---|
531 | theta, phi, psi = view |
---|
532 | theta_pd, phi_pd, psi_pd = [scale*v for v in jitter] |
---|
533 | theta_pd_n, phi_pd_n, psi_pd_n = PD_N_TABLE[(theta_pd > 0, phi_pd > 0, psi_pd > 0)] |
---|
534 | ## increase pd_n for testing jitter integration rather than simple viz |
---|
535 | #theta_pd_n, phi_pd_n, psi_pd_n = [5*v for v in (theta_pd_n, phi_pd_n, psi_pd_n)] |
---|
536 | |
---|
537 | pars = dict( |
---|
538 | theta=theta, theta_pd=theta_pd, theta_pd_type=dist, theta_pd_n=theta_pd_n, |
---|
539 | phi=phi, phi_pd=phi_pd, phi_pd_type=dist, phi_pd_n=phi_pd_n, |
---|
540 | psi=psi, psi_pd=psi_pd, psi_pd_type=dist, psi_pd_n=psi_pd_n, |
---|
541 | ) |
---|
542 | pars.update(calculator.pars) |
---|
543 | |
---|
544 | # compute the pattern |
---|
545 | qx, qy = calculator._data.x_bins, calculator._data.y_bins |
---|
546 | Iqxy = calculator(**pars).reshape(len(qx), len(qy)) |
---|
547 | |
---|
548 | # scale it and draw it |
---|
549 | Iqxy = np.log(Iqxy) |
---|
550 | if calculator.limits: |
---|
551 | # use limits from orientation (0,0,0) |
---|
552 | vmin, vmax = calculator.limits |
---|
553 | else: |
---|
554 | vmax = Iqxy.max() |
---|
555 | vmin = vmax*10**-7 |
---|
556 | #vmin, vmax = clipped_range(Iqxy, portion=portion, mode='top') |
---|
557 | #print("range",(vmin,vmax)) |
---|
558 | #qx, qy = np.meshgrid(qx, qy) |
---|
559 | if 0: |
---|
560 | level = np.asarray(255*(Iqxy - vmin)/(vmax - vmin), 'i') |
---|
561 | level[level < 0] = 0 |
---|
562 | colors = plt.get_cmap()(level) |
---|
563 | axes.plot_surface(qx, qy, -1.1, rstride=1, cstride=1, facecolors=colors) |
---|
564 | elif 1: |
---|
565 | axes.contourf(qx/qx.max(), qy/qy.max(), Iqxy, zdir='z', offset=-1.1, |
---|
566 | levels=np.linspace(vmin, vmax, 24)) |
---|
567 | else: |
---|
568 | axes.pcolormesh(qx, qy, Iqxy) |
---|
569 | |
---|
570 | def build_model(model_name, n=150, qmax=0.5, **pars): |
---|
571 | """ |
---|
572 | Build a calculator for the given shape. |
---|
573 | |
---|
574 | *model_name* is any sasmodels model. *n* and *qmax* define an n x n mesh |
---|
575 | on which to evaluate the model. The remaining parameters are stored in |
---|
576 | the returned calculator as *calculator.pars*. They are used by |
---|
577 | :func:`draw_scattering` to set the non-orientation parameters in the |
---|
578 | calculation. |
---|
579 | |
---|
580 | Returns a *calculator* function which takes a dictionary or parameters and |
---|
581 | produces Iqxy. The Iqxy value needs to be reshaped to an n x n matrix |
---|
582 | for plotting. See the :class:`sasmodels.direct_model.DirectModel` class |
---|
583 | for details. |
---|
584 | """ |
---|
585 | from sasmodels.core import load_model_info, build_model as build_sasmodel |
---|
586 | from sasmodels.data import empty_data2D |
---|
587 | from sasmodels.direct_model import DirectModel |
---|
588 | |
---|
589 | model_info = load_model_info(model_name) |
---|
590 | model = build_sasmodel(model_info) #, dtype='double!') |
---|
591 | q = np.linspace(-qmax, qmax, n) |
---|
592 | data = empty_data2D(q, q) |
---|
593 | calculator = DirectModel(data, model) |
---|
594 | |
---|
595 | # stuff the values for non-orientation parameters into the calculator |
---|
596 | calculator.pars = pars.copy() |
---|
597 | calculator.pars.setdefault('backgound', 1e-3) |
---|
598 | |
---|
599 | # fix the data limits so that we can see if the pattern fades |
---|
600 | # under rotation or angular dispersion |
---|
601 | Iqxy = calculator(theta=0, phi=0, psi=0, **calculator.pars) |
---|
602 | Iqxy = np.log(Iqxy) |
---|
603 | vmin, vmax = clipped_range(Iqxy, 0.95, mode='top') |
---|
604 | calculator.limits = vmin, vmax+1 |
---|
605 | |
---|
606 | return calculator |
---|
607 | |
---|
608 | def select_calculator(model_name, n=150, size=(10, 40, 100)): |
---|
609 | """ |
---|
610 | Create a model calculator for the given shape. |
---|
611 | |
---|
612 | *model_name* is one of sphere, cylinder, ellipsoid, triaxial_ellipsoid, |
---|
613 | parallelepiped or bcc_paracrystal. *n* is the number of points to use |
---|
614 | in the q range. *qmax* is chosen based on model parameters for the |
---|
615 | given model to show something intersting. |
---|
616 | |
---|
617 | Returns *calculator* and tuple *size* (a,b,c) giving minor and major |
---|
618 | equitorial axes and polar axis respectively. See :func:`build_model` |
---|
619 | for details on the returned calculator. |
---|
620 | """ |
---|
621 | a, b, c = size |
---|
622 | d_factor = 0.06 # for paracrystal models |
---|
623 | if model_name == 'sphere': |
---|
624 | calculator = build_model('sphere', n=n, radius=c) |
---|
625 | a = b = c |
---|
626 | elif model_name == 'sc_paracrystal': |
---|
627 | a = b = c |
---|
628 | dnn = c |
---|
629 | radius = 0.5*c |
---|
630 | calculator = build_model('sc_paracrystal', n=n, dnn=dnn, |
---|
631 | d_factor=d_factor, radius=(1-d_factor)*radius, |
---|
632 | background=0) |
---|
633 | elif model_name == 'fcc_paracrystal': |
---|
634 | a = b = c |
---|
635 | # nearest neigbour distance dnn should be 2 radius, but I think the |
---|
636 | # model uses lattice spacing rather than dnn in its calculations |
---|
637 | dnn = 0.5*c |
---|
638 | radius = sqrt(2)/4 * c |
---|
639 | calculator = build_model('fcc_paracrystal', n=n, dnn=dnn, |
---|
640 | d_factor=d_factor, radius=(1-d_factor)*radius, |
---|
641 | background=0) |
---|
642 | elif model_name == 'bcc_paracrystal': |
---|
643 | a = b = c |
---|
644 | # nearest neigbour distance dnn should be 2 radius, but I think the |
---|
645 | # model uses lattice spacing rather than dnn in its calculations |
---|
646 | dnn = 0.5*c |
---|
647 | radius = sqrt(3)/2 * c |
---|
648 | calculator = build_model('bcc_paracrystal', n=n, dnn=dnn, |
---|
649 | d_factor=d_factor, radius=(1-d_factor)*radius, |
---|
650 | background=0) |
---|
651 | elif model_name == 'cylinder': |
---|
652 | calculator = build_model('cylinder', n=n, qmax=0.3, radius=b, length=c) |
---|
653 | a = b |
---|
654 | elif model_name == 'ellipsoid': |
---|
655 | calculator = build_model('ellipsoid', n=n, qmax=1.0, |
---|
656 | radius_polar=c, radius_equatorial=b) |
---|
657 | a = b |
---|
658 | elif model_name == 'triaxial_ellipsoid': |
---|
659 | calculator = build_model('triaxial_ellipsoid', n=n, qmax=0.5, |
---|
660 | radius_equat_minor=a, |
---|
661 | radius_equat_major=b, |
---|
662 | radius_polar=c) |
---|
663 | elif model_name == 'parallelepiped': |
---|
664 | calculator = build_model('parallelepiped', n=n, a=a, b=b, c=c) |
---|
665 | else: |
---|
666 | raise ValueError("unknown model %s"%model_name) |
---|
667 | |
---|
668 | return calculator, (a, b, c) |
---|
669 | |
---|
670 | SHAPES = [ |
---|
671 | 'parallelepiped', |
---|
672 | 'sphere', 'ellipsoid', 'triaxial_ellipsoid', |
---|
673 | 'cylinder', |
---|
674 | 'fcc_paracrystal', 'bcc_paracrystal', 'sc_paracrystal', |
---|
675 | ] |
---|
676 | |
---|
677 | DRAW_SHAPES = { |
---|
678 | 'fcc_paracrystal': draw_fcc, |
---|
679 | 'bcc_paracrystal': draw_bcc, |
---|
680 | 'sc_paracrystal': draw_sc, |
---|
681 | 'parallelepiped': draw_parallelepiped, |
---|
682 | } |
---|
683 | |
---|
684 | DISTRIBUTIONS = [ |
---|
685 | 'gaussian', 'rectangle', 'uniform', |
---|
686 | ] |
---|
687 | DIST_LIMITS = { |
---|
688 | 'gaussian': 30, |
---|
689 | 'rectangle': 90/sqrt(3), |
---|
690 | 'uniform': 90, |
---|
691 | } |
---|
692 | |
---|
693 | def run(model_name='parallelepiped', size=(10, 40, 100), |
---|
694 | dist='gaussian', mesh=30, |
---|
695 | projection='equirectangular'): |
---|
696 | """ |
---|
697 | Show an interactive orientation and jitter demo. |
---|
698 | |
---|
699 | *model_name* is one of: sphere, ellipsoid, triaxial_ellipsoid, |
---|
700 | parallelepiped, cylinder, or sc/fcc/bcc_paracrystal |
---|
701 | |
---|
702 | *size* gives the dimensions (a, b, c) of the shape. |
---|
703 | |
---|
704 | *dist* is the type of dispersition: gaussian, rectangle, or uniform. |
---|
705 | |
---|
706 | *mesh* is the number of points in the dispersion mesh. |
---|
707 | |
---|
708 | *projection* is the map projection to use for the mesh: equirectangular, |
---|
709 | sinusoidal, guyou, azimuthal_equidistance, or azimuthal_equal_area. |
---|
710 | """ |
---|
711 | # projection number according to 1-order position in list, but |
---|
712 | # only 1 and 2 are implemented so far. |
---|
713 | from sasmodels import generate |
---|
714 | generate.PROJECTION = PROJECTIONS.index(projection) + 1 |
---|
715 | if generate.PROJECTION > 2: |
---|
716 | print("*** PROJECTION %s not implemented in scattering function ***"%projection) |
---|
717 | generate.PROJECTION = 2 |
---|
718 | |
---|
719 | # set up calculator |
---|
720 | calculator, size = select_calculator(model_name, n=150, size=size) |
---|
721 | draw_shape = DRAW_SHAPES.get(model_name, draw_parallelepiped) |
---|
722 | |
---|
723 | ## uncomment to set an independent the colour range for every view |
---|
724 | ## If left commented, the colour range is fixed for all views |
---|
725 | calculator.limits = None |
---|
726 | |
---|
727 | ## initial view |
---|
728 | #theta, dtheta = 70., 10. |
---|
729 | #phi, dphi = -45., 3. |
---|
730 | #psi, dpsi = -45., 3. |
---|
731 | theta, phi, psi = 0, 0, 0 |
---|
732 | dtheta, dphi, dpsi = 0, 0, 0 |
---|
733 | |
---|
734 | ## create the plot window |
---|
735 | #plt.hold(True) |
---|
736 | plt.subplots(num=None, figsize=(5.5, 5.5)) |
---|
737 | plt.set_cmap('gist_earth') |
---|
738 | plt.clf() |
---|
739 | plt.gcf().canvas.set_window_title(projection) |
---|
740 | #gs = gridspec.GridSpec(2,1,height_ratios=[4,1]) |
---|
741 | #axes = plt.subplot(gs[0], projection='3d') |
---|
742 | axes = plt.axes([0.0, 0.2, 1.0, 0.8], projection='3d') |
---|
743 | try: # CRUFT: not all versions of matplotlib accept 'square' 3d projection |
---|
744 | axes.axis('square') |
---|
745 | except Exception: |
---|
746 | pass |
---|
747 | |
---|
748 | axcolor = 'lightgoldenrodyellow' |
---|
749 | |
---|
750 | ## add control widgets to plot |
---|
751 | axes_theta = plt.axes([0.1, 0.15, 0.45, 0.04], axisbg=axcolor) |
---|
752 | axes_phi = plt.axes([0.1, 0.1, 0.45, 0.04], axisbg=axcolor) |
---|
753 | axes_psi = plt.axes([0.1, 0.05, 0.45, 0.04], axisbg=axcolor) |
---|
754 | stheta = Slider(axes_theta, 'Theta', -90, 90, valinit=theta) |
---|
755 | sphi = Slider(axes_phi, 'Phi', -180, 180, valinit=phi) |
---|
756 | spsi = Slider(axes_psi, 'Psi', -180, 180, valinit=psi) |
---|
757 | |
---|
758 | axes_dtheta = plt.axes([0.75, 0.15, 0.15, 0.04], axisbg=axcolor) |
---|
759 | axes_dphi = plt.axes([0.75, 0.1, 0.15, 0.04], axisbg=axcolor) |
---|
760 | axes_dpsi = plt.axes([0.75, 0.05, 0.15, 0.04], axisbg=axcolor) |
---|
761 | # Note: using ridiculous definition of rectangle distribution, whose width |
---|
762 | # in sasmodels is sqrt(3) times the given width. Divide by sqrt(3) to keep |
---|
763 | # the maximum width to 90. |
---|
764 | dlimit = DIST_LIMITS[dist] |
---|
765 | sdtheta = Slider(axes_dtheta, 'dTheta', 0, 2*dlimit, valinit=dtheta) |
---|
766 | sdphi = Slider(axes_dphi, 'dPhi', 0, 2*dlimit, valinit=dphi) |
---|
767 | sdpsi = Slider(axes_dpsi, 'dPsi', 0, 2*dlimit, valinit=dpsi) |
---|
768 | |
---|
769 | |
---|
770 | ## callback to draw the new view |
---|
771 | def update(val, axis=None): |
---|
772 | view = stheta.val, sphi.val, spsi.val |
---|
773 | jitter = sdtheta.val, sdphi.val, sdpsi.val |
---|
774 | # set small jitter as 0 if multiple pd dims |
---|
775 | dims = sum(v > 0 for v in jitter) |
---|
776 | limit = [0, 0.5, 5][dims] |
---|
777 | jitter = [0 if v < limit else v for v in jitter] |
---|
778 | axes.cla() |
---|
779 | draw_beam(axes, (0, 0)) |
---|
780 | draw_jitter(axes, view, jitter, dist=dist, size=size, draw_shape=draw_shape) |
---|
781 | #draw_jitter(axes, view, (0,0,0)) |
---|
782 | draw_mesh(axes, view, jitter, dist=dist, n=mesh, projection=projection) |
---|
783 | draw_scattering(calculator, axes, view, jitter, dist=dist) |
---|
784 | plt.gcf().canvas.draw() |
---|
785 | |
---|
786 | ## bind control widgets to view updater |
---|
787 | stheta.on_changed(lambda v: update(v, 'theta')) |
---|
788 | sphi.on_changed(lambda v: update(v, 'phi')) |
---|
789 | spsi.on_changed(lambda v: update(v, 'psi')) |
---|
790 | sdtheta.on_changed(lambda v: update(v, 'dtheta')) |
---|
791 | sdphi.on_changed(lambda v: update(v, 'dphi')) |
---|
792 | sdpsi.on_changed(lambda v: update(v, 'dpsi')) |
---|
793 | |
---|
794 | ## initialize view |
---|
795 | update(None, 'phi') |
---|
796 | |
---|
797 | ## go interactive |
---|
798 | plt.show() |
---|
799 | |
---|
800 | def main(): |
---|
801 | parser = argparse.ArgumentParser( |
---|
802 | description="Display jitter", |
---|
803 | formatter_class=argparse.ArgumentDefaultsHelpFormatter, |
---|
804 | ) |
---|
805 | parser.add_argument('-p', '--projection', choices=PROJECTIONS, |
---|
806 | default=PROJECTIONS[0], |
---|
807 | help='coordinate projection') |
---|
808 | parser.add_argument('-s', '--size', type=str, default='10,40,100', |
---|
809 | help='a,b,c lengths') |
---|
810 | parser.add_argument('-d', '--distribution', choices=DISTRIBUTIONS, |
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811 | default=DISTRIBUTIONS[0], |
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812 | help='jitter distribution') |
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813 | parser.add_argument('-m', '--mesh', type=int, default=30, |
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814 | help='#points in theta-phi mesh') |
---|
815 | parser.add_argument('shape', choices=SHAPES, nargs='?', default=SHAPES[0], |
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816 | help='oriented shape') |
---|
817 | opts = parser.parse_args() |
---|
818 | size = tuple(int(v) for v in opts.size.split(',')) |
---|
819 | run(opts.shape, size=size, |
---|
820 | mesh=opts.mesh, dist=opts.distribution, |
---|
821 | projection=opts.projection) |
---|
822 | |
---|
823 | if __name__ == "__main__": |
---|
824 | main() |
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