Changeset 06ee63c in sasmodels
 Timestamp:
 Dec 20, 2017 2:41:35 PM (3 years ago)
 Branches:
 master, core_shell_microgels, magnetic_model, ticket1257vesicleproduct, ticket_1156, ticket_1265_superball, ticket_822_more_unit_tests
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 8cfb486
 Parents:
 7e6bc45e
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 1 edited
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doc/guide/orientation/orientation.rst
r7e6bc45e r06ee63c 111 111 and $\Delta\phi$ are translated into latitude/longitude on the sphere. 112 112 113 Sasmodels is using the *equirectangular* projection. In this projection, 114 square patches in angular dispersity become wedgeshaped patches on the 115 sphere. To correct for the changing point density, there is a scale factor of 116 $\sin(\Delta\theta)$ that applies to each point in the integral. This is not 117 enough, though. Consider a shape which is tumbling freely around the $b$ 118 axis, with $\Delta\theta$ uniform in $[180, 180]$. At $\pm 90$, all points 119 in $\Delta\phi$ map to the pole, so the jitter will have a 120 distinct angular preference. If the spin axis is normal to the beam 121 (which will be the case for $\theta=90$ and $\Psi=90$), the scattering 122 pattern should be circularly symmetric, but it will go to zero at $\pm 90$ due 123 to the $\sin(\Delta\theta)$ correction. This problem does not appear for a shape 113 Sasmodels is using the 114 `equirectangular projection <https://en.wikipedia.org/wiki/Equirectangular_projection>`_. 115 In this projection, square patches in angular dispersity become wedgeshaped 116 patches on the sphere. To correct for the changing point density, there is a 117 scale factor of $\sin(\Delta\theta)$ that applies to each point in the 118 integral. This is not enough, though. Consider a shape which is tumbling 119 freely around the $b$ axis, with $\Delta\theta$ uniform in $[180, 180]$. At 120 $\pm 90$, all points in $\Delta\phi$ map to the pole, so the jitter will have 121 a distinct angular preference. If the spin axis is normal to the beam (which 122 will be the case for $\theta=90$ and $\Psi=90$), the scattering pattern 123 should be circularly symmetric, but it will go to zero at $\pm 90$ due to the 124 $\sin(\Delta\theta)$ correction. This problem does not appear for a shape 124 125 that is tumbling freely around the $a$ axis, with $\Delta\phi$ uniform in 125 126 $[180, 180]$, so swap the $a$ and $b$ axes so $\Delta\theta < \Delta\phi$ 126 and adjust $\Psi$ by 90. This works with the existing sasmodels shapes 127 due tosymmetry.127 and adjust $\Psi$ by 90. This works with the existing sasmodels shapes due to 128 symmetry. 128 129 129 There are alternative projections. The *sinusoidal* projection works by 130 scaling $\Delta\phi$ as $\Delta\theta$ increases, and dropping those points 131 outside $[180, 180]$. The distortions are a little less for middle ranges of 132 $\Delta\theta$, but they are still severe for large $\Delta\theta$ and the 133 model is much harder to explain. The *Guyou* projection has an excellent 134 balance with reasonable distortion in both $\Delta\theta$ and $\Delta\phi$, 135 as well as preserving small patches. However, it is considerably more 136 expensive to implement, and we have not yet computed the distortion 137 correction, measuring the degree of stretch at the 138 point $(\Delta\theta, \Delta\phi)$ in the correction. 130 There are alternative projections. The 131 `sinusoidal projection <https://en.wikipedia.org/wiki/Sinusoidal_projection>`_ 132 works by scaling $\Delta\phi$ as $\Delta\theta$ increases, and dropping those 133 points outside $[180, 180]$. The distortions are a little less for middle 134 ranges of $\Delta\theta$, but they are still severe for large $\Delta\theta$ 135 and the model is much harder to explain. The 136 `Guyou projection <https://en.wikipedia.org/wiki/Guyou_hemisphereinasquare_projection>`_ 137 has an excellent balance with reasonable distortion in both $\Delta\theta$ 138 and $\Delta\phi$, as well as preserving small patches. However, it is 139 considerably more expensive to implement, and we have not yet computed the 140 distortion correction, measuring the degree of stretch at the point 141 $(\Delta\theta, \Delta\phi)$ in the correction. 139 142 140 143 .. note::
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