Changeset 06ee63c in sasmodels


Ignore:
Timestamp:
Dec 20, 2017 2:41:35 PM (3 years ago)
Author:
Paul Kienzle <pkienzle@…>
Branches:
master, core_shell_microgels, magnetic_model, ticket-1257-vesicle-product, ticket_1156, ticket_1265_superball, ticket_822_more_unit_tests
Children:
8cfb486
Parents:
7e6bc45e
Message:

update orientation docs

File:
1 edited

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  • doc/guide/orientation/orientation.rst

    r7e6bc45e r06ee63c  
    111111and $\Delta\phi$ are translated into latitude/longitude on the sphere. 
    112112 
    113 Sasmodels is using the *equirectangular* projection. In this projection, 
    114 square patches in angular dispersity become wedge-shaped 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 
     113Sasmodels is using the 
     114`equirectangular projection <https://en.wikipedia.org/wiki/Equirectangular_projection>`_. 
     115In this projection, square patches in angular dispersity become wedge-shaped 
     116patches on the sphere. To correct for the changing point density, there is a 
     117scale factor of $\sin(\Delta\theta)$ that applies to each point in the 
     118integral. This is not enough, though. Consider a shape which is tumbling 
     119freely 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 
     121a distinct angular preference. If the spin axis is normal to the beam (which 
     122will be the case for $\theta=90$ and $\Psi=90$), the scattering pattern 
     123should 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 
    124125that is tumbling freely around the $a$ axis, with $\Delta\phi$ uniform in 
    125126$[-180, 180]$, 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 to symmetry. 
     127and adjust $\Psi$ by 90. This works with the existing sasmodels shapes due to 
     128symmetry. 
    128129 
    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. 
     130There are alternative projections. The 
     131`sinusoidal projection <https://en.wikipedia.org/wiki/Sinusoidal_projection>`_ 
     132works by scaling $\Delta\phi$ as $\Delta\theta$ increases, and dropping those 
     133points outside $[-180, 180]$. The distortions are a little less for middle 
     134ranges of $\Delta\theta$, but they are still severe for large $\Delta\theta$ 
     135and the model is much harder to explain. The 
     136`Guyou projection <https://en.wikipedia.org/wiki/Guyou_hemisphere-in-a-square_projection>`_ 
     137has an excellent balance with reasonable distortion in both $\Delta\theta$ 
     138and $\Delta\phi$, as well as preserving small patches. However, it is 
     139considerably more expensive to implement, and we have not yet computed the 
     140distortion correction, measuring the degree of stretch at the point 
     141$(\Delta\theta, \Delta\phi)$ in the correction. 
    139142 
    140143.. note:: 
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