# Changeset 06ee63c in sasmodels

Ignore:
Timestamp:
Dec 20, 2017 2:41:35 PM (3 years ago)
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

### Legend:

Unmodified
 r7e6bc45e and $\Delta\phi$ are translated into latitude/longitude on the sphere. Sasmodels is using the *equirectangular* projection. In this projection, square patches in angular dispersity become wedge-shaped patches on the sphere. To correct for the changing point density, there is a scale factor of $\sin(\Delta\theta)$ that applies to each point in the integral. This is not enough, though. Consider a shape which is tumbling freely around the $b$ axis, with $\Delta\theta$ uniform in $[-180, 180]$. At $\pm 90$, all points in $\Delta\phi$ map to the pole, so the jitter will have a distinct angular preference. If the spin axis is normal to the beam (which will be the case for $\theta=90$ and $\Psi=90$), the scattering pattern should be circularly symmetric, but it will go to zero at $\pm 90$ due to the $\sin(\Delta\theta)$ correction. This problem does not appear for a shape Sasmodels is using the equirectangular projection _. In this projection, square patches in angular dispersity become wedge-shaped patches on the sphere. To correct for the changing point density, there is a scale factor of $\sin(\Delta\theta)$ that applies to each point in the integral. This is not enough, though. Consider a shape which is tumbling freely around the $b$ axis, with $\Delta\theta$ uniform in $[-180, 180]$. At $\pm 90$, all points in $\Delta\phi$ map to the pole, so the jitter will have a distinct angular preference. If the spin axis is normal to the beam (which will be the case for $\theta=90$ and $\Psi=90$), the scattering pattern should be circularly symmetric, but it will go to zero at $\pm 90$ due to the $\sin(\Delta\theta)$ correction. This problem does not appear for a shape that is tumbling freely around the $a$ axis, with $\Delta\phi$ uniform in $[-180, 180]$, so swap the $a$ and $b$ axes so $\Delta\theta < \Delta\phi$ and adjust $\Psi$ by 90. This works with the existing sasmodels shapes due to symmetry. and adjust $\Psi$ by 90. This works with the existing sasmodels shapes due to symmetry. There are alternative projections. The *sinusoidal* projection works by scaling $\Delta\phi$ as $\Delta\theta$ increases, and dropping those points outside $[-180, 180]$. The distortions are a little less for middle ranges of $\Delta\theta$, but they are still severe for large $\Delta\theta$ and the model is much harder to explain. The *Guyou* projection has an excellent balance with reasonable distortion in both $\Delta\theta$ and $\Delta\phi$, as well as preserving small patches. However, it is considerably more expensive to implement, and we have not yet computed the distortion correction, measuring the degree of stretch at the point $(\Delta\theta, \Delta\phi)$ in the correction. There are alternative projections. The sinusoidal projection _ works by scaling $\Delta\phi$ as $\Delta\theta$ increases, and dropping those points outside $[-180, 180]$. The distortions are a little less for middle ranges of $\Delta\theta$, but they are still severe for large $\Delta\theta$ and the model is much harder to explain. The Guyou projection _ has an excellent balance with reasonable distortion in both $\Delta\theta$ and $\Delta\phi$, as well as preserving small patches. However, it is considerably more expensive to implement, and we have not yet computed the distortion correction, measuring the degree of stretch at the point $(\Delta\theta, \Delta\phi)$ in the correction. .. note::