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From "Martin Desruisseaux (JIRA)" <>
Subject [jira] [Created] (SIS-451) Verify map projection derivative by comparaison with Snyder terms
Date Tue, 30 Apr 2019 12:57:00 GMT
Martin Desruisseaux created SIS-451:

             Summary: Verify map projection derivative by comparaison with Snyder terms
                 Key: SIS-451
             Project: Spatial Information Systems
          Issue Type: Test
          Components: Referencing
            Reporter: Martin Desruisseaux

Map projections in Apache SIS can provide derivatives as Jacobian matrices. However those
formulas do not appear in the John Parr Snyder's book _Map Projections: A Working Manual_
(1987); we had to derive them ourselves, with the risk of errors. However Snyder provides
the following terms in addition of _x_ and _y_ projection results:

* _h_: scale factor along meridian.
* _k_: scale factor along parallel.
* _ω_: maximal angular deformation.

Those terms are related to Jacobian matrix terms on sphere as below (Snyder 4-10 and -11):

* _h_ = hypot(∂x/∂φ, ∂y/∂φ)
* _k_ = hypot(∂x/∂λ, ∂y/∂λ)

Ellipsoidal case (Snyder 4-27 and -28):

* _h_ = hypot(∂x/∂φ, ∂y/∂φ) ⋅ (1 - ℯ²⋅sin²φ)^1.5 / (a⋅(1 - ℯ²))
* _k_ = hypot(∂x/∂λ, ∂y/∂λ) ⋅ (1 - ℯ²⋅sin²φ)^0.5 / (a⋅cos φ)

Formulas for _h_ and _k_ are provided for most map projections. We should implement them,
which would allow us to compare with Jacobian matrix terms at least on a row-by-row basis.

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