Details
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Test
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Status: Open
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Major
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Resolution: Unresolved
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Description
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.