# [Proj] Exact transverse Mercator code

strebe strebe at aol.com
Tue Feb 3 13:17:01 EST 2009

```Charles & the interested:

Daylight casts unflattering light on blemishes. This line is not cogent:

>X⁻¹ inverts X, but acts on (ψ, 0.0), ψ being the complex "latitude"

Amending just that line in the complete description:

Define
Project to complex plane, ψ = X + iΥ
Wallis I TM, complex ζ = π/2 - X⁻¹(ψ)
Wallis II TM, complex ξ = η⁻¹(ζ)
Gauß-Krüger, complex F(ξ)

where
X(φ, λ), Υ(φ, λ) is any conformal mapping to the plane; e.g.,
X = r cos λ
Υ = r sin λ
r = [(1-sin φ)/cos φ][(1+e sin φ)/(1-e sin φ)]^(e/2).
X⁻¹ finds μ such that ψ = X(μ, 0), μ being a complex "latitude"
η⁻¹ is inverse parametric latitude, acting on a complex "latitude"
η⁻¹(p) = tan⁻¹ [tan(p)/√(1-e²)]
F is the elliptic integral of the second kind with parameter e.
e is eccentricity

Regards,
— daan Strebe

On Feb 3, 2009, at 2:54:30 AM, strebe <strebe at aol.com> wrote:
Wallis happens to project to the ellipsoidal stereographic to arrive onto the plane from the ellipsoid. The specific projection to the plane is not important, other than the simpler the better for practical purposes. The point is to arrive at Wallis I, where parallels are evenly spaced on the complex plane.

Define
Project to complex plane, ψ = X + iΥ
Wallis I TM, complex ζ = π/2 - X⁻¹ (ψ)
Wallis II TM, complex ξ = η⁻¹ (ζ)
Gauß-Krüger, complex F (ξ)

where
X (φ, λ), Υ (φ, λ) is any conformal mapping to the plane; e.g.,
X = r cos λ
Υ = r sin λ
r = [(1-sin φ)/cos φ][(1+e sin φ)/(1-e sin φ)]^(e/2).
X⁻¹ inverts X, but acts on (ψ, 0.0), ψ being the complex "latitude"
η⁻¹ is inverse parametric latitude, acting on a complex "latitude"
η⁻¹ (p) = tan⁻¹ [tan (p)/√(1-e²)]
F is the elliptic integral of the second kind with parameter e.
e is eccentricity

As usual, the devil is in the details.

Regards,
— daan Strebe
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