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@@ -1 +1,52 @@ Finish libtellurian_vincenty_inverse__ +Document the details of NaN returns in libtellurian_distance.3 +Document what happens to the azimuths when the starting point is a pole. + +libtellurian_coarse_distance can be simplified to (d/2R)² +which would be an good alternative for simply sorting distances, +and it would still be easy to afterwards convert the values for +a selection of them into approximate distances. + +Add inverse function of libtellurian_effective_gravity_radians + +Add inverse function of libtellurian_elevated_gravity_radians + +And functions for converting between coordinate systems + Geodetic (h, φ, λ) : Angle between normal and equatorial plane + Inverse of Cartesian (see Cartesian for definition of N) + Iteration is required unless φ or h is known + λ = atan2(Y, X) + h = p / cos φ - N + φ = tan⁻¹(Zp⁻¹/(1 - e²N/(N + h))) + where p = √(X² + Y²) + https://en.wikipedia.org/wiki/Geographic_coordinate_conversion#From_ECEF_to_geodetic_coordinates + When h=0 + λ = atan2(Y, X) + φ = tan⁻¹(Zp⁻¹/(1 - e²)) + where p = √(X² + Y²) + wouldn't it easier to convert to geocentric intermittently + Geocentric ϕ : Angle from centre of earth + ϕ = tan⁻¹((1 - e²) tan φ) + Geometric : Arc length = geodetic + Parametric/reduced β : spherical angle resulting in same distance from polar axis as + β = tan⁻¹((1 - f) tan φ) + Ellipsoidal-harmonic + TODO + Rectifying μ + μ = πm(φ) / 2m(½π), where m(u) = a(1 - e²) ∫{0→u} √(1 - e² sin² v)⁻³ dv + Authalic ξ + ξ = sin⁻¹(q(φ) / q(½π)), where q(u) = ((1 - e²) sin u)/(1 - e² sin u) + (1 - e²) e⁻¹ tanh⁻¹ (e sin u) + q(½π) = 1 + (1 - e²) e⁻¹ tanh⁻¹ e + Conformal χ + χ = tan⁻¹ (sinh [sinh⁻¹ tan φ - e tanh⁻¹ (e sin φ)]) + Isometric ψ + ψ = sinh⁻¹ tan φ - e than⁻¹ (e sin φ) + Astronomical Φ : angle between equatorial plane and the true vertical direction + the true vertical direction, is the direction of gravity which is affect + byu the centrifugal acceleration in addition to the gravitational acceleration. + Cartesian (geocentric Cartesian) + X = (N + h) cos φ cos λ + Y = (N + h) cos φ sin λ + Z = (Nb²/a² + h) sin φ + where + N = a²/√(a² cos² φ + b² sin² φ) |