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The convergence angle γ at a point on the projection is defined by the angle measured from the projected meridian, which defines true north, to a grid line of constant x, defining grid north. Therefore, γ is positive in the quadrant north of the equator and east of the central meridian and also in the quadrant south of the equator and west of ...
In each zone the scale factor of the central meridian reduces the diameter of the transverse cylinder to produce a secant projection with two standard lines, or lines of true scale, about 180 km on each side of, and about parallel to, the central meridian (Arc cos 0.9996 = 1.62° at the Equator). The scale is less than 1 inside the standard ...
The analysis of small terms depends on the latitude and longitude ranges in the various grids. There are also slight differences in the formulae utilised for meridian distance: one extra term is sometimes added to the formula specified above but such a term is less than 0.1mm.
until convergence. A suitable starting guess is given by φ 0 = μ where = is the rectifying latitude. Note that it there is no need to differentiate the series for m(φ), since the formula for the meridian radius of curvature M(φ) can be used instead.
Spherical coordinates (r, θ, φ) as commonly used: (ISO 80000-2:2019): radial distance r (slant distance to origin), polar angle θ (angle with respect to positive polar axis), and azimuthal angle φ (angle of rotation from the initial meridian plane). This is the convention followed in this article.
If one standard parallel is used (i.e. =), the formula for n above is indeterminate, but then = (). [11] Formulae for ellipsoidal datums are more involved. ...
Image of a loxodrome, or rhumb line, spiraling towards the North Pole. In navigation, a rhumb line, rhumb (/ r ʌ m /), or loxodrome is an arc crossing all meridians of longitude at the same angle, that is, a path with constant azimuth (bearing as measured relative to true north).
Vincenty suggested a method of accelerating the convergence in such cases (Rapp, 1993). An example of a failure of the inverse method to converge is (Φ 1, L 1) = (0°, 0°) and (Φ 2, L 2) = (0.5°, 179.7°) for the WGS84 ellipsoid. In an unpublished report, Vincenty (1975b) gave an alternative iterative scheme to handle such cases.