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Latitude (i.e., the angle of latitude) may be either geocentric latitude, measured (rotated) from the Earth's center—and designated variously by ψ, q, φ′, φ c, φ g —or geodetic latitude, measured (rotated) from the observer's local vertical, and typically designated φ.
Consider a point on the globe of radius R with longitude λ and latitude φ. If φ is increased by an infinitesimal amount, dφ, the point moves R dφ along a meridian of the globe of radius R, so the corresponding change in y, dy, must be hR dφ = R sec φ dφ. Therefore y′(φ) = R sec φ.
The latitude ϕ of a point on Earth's surface is the angle between the equatorial plane and the straight line that passes through that point and through (or close to) the center of the Earth. [note 2] Lines joining points of the same latitude trace circles on the surface of Earth called parallels, as they are parallel to the Equator and to each ...
The geocentric latitude θ is the complement of the polar angle or colatitude θ′ in conventional spherical polar coordinates in which the coordinates of a point are P(r,θ′,λ) where r is the distance of P from the centre O, θ′ is the angle between the radius vector and the polar axis and λ is longitude.
Earth radius (denoted as R 🜨 or R E) is the distance from the center of Earth to a point on or near its surface. Approximating the figure of Earth by an Earth spheroid (an oblate ellipsoid), the radius ranges from a maximum (equatorial radius, denoted a) of nearly 6,378 km (3,963 mi) to a minimum (polar radius, denoted b) of nearly 6,357 km (3,950 mi).
As with latitude and longitude, the values are bounded by ±90° and ±180° respectively. Positive latitudes are north of the equator, negative latitudes are south of the equator. Positive longitudes are east of the Prime Meridian; negative longitudes are west of the Prime Meridian. Latitude and longitude are usually expressed in that sequence ...
The calculation is essentially the conversion of the equatorial polar coordinates of Mecca (i.e. its longitude and latitude) to its polar coordinates (i.e. its qibla and distance) relative to a system whose reference meridian is the great circle through the given location and the Earth's poles and whose polar axis is the line through the ...
This table shows the effect of each bit. At each stage, the relevant half of the range is highlighted in green; a low bit selects the lower range, a high bit selects the upper range. The column "mean value" shows the latitude, simply the mean value of the range. Each subsequent bit makes this value more precise.