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  2. Haversine formula - Wikipedia

    en.wikipedia.org/wiki/Haversine_formula

    In that case, a and b are ⁠ π / 2 ⁠ − φ 1,2 (that is, the, co-latitudes), C is the longitude separation λ 2 − λ 1, and c is the desired ⁠ d / R ⁠. Noting that sin(⁠ π / 2 ⁠ − φ) = cos(φ), the haversine formula immediately follows. To derive the law of haversines, one starts with the spherical law of cosines:

  3. List of common coordinate transformations - Wikipedia

    en.wikipedia.org/wiki/List_of_common_coordinate...

    Let (x, y, z) be the standard Cartesian coordinates, and (ρ, θ, φ) the spherical coordinates, with θ the angle measured away from the +Z axis (as , see conventions in spherical coordinates). As φ has a range of 360° the same considerations as in polar (2 dimensional) coordinates apply whenever an arctangent of it is taken. θ has a range ...

  4. Polar coordinate system - Wikipedia

    en.wikipedia.org/wiki/Polar_coordinate_system

    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 ...

  5. Geographic coordinate conversion - Wikipedia

    en.wikipedia.org/wiki/Geographic_coordinate...

    Coordinate conversion is composed of a number of different types of conversion: format change of geographic coordinates, conversion of coordinate systems, or transformation to different geodetic datums.

  6. Geographical distance - Wikipedia

    en.wikipedia.org/wiki/Geographical_distance

    In the given forms of the formulae below, one or more values must be expressed in the specified units to obtain the correct result. Where geographic coordinates are used as the argument of a trigonometric function, the values may be expressed in any angular units compatible with the method used to determine the value of the trigonometric function.

  7. Astronomical coordinate systems - Wikipedia

    en.wikipedia.org/wiki/Astronomical_coordinate...

    Angles greater than 360° (2 π) or less than 0° may need to be reduced to the range 0°−360° (0–2 π) depending upon the particular calculating machine or program. The cosine of a latitude (declination, ecliptic and Galactic latitude, and altitude) are never negative by definition, since the latitude varies between −90° and +90°.

  8. Exact trigonometric values - Wikipedia

    en.wikipedia.org/wiki/Exact_trigonometric_values

    In contrast, by the Lindemann–Weierstrass theorem, the sine or cosine of any non-zero algebraic number is always transcendental. [4] The real part of any root of unity is a trigonometric number. By Niven's theorem, the only rational trigonometric numbers are 0, 1, −1, 1/2, and −1/2. [5]

  9. Coordinate systems for the hyperbolic plane - Wikipedia

    en.wikipedia.org/wiki/Coordinate_systems_for_the...

    The Poincaré coordinates of a point are the Cartesian coordinates of the point when the point is mapped in the Poincaré disk model of the hyperbolic plane, [1] the x-axis is mapped to the segment (−1,0) − (1,0) and the origin is mapped to the centre of the boundary circle. The Poincaré coordinates, in terms of the Beltrami coordinates, are: