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Formulas for the Web Mercator are fundamentally the same as for the standard spherical Mercator, but before applying zoom, the "world coordinates" are adjusted such that the upper left corner is (0, 0) and the lower right corner is ( , ): [7] = ⌊ (+) ⌋ = ⌊ ( [ (+)]) ⌋ where is the longitude in radians and is geodetic latitude in radians.
with x(λ 0) = 0 and y(0) = 0, gives x(λ) and y(φ). The value λ 0 is the longitude of an arbitrary central meridian that is usually, but not always, that of Greenwich (i.e., zero). The angles λ and φ are expressed in radians. By the integral of the secant function, [30] [31]
In the following equations, denotes the sagitta (the depth or height of the arc), equals the radius of the circle, and the length of the chord spanning the base of the arc. As 1 2 l {\displaystyle {\tfrac {1}{2}}l} and r − s {\displaystyle r-s} are two sides of a right triangle with r {\displaystyle r} as the hypotenuse , the Pythagorean ...
A quadratrix in the first quadrant (x, y) is a curve with y = ρ sin θ equal to the fraction of the quarter circle with radius r determined by the radius through the curve point. Since this fraction is 2 r θ π {\displaystyle {\frac {2r\theta }{\pi }}} , the curve is given by ρ ( θ ) = 2 r θ π sin θ {\displaystyle \rho (\theta ...
In land navigation, a 'bearing' is ordinarily calculated in a clockwise direction starting from a reference direction of 0° and increasing to 359.9 degrees. [5] Measured in this way, a bearing is referred to as an azimuth by the US Army but not by armies in other English speaking nations, which use the term bearing. [6]
1°5 ′ by 1° Width of little finger with arm stretched out 1° 17.5 meter at 1 km distance The Sun in the sky of Venus: 0.7° [13] [14] Io (as seen from the “surface” of Jupiter) 35’ 35” Moon: 34 ′ 6″ – 29 ′ 20″ 32.5–28 times the maximum value for Venus (orange bar below) / 2046–1760″ the Moon has a diameter of 3,474 ...
Vincenty's formulae are two related iterative methods used in geodesy to calculate the distance between two points on the surface of a spheroid, developed by Thaddeus Vincenty (1975a). They are based on the assumption that the figure of the Earth is an oblate spheroid, and hence are more accurate than methods that assume a spherical Earth, such ...
A diagram illustrating great-circle distance (drawn in red) between two points on a sphere, P and Q. Two antipodal points, u and v are also shown.. The great-circle distance, orthodromic distance, or spherical distance is the distance between two points on a sphere, measured along the great-circle arc between them.