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by 1.3 × 10 −11 and the 16-point Gaussian quadrature rule estimate of 1.570 796 326 794 727 differs from the true length by only 1.7 × 10 −13. This means it is possible to evaluate this integral to almost machine precision with only 16 integrand evaluations.
When calculating the length of a short north-south line at the equator, the circle that best approximates that line has a radius of (which equals the meridian's semi-latus rectum), or 6335.439 km, while the spheroid at the poles is best approximated by a sphere of radius , or 6399.594 km, a 1% difference. So long as a spherical Earth is assumed ...
2.1 To Cartesian coordinates. 2.1.1 From spherical coordinates. ... 2.4 Arc-length, curvature and torsion from Cartesian coordinates. 3 See also. 4 References.
Geographical distance or geodetic distance is the distance measured along the surface of the Earth, or the shortest arch length. The formulae in this article calculate distances between points which are defined by geographical coordinates in terms of latitude and longitude. This distance is an element in solving the second (inverse) geodetic ...
Using radians, the formula for the arc length s of a circular arc of radius r and subtending a central angle of measure 𝜃 is =, and the formula for the area A of a circular sector of radius r and with central angle of measure 𝜃 is A = 1 2 θ r 2 . {\displaystyle A={\frac {1}{2}}\theta r^{2}.}
This is a 2-d problem in span{^, ^}, which will be solved with the help of the arc length formula above. If the arc length, s 12 {\displaystyle s_{12}} is given then the problem is to find the corresponding change in the central angle θ 12 {\displaystyle \theta _{12}} , so that θ 2 = θ 1 + θ 12 {\displaystyle \theta _{2}=\theta _{1}+\theta ...
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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: