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The slant distance s (chord length) between two points can be reduced to the arc length on the ellipsoid surface S as: [21] = (+) / / where R is evaluated from Earth's azimuthal radius of curvature and h are ellipsoidal heights are each point. The first term on the right-hand side of the equation accounts for the mean elevation and the second ...
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 ...
The shortest path between two points on a spheroid is known as a geodesic. Such paths are developed using differential geometry. The equator and meridians are great ellipses that are also geodesics [a]. The maximum difference in length between a great ellipse and the corresponding geodesic of length 5,000 nautical miles is about 10.5 meters.
This is analogous to the Earth's surface, where the geodesic between two points along a great circle is the shortest route only up to the antipodal point; beyond that, there are shorter paths. Beyond a conjugate point, a geodesic in Lorentzian geometry may not be maximizing proper time (for timelike geodesics), and the geodesic may enter a ...
The haversine formula determines the great-circle distance between two points on a sphere given their longitudes and latitudes. Important in navigation , it is a special case of a more general formula in spherical trigonometry , the law of haversines , that relates the sides and angles of spherical triangles.
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. This arc is the shortest path ...
Going the "long way round" on a great circle between two points on a sphere is a geodesic but not the shortest path between the points. The map t → t 2 {\displaystyle t\to t^{2}} from the unit interval on the real number line to itself gives the shortest path between 0 and 1, but is not a geodesic because the velocity of the corresponding ...
For example, to find the midpoint of the path, substitute σ = 1 ⁄ 2 (σ 01 + σ 02); alternatively to find the point a distance d from the starting point, take σ = σ 01 + d/R. Likewise, the vertex, the point on the great circle with greatest latitude, is found by substituting σ = + 1 ⁄ 2 π. It may be convenient to parameterize the ...