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Angular distance or angular separation is the measure of the angle between the orientation of two straight lines, rays, or vectors in three-dimensional space, or the central angle subtended by the radii through two points on a sphere.
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 ...
Difference in longitude of the points on the auxiliary sphere; α 1, α 2: forward azimuths at the points; α: forward azimuth of the geodesic at the equator, if it were extended that far; s: ellipsoidal distance between the two points; σ: angular separation between points: σ 1: angular separation between the point and the equator: σ m
d is the distance between the two points along a great circle of the sphere (see spherical distance), r is the radius of the sphere. The haversine formula allows the haversine of θ to be computed directly from the latitude (represented by φ) and longitude (represented by λ) of the two points:
The normalized angle, referred to as angular distance, between any two vectors and is a formal distance metric and can be calculated from the cosine similarity. [5] The complement of the angular distance metric can then be used to define angular similarity function bounded between 0 and 1, inclusive.
The distance between two points 1 degree apart on the same circle of latitude, measured along that circle of latitude, is slightly more than the shortest distance between those points (unless on the equator, where these are equal); the difference is less than 0.6 m (2 ft).
The distance (or perpendicular distance) from a point to a line is the shortest distance from a fixed point to any point on a fixed infinite line in Euclidean geometry. It is the length of the line segment which joins the point to the line and is perpendicular to the line. The formula for calculating it can be derived and expressed in several ways.
Alternatively, for points on rays that are nearly opposite, it is shorter to follow one ray all the way to the origin and then follow the other ray back out. Therefore, the Karlsruhe distance between two points d k ( p 1 , p 2 ) {\displaystyle d_{k}(p_{1},p_{2})} is the minimum of the two lengths that would be obtained for these two types of path.