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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 ...
For example, one sphere that is described in Cartesian coordinates with the equation x 2 + y 2 + z 2 = c 2 can be described in spherical coordinates by the simple equation r = c. (In this system—shown here in the mathematics convention—the sphere is adapted as a unit sphere, where the radius is set to unity and then can generally be ignored ...
A sphere (from Greek σφαῖρα, sphaîra) [1] is a geometrical object that is a three-dimensional analogue to a two-dimensional circle.Formally, a sphere is the set of points that are all at the same distance r from a given point in three-dimensional space. [2]
The sphere's radius is taken as unity. For specific practical problems on a sphere of radius R the measured lengths of the sides must be divided by R before using the identities given below. Likewise, after a calculation on the unit sphere the sides a, b, and c must be multiplied by R.
In vector notation, the equations are as follows: . Equation for a sphere ‖ ‖ = : points on the sphere : center point : radius of the sphere; Equation for a line starting at
For example, assuming the Earth is a sphere of radius 6371 km, the surface area of the arctic (north of the Arctic Circle, at latitude 66.56° as of August 2016 [7]) is 2π ⋅ 6371 2 | sin 90° − sin 66.56° | = 21.04 million km 2 (8.12 million sq mi), or 0.5 ⋅ | sin 90° − sin 66.56° | = 4.125% of the total surface area of the Earth.
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.
If the sphere is isometrically embedded in Euclidean space, the sphere's intersection with a plane is a circle, which can be interpreted extrinsically to the sphere as a Euclidean circle: a locus of points in the plane at a constant Euclidean distance (the extrinsic radius) from a point in the plane (the extrinsic center). A great circle lies ...