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The Book of Unknown Arcs of a Sphere written by the Islamic mathematician Al-Jayyani is considered to be the first treatise on spherical trigonometry. The book contains formulae for right-handed triangles, the general law of sines, and the solution of a spherical triangle by means of the polar triangle. [5]
The sphere has a radius of 1, and so the side lengths and lower case angles are equivalent (see arc length). The angle A (respectively, B and C ) may be regarded either as the dihedral angle between the two planes that intersect the sphere at the vertex A , or, equivalently, as the angle between the tangents of the great circle arcs where they ...
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 ...
For any natural number n, an n-sphere, often denoted S n, is the set of points in (n + 1)-dimensional Euclidean space that are at a fixed distance r from a central point of that space, where r is, as before, a positive real number. In particular: S 0: a 0-sphere consists of two discrete points, −r and r; S 1: a 1-sphere is a ...
The unit sphere S 2 in three-dimensional space R 3 is the set of points (x, y, z) such that x 2 + y 2 + z 2 = 1. Let N = (0, 0, 1) be the "north pole", and let M be the rest of the sphere. The plane z = 0 runs through the center of the sphere; the "equator" is the intersection of the sphere with this plane.
For any natural number , an -sphere of radius is defined as the set of points in (+) -dimensional Euclidean space that are at distance from some fixed point , where may be any positive real number and where may be any point in (+) -dimensional space.
In mathematics, a unit sphere is a sphere of unit radius: the set of points at Euclidean distance 1 from some center point in three-dimensional space. More generally, the unit n {\displaystyle n} -sphere is an n {\displaystyle n} -sphere of unit radius in ( n + 1 ) {\displaystyle (n+1)} - dimensional Euclidean space ; the unit circle is a ...
Consider the projection centered at S = (0, 0, −1) on the unit sphere, which is the set of points (x, y, z) in three-dimensional space R 3 such that x 2 + y 2 + z 2 = 1. In Cartesian coordinates (x, y, z) on the sphere and (X, Y) on the plane, the projection and its inverse are then described by