Spherical+geometry

= Spherical geometry =

In plane geometry the basic concepts are points and (straight) lines. On the sphere, points are defined in the usual sense. The equivalents of lines are not defined in the usual sense of "straight line" but in the sense of "the shortest paths between points," which are called geodesics. On the sphere the geodesics are the great circles; other geometric concepts are defined as in plane geometry but with straight lines replaced by great circles. Thus, in spherical geometry angles are defined between great circles, resulting in a spherical trigonometry that differs from ordinary trigonometry in many respects; for example, the sum of the interior angles of a triangle exceeds 180 degrees. Spherical geometry is the simplest form of elliptic geometry, in which a line has no parallels through a given point. Contrast this with Euclidean geometry, in which a line has one parallel, and hyperbolic geometry, in which a line has two parallels and an infinite number of ultraparallels through a given point. An important geometry related to that of the sphere is that of the real projective plane; it is obtained by identifying antipodal points (pairs of opposite points) on the sphere. (This is another kind of elliptic geometry.) Locally, the projective plane has all the properties of spherical geometry, but it has different global properties. In particular, it is non-orientable, or one-sided. Concepts of spherical geometry may also be applied to the oblong sphere, though minor modifications must be implemented on certain formulas. On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is very nearly 180. The surface of a sphere can be represented by a collection of two dimensional maps. Therefore it is a two dimensional manifold.
 * Spherical geometry ** is the geometry of the two-dimensional surface of a sphere. It is an example of a non-Euclidean geometry. Two practical applications of the principles of spherical geometry are to navigation and astronomy.

History
Spherical trigonometry was studied by early Greek mathematicians such as Menelaus of Alexandria, who wrote a book on spherical trigonometry called //Sphaerica// and developed Menelaus' theorem. // The book of unknown arcs of a sphere // written by 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. The book //On Triangles// by Regiomontanus, written around 1463, is the first pure trigonometrical work in Europe. However, Gerolamo Cardano noted a century later that much of the material there on spherical trigonometry was taken from the twelfth-century work of the Spanish Islamic scholar Jabir ibn Aflah.