What is "spherical geometry"

Spherical \Spher"ic*al\, Spheric \Spher"ic\, a. [L. sphaericus, Gr. ???: cf. F. sph['e]rique.]

1. Having the form of a sphere; like a sphere; globular; orbicular; as, a spherical body.

2. Of or pertaining to a sphere.

3. Of or pertaining to the heavenly orbs, or to the sphere or spheres in which, according to ancient astronomy and astrology, they were set.

   Knaves, thieves, and treachers by spherical predominance.
   --Shak.

   Though the stars were suns, and overburned Their spheric limitations.
   --Mrs. Browning.

   Spherical angle, Spherical co["o]rdinate, Spherical excess, etc. See under Angle, Coordinate, etc.

   Spherical geometry, that branch of geometry which treats of spherical magnitudes; the doctrine of the sphere, especially of the circles described on its surface.

   Spherical harmonic analysis. See under Harmonic, a.

   Spherical lune,portion of the surface of a sphere included between two great semicircles having a common diameter.

   Spherical opening, the magnitude of a solid angle. It is measured by the portion within the solid angle of the surface of any sphere whose center is the angular point.

   Spherical polygon,portion of the surface of a sphere bounded by the arcs of three or more great circles.

   Spherical projection, the projection of the circles of the sphere upon a plane. See Projection.

   Spherical sector. See under Sector.

   Spherical segment, the segment of a sphere. See under Segment.

   Spherical triangle,re on the surface of a sphere, bounded by the arcs of three great circles which intersect each other.

   Spherical trigonometry. See Trigonometry. [1913 Webster] -- Spher"ic*al*ly, adv. -- Spher"ic*al*ness, n.

n. (context geometry English) The non-Euclidean geometry on the surface of a sphere.

n. the geometry of figures on the surface of a sphere

Spherical geometry is the geometry of the two- dimensional surface of a sphere. It is an example of a geometry that is not Euclidean. Two practical applications of the principles of spherical geometry are navigation and astronomy.

In plane geometry, the basic concepts are points and (straight) lines. On a sphere, points are defined in the usual sense. The equivalents of lines are not defined in the usual sense of "straight line" in Euclidean geometry, but in the sense of "the shortest paths between points", which are called geodesics. On a 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 not elliptic geometry, but shares with that geometry the property that a line has no parallels through a given point. Contrast this with Euclidean geometry, in which a line has one parallel through a given point, 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 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.

Higher-dimensional spherical geometries exist; see elliptic geometry.