# Spherical geometry

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Spherical geometry

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 navigation and astronomy.

In plane geometry the basic concepts are points and line. 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 is called a geodesic. On the sphere the geodesics are the great circles, so the other geometric concepts are defined like in plane geometry but with 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 model of elliptic geometry, in which a line has no parallels through a given point. Contrast this with hyperbolic geometry, in which a line has two parallels, and an infinite number of ultra-parallels, through a given point.

An important related geometry related to that modeled by the sphere is called the real projective plane; it is obtained by identifying antipodes (pairs of opposite points) on the sphere. Locally, the projective plane has all the properties of spherical geometry, but it has different global properties. In particular, it is non-orientable.

ee also

*Spherical distance
*Spherical polyhedron
*Spherical trigonometry
*SIGI

* [http://www.math.uncc.edu/~droyster/math3181/notes/hyprgeom/node5.html Spherical Geometry] UNCC
* [http://math.rice.edu/~pcmi/sphere/ The Geometry of the Sphere] Rice University
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• 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. [1913 Webster] 2. Of or pertaining to a sphere. [1913 Webster] 3 …   The Collaborative International Dictionary of English

• spherical geometry — n. the study of the geometry of figures drawn on a sphere …   English World dictionary

• spherical geometry — noun (mathematics) the geometry of figures on the surface of a sphere • Topics: ↑mathematics, ↑math, ↑maths • Hypernyms: ↑geometry * * * noun : the geometry of figures on …   Useful english dictionary

• spherical geometry — the branch of geometry that deals with figures on spherical surfaces. [1720 30] * * * …   Universalium

• spherical geometry — spher′ical geom′etry n. math. the branch of geometry that deals with figures on spherical surfaces • Etymology: 1720–30 …   From formal English to slang

• spherical geometry — noun Date: 1728 the geometry of figures on a sphere …   New Collegiate Dictionary

• spherical geometry — noun The non Euclidean geometry on the surface of a sphere …   Wiktionary

• Spherical trigonometry — Spherical triangle Spherical trigonometry is a branch of spherical geometry which deals with polygons (especially triangles) on the sphere and the relationships between the sides and the angles. This is of great importance for calculations in… …   Wikipedia

• 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. [1913 Webster] 2. Of or pertaining to a sphere. [1913 Webster] 3. Of or… …   The Collaborative International Dictionary of English

• Spherical angle — 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. [1913 Webster] 2. Of or pertaining to a sphere. [1913 Webster] 3 …   The Collaborative International Dictionary of English