Spherical geometry
From The Art and Popular Culture Encyclopedia
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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.
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.
Higher-dimensional spherical geometries exist; see elliptic geometry.
