spherical geometry

plural spherical geometries

1. uncountableusuallygeometryThe geometry of the 2-dimensional surface of a sphere;

   The basic concepts of Euclidean geometry of the plane are the point and the (straight) line; in spherical geometry, the corresponding concepts are the point and the great circle.

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   Due to the way the geometry of a sphere's surface differs from that of the plane, spherical geometry has some features of a non-Euclidean geometry and is sometimes described as being one. Historically, however, spherical geometry was not considered a fully fledged (non-Euclidean) geometry capable of resolving the question of whether the parallel postulate is a logical consequence of the rest of Euclid's axioms of plane geometry.

   1972, Morris Kline, Mathematical Thought from Ancient to Modern Times: Volume 1, Oxford University Press, 1990, paperback, page 119, Spherical trigonometry presupposes spherical geometry, for example the properties of great circles and spherical triangles, much of which was already known; it had been investigated as soon as astronomy became mathematical, during the time of the later Pythagoreans.

   We start with spherical geometries. The two geometries on spheres of radiuses R₁ and R₂ are obviously identical if R₁ = R₂; moreover, the converse also holds.

   1994, Viacheslav V. Nikulin, Igor R. Shafarevich, translated by Miles Reid, Geometries and Groups, Springer, page 194

   To collect the boundary of the point set in the tree we pass a bounding box (for Euclidean geometries) or the universal set (for spherical geometries) to the root of the tree.

   2008, Sherif Ghali, Introduction to Geometric Computing, Springer, page 272

   2020, Marshall A. Whittlesey, Spherical Geometry and Its Applications, Taylor & Francis (CRC Press), unnumbered page, It has been at least fifty years since spherical geometry and spherical trigonometry have been a regular part of the high school or undergraduate curriculum.

2. The geometry of the 2-dimensional surface of a sphere; (countable) a given geometry of the surface of a sphere; a geometry of the surface of a given sphere, regarded as distinct from that of other spheres.
   1. countableusuallygeometrya given geometry of the surface of a sphere; a geometry of the surface of a given sphere, regarded as distinct from that of other spheres.