Elliptic Geometry(redirected from Elliptic or Riemannian geometry)
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elliptic geometry[ə′lip·tik jē′äm·ə·trē]
a type of non-Euclidean geometry, that is, a geometric theory based on axioms whose requirements are to a considerable degree different from the requirements of the axioms of Euclidean geometry. The basic objects, or elements, of three-dimensional elliptic geometry are points, lines, and planes; the basic concepts of elliptic geometry are the concepts of incidence (a point is on a line, a line is in a plane), order (for example, the order of points on a line or the order of lines passing through a given point in a given plane), and congruence (of figures).
The requirements of the axioms of elliptic geometry concerning membership and order completely coincide with the requirements of the axioms of projective geometry. Accordingly, such propositions as the following are found in elliptic geometry: through any two points there passes one line; any two planes intersect in one line; any two coplanar lines intersect in one point; and the points on a line are arranged in a cyclic order, as are lines lying in the same plane and passing through the same point.
The requirements of the axioms of elliptic geometry concerning congruence are similar to the corresponding requirements of the axioms of Euclidean geometry; in any case, they provide for motions of figures in the elliptic plane and in elliptic space that are just as free as in the plane and space of Euclidean geometry.
The metric properties of the elliptic plane coincide “in the small” with the metric properties of the ordinary sphere. More precisely, for any point of the elliptic plane there exists a part of the plane, containing this point, that is isometric to some part of the sphere; the radius R of this sphere is the same for all planes in the given elliptic space. The number K = 1/R2 is called the curvature of the elliptic space. The smaller is K, the closer are the properties of this space to those of Euclidean space.
The properties of the elliptic plane “in the large” differ from the properties of the sphere as a whole. For example, in the elliptic plane, two lines intersect in one point; on the sphere, two great circles, which play the role of lines in spherical geometry, intersect in two points. A line in a plane does not separate the plane—that is, if the line a is in the plane α, then any two points of α not on a can be joined by a line segment that does not intersect a.
Elliptic geometry was apparently first discussed by B. Riemann in his lecture “Über die Hypothesen, welche der Geometrie zu Grunde liegen” (On the Hypotheses That Form the Foundations of Geometry), which was delivered in 1854 and published in 1867. In this lecture elliptic geometry was examined as a special case of Riemannian geometry, which is the theory of Riemannian spaces in the broad sense. Elliptic geometry belongs to the theory of spaces of constant positive curvature.
N. V. EFIMOV