Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point (rather than two). Because of this, the elliptic geometry described in this article is sometimes referred to as single elliptic geometry whereas spherical geometry is sometimes referred to as double elliptic geometry.
The appearance of this geometry in the nineteenth century stimulated the development of non-Euclidean geometry generally, including hyperbolic geometry.
Elliptic geometry has a variety of properties that differ from those of classical Euclidean plane geometry. For example, the sum of the interior angles of any triangle is always greater than 180°.
Ellipticgeometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel...
ellipticgeometries, that have no parallel lines. Absolute geometry is an extension of ordered geometry, and thus, all theorems in ordered geometry hold...
plane intersect in two antipodal points, unlike coplanar lines in Ellipticgeometry. In the extrinsic 3-dimensional picture, a great circle is the intersection...
now rarely used sequence ellipticgeometry (spherical geometry), parabolic geometry (Euclidean geometry), and hyperbolic geometry. In the former Soviet Union...
the former case, one obtains hyperbolic geometry and ellipticgeometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed...
properties of elliptic, Euclidean and hyperbolic geometries contrast as follows: Given a line l and a point P not on the line, Elliptic there exists no...
section). The intersection of an elliptic cone with a concentric sphere is a spherical conic. In projective geometry, a cylinder is simply a cone whose...
one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also...
not all equal, though opposite angles are equal. Other geometries, such as spherical, elliptic, and hyperbolic, have so-called rectangles with opposite...
absolute geometry, while negating it yields hyperbolic geometry. Other consistent axiom sets can yield other geometries, such as projective, elliptic, spherical...
since antiquity was a non-Euclidean geometry, an ellipticgeometry. The development of intrinsic differential geometry in the language of Gauss was spurred...
resulting geometries were later developed by Lobachevsky, Riemann and Poincaré into hyperbolic geometry (the acute case) and ellipticgeometry (the obtuse...
algebraic geometry, it is possible to describe some features of elliptic curves over the real numbers using only introductory algebra and geometry. In this...
However, the Pythagorean theorem remains true in hyperbolic geometry and ellipticgeometry if the condition that the triangle be right is replaced with...
arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around...
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements...
sphere. Spherical geometry is a form of ellipticgeometry, which together with hyperbolic geometry makes up non-Euclidean geometry. The sphere is a smooth...
(1824–1873) – differential geometry Bernhard Riemann (1826–1866) – ellipticgeometry (a non-Euclidean geometry) and Riemannian geometry Julius Wilhelm Richard...
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