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Encyclopedia > Elliptic geometry

Elliptic geometry (sometimes known as Riemannian geometry) is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. Behavior of lines with a common perpendicular in each of the three types of geometry The term non-Euclidean geometry (also spelled: non-Euclidian geometry) describes both hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. ... A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ... The word point can refer to: a location in physical space a unit of angular measurement; see navigation point is a typographic unit of measure in typography equal inch or sometimes approximated as inch; on computer displays it should be equal to point in typography if the correct display resolution... The term Parallel has a number of important meanings: Parallel (geometry) occurs in geometry. ...

Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which asserts that there is exactly one line parallel to L passing through p. In elliptic geometry, there are no parallel lines at all. Elliptic geometry has other unusual properties. For example, the sum of the angles of any triangle is always greater than 180°. A triangle immersed in a saddle-shape plane, as well as two diverging parallel lines. ... a and b are parallel, the transversal t produces congruent angles. ... This article is about angles in geometry. ... A triangle is one of the basic shapes of geometry: a two-dimensional figure with three vertices and three sides which are straight line segments. ...

The simplest model of elliptic geometry is that of spherical geometry, where points are points on the sphere, and lines are great circles through those points. On the sphere, such as the surface of the Earth, it is easy to give an example of a triangle that requires more than 180°: For two of the sides, take lines of longitude that differ by 90°. These form an angle of 90° at the North Pole. For the third side, take the equator. The angle of any longitude line makes with the equator is again 90°. This gives us a triangle with an angle sum of 270°, which would be impossible in Euclidean geometry. Spherical geometry is the geometry of the two-dimensional surface of a sphere. ... A sphere is a perfectly symmetrical geometrical object. ... A great circle on a sphere A great circle is a circle on the surface of a sphere that has the same diameter as the sphere, dividing the sphere into two equal hemispheres. ... Map of Earth showing curved lines of longitude Longitude, sometimes denoted by the Greek letter Î», describes the location of a place on Earth east or west of a north-south line called the Prime Meridian. ... The North Pole is the northernmost point on any planet. ... The equator is an imaginary circle drawn around a planet at a distance halfway between the poles. ... In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. ...

Elliptic geometry is sometimes called Riemannian geometry, in honor of Bernhard Riemann, but this term is usually used for a vast generalization of elliptic geometry. In mathematics, Riemannian geometry has at least two meanings, one of which is described in this article and another also called elliptic geometry. ... Bernhard Riemann. ... Results from FactBites:

 Elliptic geometry - Wikipedia, the free encyclopedia (237 words) Elliptic geometry (sometimes known as Riemannian geometry) is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which asserts that there is exactly one line parallel to L passing through p. Elliptic geometry is sometimes called Riemannian geometry, in honor of Bernhard Riemann, but this term is usually used for a vast generalization of elliptic geometry.
 Non-Euclidean geometry - Wikipedia, the free encyclopedia (1062 words) In Euclidean geometry, however, the lines remain at a constant distance, while in hyperbolic geometry they "curve away" from each other, increasing their distance as one moves farther from the point of intersection with the common perpendicular. Sometimes he is unjustly credited with only discovering elliptic geometry; but in fact, this construction shows that his work was far-reaching, with his theorems holding for all geometries. Euclidean geometry is modelled by our notion of a "flat plane." The simplest model for elliptic geometry is a sphere, where lines are "great circles" (such as the equator or the meridians on a globe), and points opposite each other are identified (considered to be the same).
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