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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. 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°. 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 Euclidian geometry. 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.. ,Elliptic geometry is anon Euclidian 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 hyperbollic geometry, violates Euclid's parallel postulate, which can be interpreted as asserting 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 a variety of properties that differ from those of classical Euclidean plane geometry. For example, the sum of the angles of any triangle is always greater than 180°. Types of elliptic geometry
The two main types of elliptic geometry may be called spherical elliptic geometry and projective elliptic geometry. These two geometries are locally identical but taken as a whole, they are essentially different from each other. Thus, unlike with Euclidean geometry, there is not one single elliptic geometry in each dimension. Spherical elliptic geometry is modelled by the surface of a sphere and, in higher dimensions, a hypersphere, or alternatively by the Euclidean plane or higher euclidian space with the addition of a point at infinity. Projective elliptic geometry is modelled by real projective spaces. These three models are described below. On a sphere, the sum of the angles of a triangle is not equal to 180°. The surface of a sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is very nearly 180°. A simple way to picture elliptic geometry is to look at a globe. Neighboring lines of longitude appear to be parallel at the equator, yet they intersect at the poles. More precisely, the surface of a sphere is a model of elliptic geometry if lines are modeled by great circles, and points at each other's antipodes are considered to be the same point. With this identification of antipodal points, the model satisfies Euclid's first postulate, which states that two points uniquely determine a line. If the antipodal points were considered to be distinct, as in spherical geometry, then uniqueness would be violated, e.g., the lines of longitude on the Earth's surface all pass through both the north pole and the south pole. Although models such as the spherical model are useful for visualization and for proof of the theory's self-consistency, neither a model nor an embedding in a higher-dimensional space is logically necessary. For example, Einstein's theory of geneal relativity has static solutions in which space containing a gravitational field is (locally) described by three-dimensional elliptic geometry, but the theory does...
The two main types of elliptic geometry may be called spherical elliptic geometry and projective elliptic geometry. These two geometries are locally identical but taken as a whole, they are essentially different from each other. Thus, unlike with Euclidean geometry, there is not one single elliptic geometry in each dimension. Spherical elliptic geometry is modelled by the surface of a sphere and, in higher dimensions, a hypersphere, or alternatively by the Euclidean plane or higher euclidian space with the addition of a point at infinity. Projective elliptic geometry is modelled by real projective spaces. These three models are described below. On a sphere, the sum of the angles of a triangle is not equal to 180°. The surface of a sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is very nearly 180°. A simple way to picture elliptic geometry is to look at a globe. Neighboring lines of longitude appear to be parallel at the equator, yet they intersect at the poles. More precisely, the surface of a sphere is a model of elliptic geometry if lines are modeled by great circles, and points at each other's antipodes are considered to be the same point. With this identification of antipodal points, the model satisfies Euclid's first postulate, which states that two points uniquely determine a line. If the antipodal points were considered to be distinct, as in spherical geometry, then uniqueness would be violated, e.g., the lines of longitude on the Earth's surface all pass through both the north pole and the south pole. Although models such as the spherical model are useful for visualization and for proof of the theory's self-consistency, neither a model nor an embedding in a higher-dimensional space is logically necessary. For example, Einstein's theory of geneal relativity has static solutions in which space containing a gravitational field is (locally) described by three-dimensional elliptic geometry, but the theory does...
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