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...

## Share this Document

Let your classmates know about this document and more at StudyMode.com