Types of Elliptic Geometry

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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.

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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 not posit the existence of a fourth spatial dimension, or even suggest any way in which the existence of a higher-dimensional space could be detected. (This is unrelated to the treatment of time as a fourth dimension in relativity. ) Metaphorically, we can imagine geometers who are like ants living on the surface of a sphere.

Even if the ants are unable to move off of the surface, they can still construct lines and verify that parallels do not exist. The existence of a third dimension is irrelevant to the ants’ ability to do geometry, and its existence is neither verifiable nor necessary from their point of view. Another way of putting this is that the language of the theory’s axioms is incapable of expressing the distinction between one model and another. Comparison with Euclidean geometry In Euclidean geometry, a figure can be scaled up or scaled down indefinitely, and the resulting figures are similar, i. . , they have the same angles and the same internal proportions. In elliptic geometry this is not the case. For example, in the spherical model we can see that the distance between any two points must be strictly less than half the circumference of the sphere (because antipodal points are identified). A line segment therefore cannot be scaled up indefinitely. A geometer measuring the geometrical properties of the space he or she inhabits can detect, via measurements, that there is a certain distance scale that is a property of the space.

On scales much smaller than this one, the space is approximately flat, geometry is approximately Euclidean, and figures can be scaled up and down while remaining approximately similar. A great deal of Euclidean geometry carries over directly to elliptic geometry. For example, the first and fourth of Euclid’s postulates, that there is a unique line between any two points and that all right angles are equal, hold in elliptic geometry. Postulate 3, that one can construct a circle with any given center and radius, fails if “any radius” is taken to mean “any real number,” but holds if it is taken to mean “the length of any given line segment. Therefore any result in Euclidean geometry that follows from these three postulates will hold in elliptic geometry, such as proposition 1 from book I of the Elements, which states that given any line segment, an equilateral triangle can be constructed with the segment as its base.

Elliptic geometry is also like Euclidean geometry in that space is continuous, homogeneous, isotropic, and without boundaries. Isotropy is guaranteed by the fourth postulate, that all right angles are equal. For an example of homogeneity, note that Euclid’s proposition I. implies that the same equilateral triangle can be constructed at any location, not just in locations that are special in some way. The lack of boundaries follows from the second postulate, extensibility of a line segment. One way in which elliptic geometry differs from Euclidean geometry is that the sum of the interior angles of a triangle is greater than 180 degrees. In the spherical model, for example, a triangle can be constructed with vertices at the locations where the three positive Cartesian coordinate axes intersect the sphere, and all three of its internal angles are 90 degrees, summing to 270 degrees.

For sufficiently small triangles, the excess over 180 degrees can be made as small as desired. The Pythagorean theorem fails in elliptic geometry. In the 90-90-90 triangle described above, all three sides have the same length, and they therefore do not satisfy a2 + b2 = c2. The Pythagorean result is recovered in the limit of small triangles. The ratio of a circle’s circumference to its area is smaller than in Euclidean geometry. In general, area and volume do not scale as the second and third powers of linear dimensions.

Higher-dimensional spaces Hyperspherical model The hyperspherical model is the generalization of the spherical model to higher dimensions. The points of n-dimensional elliptic space are the unit vectors in Rn+1, that is, the points on the surface of the unit ball in (n+1)-dimensional space. These points are called the n-dimensional hypersphere. Lines in this model are great circles, i. e. , intersections of the hypersphere with flat hypersurfaces of dimension n passing through the origin.

Projective elliptic geometry. In the projective model of elliptic geometry, the points of n-dimensional real projective space are used as points of the model. This models an abstract elliptic geometry that is also known as projective geometry. The points of n-dimensional projective space can be identified with lines through the origin in (n+1)-dimensional space, and can be represented non-uniquely by nonzero vectors in Rn+1, with the understanding that u and ?u, for any non-zero scalar ?, represent the same point. Distance is defined using the metric hat is, the distance between two points is the angle between their corresponding lines in Rn+1. The distance formula is homogeneous in each variable, with d(?u, ?v) = d(u, v) if ? and ? are non-zero scalars, so it does define a distance on the points of projective space. A notable property of the projective elliptic geometry is that for even dimensions, such as the plane, the geometry is nonorientable. It erases the distinction between clockwise and counterclockwise rotation by identifying them.

Stereographic model. A model representing the same space as the hyperspherical model can be obtained by means of steriographic projection. Let En represent Rn ? {?}, that is, n-dimensional real space extended by a single point at infinity. We may define a metric, the chordal metric, on En by where u and v are any two vectors in Rn and ||*|| is the usual Euclidean norm. We also define The result is a metric space on En, which represents the distance along a chord of the corresponding points on the hyperspherical model, to which it maps bijectively by stereographic projection. We obtain a model of elliptic geometry if we use the metric

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