Types of elliptic geometry
Describe how it is possible to have a triangle with three right angles.
From these properties of a sphere, we see that in order to formulate a consistent axiomatic system, several of the axioms from a neutral geometry need to be dropped or modified, whether using either Hilbert's or Birkhoff's axioms. Hence, the Elliptic Parallel Postulate is inconsistent with the axioms of a neutral geometry.
The reason for doing this is that it allows elliptic geometry to satisfy the axiom that there is a unique line passing through any two points. Comparison with Euclidean geometry[ edit ] In Euclidean geometry, a figure can be scaled up or scaled down indefinitely, and the resulting figures are similar, i. 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. It does not imply any direct connection with the curve called an ellipse, but only a rather far-fetched analogy. What does it mean for a point to be between two other points on a line great circle? The lack of boundaries follows from the second postulate, extensibility of a line segment. 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. Note that with this model, a line no longer separates the plane into distinct half-planes, due to the association of antipodal points as a single point. This is because there are no antipodal points in elliptic geometry. Every point corresponds to an absolute polar line of which it is the absolute pole. One problem with the spherical geometry model is that two lines intersect in more than one point. A great deal of Euclidean geometry carries over directly to elliptic geometry. In a spherical model: Two lines, which are great circles, intersect in two points called poles or antipodal points. The hyperbolic plane is represented by one-half of the Euclidean plane, as defined by a given Euclidean line l, where l is not considered part of the hyperbolic 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.
Moreover, the angle sums of two distinct triangles are not necessarily the same. It does not imply any direct connection with the curve called an ellipse, but only a rather far-fetched analogy. However, unlike in spherical geometry, the poles on either side are the same.
In elliptic geometry this is not the case.
Non euclidean geometry
For instance, it turns out that the shortest flying distance from Florida to the Philippine Islands is a path across Alaska — even though the Philippines are at a more southerly latitude than Florida! Spherical astronomy, of importance in positional astronomy and space exploration, is the application of spherical trigonometry to determinations of stellar positions on the celestial sphere. Hyperbolic Geometry[ edit ] Hyperbolic geometry is also known as saddle geometry or Lobachevskian geometry. The model can be viewed as taking the Modified Riemann Sphere and flattening onto a Euclidean plane. Comparison with Euclidean geometry[ edit ] In Euclidean geometry, a figure can be scaled up or scaled down indefinitely, and the resulting figures are similar, i. Definitions[ edit ] In elliptic geometry, two lines perpendicular to a given line must intersect. Elliptic Geometry[ edit ] Models of Elliptic Space[ edit ] Spherical geometry gives us perhaps the simplest model of elliptic geometry. Lines are represented by circles through the points. Remember the sides of the quadrilateral must be segments of great circles. A line has finite length. Elliptic geometry requires a different set of axioms for the axiomatic system to be consistent and contain an elliptic parallel postulate. In spherical geometry any two great circles always intersect at exactly two points. Generalization to elliptical geometry It was Felix Klein who first saw clearly how to rid spherical geometry of its one blemish: the fact that two lines have not one but two common points. A central conic is called an ellipse or a hyperbola according as it has no asymptote or two asymptotes.
Is the length of the summit more or less than the length of the base? 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.
Every point corresponds to an absolute polar line of which it is the absolute pole. The concept of betweenness of points does not make sense.
A great circle has no beginning and no end. This is always the case on a surface that bulges out or, in mathematical parlance, has positive curvature. An important note is how elliptic geometry differs in an important way from either Euclidean geometry or hyperbolic geometry.
based on 44 review