Elliptic geometry

An elliptical geometry is a non-Euclidean geometry in which it line g and a point P that does not lie on the straight line, not to g parallel line are in the plane case to a given line passing through P.

In elliptic geometry certain axioms of absolute geometry, for exact details apply this further below in this article. In addition, is in lieu of the parallel postulate of Euclidean geometry the axiom:

This means that there are no parallel in an elliptical geometry. Another alternative to the Euclidean parallel axiom leads to hyperbolic geometry.

• 2.1 The real elliptical level and their representation on the sphere
• 2.2 planes and straight lines in three-dimensional vector space
• 2.3 Examples of rational elliptic levels

Properties of elliptical planes

Problem of the axioms of absolute geometry

There is no general consensus in the literature about how a Absolute geometry is to be characterized by axioms. The mentioned in the introduction geometric axioms for an "absolute geometry" that Felix Bachmann has formulated are quoted in full in Metric absolute geometry. They represent a certain minimum consensus, if one of the absolute geometry based on an orthogonality relation defines as equal to the incidence relation. It is not trivial to compare these axioms with systems that do not include the orthogonality as a basic relation.

Identification of elliptic levels

On that basis Bachmann characterizes the elliptical geometry by the axiom

In short: There is a Polardreiseit. Under these axiomatic conditions he proves a statement which is stronger, as said in the introduction to " elliptic parallel postulate ":

The idiosyncratic formulation due, on one hand, the fact that Bachmann is based defines a group-theoretical approach ( lines are axis reflections and points point reflections), in which this statement "points" and " lines " only must artificially be made ​​distinguishable, on the other hand, the fact that he Projective the conceptual level close regarded as common today: a projective plane is at Bachmann always a two-dimensional projective space over a field whose characteristic is not 2, so a pappussche projective Fano plane. In other words, the theorem states:

An elliptical level in the sense of the above mentioned axiomatic therefore is always a projective plane. Conversely, under certain necessary conditions of a projective plane has an elliptical level make:

• In a finite projective plane pappusschen (see correlation ( projective geometry) # polarities over finite spaces )
• In levels over algebraically closed fields and
• More generally in planes over fields with only one square class.

A sufficient condition for the existence of (at least) an elliptic plane: Let K be a formally real body, then is defined by the symmetric bilinear form on a projective polarity of the projective plane, with this plane is an elliptic plane. The body K need not be Archimedean.

Level models

The real elliptical level and their representation on the sphere

Over the field of real numbers exists up to isomorphism only one elliptical level: A well-known representation of this real model provides the spherical geometry, which can be understood as an illustration of the projective plane over the real numbers, if one identifies opposite points. The additional, elliptical structure is given by the elliptic polarity described here.

• The " level " is a sphere,
• A "point" is a pair of two points on the spherical surface, which are opposed to each other, and
• Is a " straight line" is a circle on the sphere whose center is the center of the sphere is ( a great circle ).

As an illustrative contrast to Euclidean geometry can be considered the sum of the angles of triangles, which in this model is always above 180 ° - the solid angle sum of 180 ° in the Euclidean geometry is equivalent to the parallel postulate. If you select two lines through the North Pole, which together form the angle a, then cut them to the equator at an angle of 90 °. So the resulting triangle has an angle sum of 180 ° a Compare the picture is right there.

First, the angle between the great circles are " Euclidean " angle between the planes on which the great circles are (or between corresponding normal vectors). In the real case, but that is not difficult, as long as only figures on the sphere are considered, which are contained in the ball without glued opposite poles all in a " hemisphere without their edge ."

The right visible " small " triangle on the map should make clear that exactly the familiar angle sum is small triangles on the sphere approximation or in the Euclidean plane ( then distorted) map of a spherical section of 180 °, this second statement is true about.

The first statement that sufficiently small parts of a world map actually behave approximately Euclidean, is also true. This can only be illustrated by a map but if the sides of the triangle shown correspond to great circles on the sphere, which for more than two of the three sides of the triangle to a representation as accurately can lead routes at the usual map projections just (compare Mercator projection ) without the angle at issue, but are distorted.

→ For area calculations and Dreieckskongruenzsätzen for the real elliptic plane see ball triangle with the explained in the next section restricting the lengths and angles must be considered.

Planes and straight lines in three-dimensional vector space

The projective plane over a portion of the field K of real numbers can be illustrated as a set of lines (as projective points) and levels ( as straight lines ) in the vector space. If the elliptical polarity defined by ( " real elliptical standard polarity " ), then a vector space line ( ie a point on the elliptic geometry) in the usual Euclidean sense is exactly perpendicular to a plane ( a straight elliptic geometry ). Each such pair (straight, vertical plane ) in the vector space is an elliptic geometry ( pole, polar ) pair. Two elliptical lines are perpendicular to each other if and only if their associated levels are in the vector space in the Euclidean sense perpendicular to each other. So you can as described in the above representation of ball transfer more generally for subfield of the Euclidean angle measure.

The angles and lengths of the sphere representation are here rotation angle between two-dimensional subspaces (elliptical angle ) or between one-dimensional subspaces (elliptical path lengths ) of the. This concept of length coincides with the length measurement of the spherical geometry, if one uses the Einheitsspäre and only lengths and angles considered less than or equal to 1 Right angle (90 ° or ) are, at larger angles between planes of adjacent angles is to be taken in longer intervals between points, as well as determining yes ( angular) distances between lines through the origin in. It is then also, if you follow these restrictions:

This angle and distance term can also be applied to elliptic planes over subfields of the real numbers, provided that the symmetric bilinear form B. Projective elliptic polarity defined, to that described in this section, " standard elliptical shape " is equivalent Compare the following examples.

Examples of rational elliptic levels

Looking specifically, the field of rational numbers with the real elliptical Standard polarity is determined by the bilinear form, then this is an elliptic partial level of real elliptic plane. Starting from the polar triangle, we form route centers, compare the figure to the right. is a real and hence rational center of the " distance". - There is a second focus on the elliptic lines AB: It is the mirror point of M in the mirror at B, but this need not be further considered in the following. The " route ", better the ordered pair of points has no rational center: The real focus is not a rational straight line. So the track has no center. The example shows, as in the case of a partial body (here) a real non-existence proof can be given, and that a route in an elliptical plane must have no center. Applying the triangle to the center vertical set of absolute geometry, more precisely, only the ( Euclidean uninteresting ) existence conclusion: If two sides of a triangle a perpendicular bisector, then the third, then given the fact that also at least one of the pairs or no center has. Since, according to the analogous calculation as above for a center that has, like, the same " length", no middle. Considering all of this on the Euclidean unit ball, then you already no longer sees that the starting points A, B, C ( projective straight line! ) Cut this sphere in two opposite rational points, the projective point though. One must therefore carefully argue with the ball model for body part of the real number.

If one chooses a fixed positive integer k, then the shape equation can not be solved by a triple of integers with no common divisor, as can only be solved by just three numbers. Therefore, the bilinear form determines an elliptic polarity. With this polarity, the elliptical rational level but can not be embedded into the real elliptical plane, because of the shape is hyperbolic!

This one has at least two non- isomorphic elliptic planes over the same body. - In fact, infinitely many, because forms arise only rational elliptic geometries isomorphic if and only if, k and l so square are equivalent.

Higher-Dimensional elliptic spaces

Three-and higher-dimensional elliptic geometries are described in the article Metric absolute geometry axiomatically. You are always projective elliptic spaces. This means that over a field K with has an appropriate elliptic polarity by a zero -part, symmetric bilinear form B of rank to be explained. This can then be defined over K an n-dimensional projective elliptic space. This construction is explained in the article projective metric geometry.