Affine Geometry

Affine geometry is a form of geometry featuring the unique parallel line property (see the parallel postulate) where the notion of angle is undefined and lengths cannot be compared in different directions (that is, Euclid's third and fourth postulates are meaningless)

First identified by Euler, many affine properties are familiar from Euclidean geometry, but also apply in Minkowski space

In effect, affine geometry is a generalization of Euclidean geometry characterized by slant and scale distortions

Affine geometry can be developed in terms of the geometry of vectors, with or without the notion of coordinates. An affine space is distinguished from a vector space of the same dimension by 'forgetting' the origin 0 (sometimes known as free vectors). Thus, affine geometry can be seen as part of linear algebra

Ordered geometry Ordered geometry is a form of geometry featuring the concept of intermediacy (or "betweenness") but, like projective geometry, omitting the basic notion of measurement. Ordered geometry is a fundamental geometry forming a common framework for affine, Euclidean, absolute, and hyperbolic geometry (but not for projective geometry).

History Moritz Pasch first defined a geometry without reference to measurement in 1882. His axioms were improved upon by Peano (1889), Hilbert (1899), and Veblen (1904) [1]. Euclid anticipated Pasch's approach in definition 4 of The Elements: "a straight line is a line which lies evenly with the points on itself"

Primitive concepts

The only primitive notions in ordered geometry are points A, B, C, ... and the relation of intermediacy [ABC] which can be read as "B is between A and C".

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The segment AB is the set of points P such that [APB]. The interval AB is the segment AB and its end points A and B. The ray A/B (read as "the ray from A away from B") is the set of points P such that [PAB]. The line AB is the interval AB and the two rays A/B and B/A. Points on the line AB are said to be collinear. An angle consists of a point O (the vertex) and two non-collinear rays out from O (the sides). A triangle is given by three non-collinear points (called vertices) and their three segments AB, BC, and CA. If three points A, B, and C are non-collinear, then a plane ABC is the set of all points collinear with pairs of points on one or two of the sides of triangle ABC. If four points A, B, C, and D are non-coplanar, then a space ( 3-space) ABCD is the set of all points collinear with pairs of points selected from any of the four faces (planar regions) of the tetrahedron ABCD.

Axioms of ordered geometry There exist at least two points. If A and B are distinct points, there exists a C such that [ABC]. If [ABC], then A and C are distinct (A≠C). If [ABC], then [CBA] but not [CAB]. If C and D are distinct points on the line AB, then A is on the line CD. If AB is a line, there is a point C not on the line AB. (Axiom of Pasch) If ABC is a triangle and [BCD] and [CEA], then there exists a point F on the line DE for which [AFB]. Axiom of dimensionality:

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For planar ordered geometry, all points are in one plane. Or If ABC is a plane, then there exists a point D not in the plane ABC.

All points are in the same plane, space, etc. (depending on the dimension one chooses to work within). (Dedekind's Axiom) For every partition of all the points on a line into two nonempty sets such that no point of either lies between two points of the other, there is a point of one set which lies between every other point of that set and every point of the other set.

Parallelism Gauss, Bolyai, and Lobachevsky developed a notion of parallelism which can be expressed in ordered geometry.

Theorem (existence of parallelism): Given a point A and a line r, not through A, there exist exactly two rays from A in the plane Ar which do not meet r. So there is a parallel line through A which does not meet r. Theorem (transmissibility of parallelism): The parallelism of a ray and a line is preserved by adding or subtracting a segment from the beginning of a ray.

The symmetry of parallelism cannot be proven in ordered geometry[5]. Therefore, the "ordered" concept of parallelism does not form an equivalence relation on lines.

Non-Euclidean In mathematics, non-Euclidean geometry describes hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. The essential difference between Euclidean and non-Euclidean geometry is the nature of parallel lines. Euclid 's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate, which states that, within a two-dimensional plane, for any given line l and a point A, which is not on l, there is exactly one line through A that does not intersect l. In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting l, while in elliptic geometry, any line through A intersects l. (See the entries on hyperbolic geometry and

Another way to describe the differences between these geometries is as follows: Consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line. In Euclidean geometry the lines remain at a constant distance from each other, and are known as parallels. In hyperbolic geometry they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called ultraparallels. In elliptic geometry the lines "curve toward" each other and eventually intersect.

Concepts of non-Euclidean geometry Non-Euclidean geometry systems differ from Euclidean geometry in that they modify Euclid's fifth postulate, which is also known as the parallel postulate.

In general, there are two forms of (homogeneous) non-Euclidean geometry : 

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Hiperbolic: In hyperbolic geometry there are many more than one distinct line through a particular point that will not intersect with another given line In elliptic geometry there are no lines that will not intersect, as all that start to separate will converge. In addition, elliptic geometry modifies Euclid's first postulate so that two points determine at least one line

Models of non-Euclidean geometry Elliptic geometry The simplest model for elliptic geometry is a sphere, where lines are "great circles" (such as the equator or the meridians on a globe), and points opposite each other are identified (considered to be the same). In the elliptic model, for any given line l and a point A, which is furthest from l, all lines through A will intersect l. 

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Hyperbolic geometry Even after the work of Lobachevsky, Gauss, and Bolyai, the question remained: does such a model exist for hyperbolic geometry? The model for hyperbolic geometry was answered by Eugenio Beltrami, in 1868, who first showed that a surface called the pseudosphere has the appropriate curvature to model a portion of hyperbolic space, and in a second paper in the same year, defined the Klein model, the Poincaré disk model, and the Poincaré half-plane model which model the entirety of hyperbolic space, and used this to show that Euclidean geometry and hyperbolic geometry were equiconsistent, so that hyperbolic geometry was logically consistent if Euclidean geometry was. (The reverse implication follows from the horosphere model of Euclidean geometry.) In the hyperbolic model, for any given line l and a point A, which is not on l, there are infinitely many lines through A that do not intersect l.

Absolute geometry Absolute geometry is a geometry based on an axiom system that does not assume the parallel postulate or any of its alternatives. The term was introduced by János Bolyai in 1832. [1] It is sometimes referred to as neutral geometry[2], as it is neutral with respect to the parallel postulate.

Relation to Other Geometries 

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The theorems of absolute geometry hold in some non-Euclidean geometries, such as hyperbolic geometry, as well as in Euclidean geometry.[3] Absolute geometry is inconsistent with elliptic geometry: in that theory, there are no parallel lines at all, so Euclid's parallel postulate can be immediately disproved; on the other hand, it is a theorem of absolute geometry that parallel lines do exist.[4] It might be imagined that absolute geometry is a rather weak system, but that is not the case. Indeed, in Euclid's Elements, the first 28 Propositions avoid using the parallel postulate, and therefore are valid in absolute geometry. One can also prove in absolute geometry the exterior angle theorem (an exterior angle of a triangle is larger than either of the remote angles), as well as the Saccheri-Legendre theorem, which states that a triangle has at most 180°

Incompleteness Absolute geometry is an incomplete axiomatic system, in the sense that one can add extra axioms without making the axiom system inconsistent. One can extend absolute geometry by adding different axioms about parallel lines and get incompatible but consistent axiom systems, giving rise to Euclidean, ordered and hyperbolic geometry. Thus every theorem of absolute geometry is a theorem of hyperbolic geometry, Euclidean geometry and ordered geometry. However the converse is not true.