In mathematics, non-Euclidean geometry consists of two geometries based on axiomsclosely related to those specifying Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises when either the metric requirement is relaxed, or the parallel postulate is replaced with an alternative one. In the latter case one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras which give rise to kinematic geometries that have also been called non-Euclidean geometry.
The essential difference between the metric geometries 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.
Another way to describe the differences between these geometries is to 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 (meaning that a line drawn perpendicular to one line at any point will intersect the other line and the length of the line segment joining the points of intersection remains constant) 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 intersect.
Axiomatic basis of non-Euclidean geometry
Euclidean geometry can be axiomatically described in several ways. Unfortunately, Euclid’s original system of five postulates (axioms) is not one of these as his proofs relied on several unstated assumptions which should also have been taken as axioms. Hilbert’s system consisting of 20 axiomsmost closely follows the approach of Euclid and provides the justification for all of Euclid’s proofs. Other systems, using different sets of undefined terms obtain the same geometry by different paths. In all approaches, however, there is an axiom which is logically equivalent to Euclid’s fifth postulate, the parallel postulate. Hilbert uses the Playfair axiom form, while Birkhoff, for instance, uses the axiom which says that “there exists a pair of similar but not congruent triangles.” In any of these systems, removal of the one axiom which is equivalent to the parallel postulate, in whatever form it takes, and leaving all the other axioms intact, produces absolute geometry. As the first 28 propositions of Euclid (in The Elements) do not require the use of the parallel postulate or anything equivalent to it, they are all true statements in absolute geometry.
To obtain a non-Euclidean geometry, the parallel postulate (or its equivalent) must be replaced by its negation. Negating the Playfair’s axiom form, since it is a compound statement (… there exists one and only one …), can be done in two ways:
- Either there will exist more than one line through the point parallel to the given line or there will exist no lines through the point parallel to the given line. In the first case, replacing the parallel postulate (or its equivalent) with the statement “In a plane, given a point P and a line ℓ not passing through P, there exist two lines through P which do not meet ℓ” and keeping all the other axioms, yields hyperbolic geometry.
- The second case is not dealt with as easily. Simply replacing the parallel postulate with the statement, “In a plane, given a point P and a line ℓ not passing through P, all the lines through P meet ℓ”, does not give a consistent set of axioms. This follows since parallel lines exist in absolute geometry, but this statement says that there are no parallel lines. This problem was known (in a different guise) to Khayyam, Saccheri and Lambert and was the basis for their rejecting what was known as the “obtuse angle case”. In order to obtain a consistent set of axioms which includes this axiom about having no parallel lines, some of the other axioms must be tweaked. The adjustments to be made depend upon the axiom system being used. Among others these tweaks will have the effect of modifying Euclid’s second postulate from the statement that line segments can be extended indefinitely to the statement that lines are unbounded. Riemann’s elliptic geometry emerges as the most natural geometry satisfying this axiom.
Non-Euclidean geometry - Wikipedia
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying…
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同論文の定理 9 および定理 15 により、各仮定をより分かりやすく言い換えるなら次の通りである。
鋭角仮定三角形の内角の和は 2 直角よりも小さい直角仮定三角形の内角の和は 2 直角に等しい鈍角仮定三角形の内角の和は 2 直角よりも大きい
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Models of non-Euclidean geometry
For more details on this topic, see Models of non-Euclidean geometry.
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°.
Main article: 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 (called antipodal points) are identified (considered to be the same). This is also one of the standard models of the real projective plane. The difference is that as a model of elliptic geometry a metric is introduced permitting the measurement of lengths and angles, while as a model of the projective plane there is no such metric.
In the elliptic model, for any given line ℓ and a point A, which is not on ℓ, all lines through A will intersect ℓ.
Main article: 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 which models 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 and only if Euclidean geometry was. (The reverse implication follows from the horosphere model of Euclidean geometry.)
In the hyperbolic model, within a two-dimensional plane, for any given line ℓ and a point A, which is not on ℓ, there are infinitely many lines through Athat do not intersect ℓ.
In these models the concepts of non-Euclidean geometries are being represented by Euclidean objects in a Euclidean setting. This introduces a perceptual distortion wherein the straight lines of the non-Euclidean geometry are being represented by Euclidean curves which visually bend. This “bending” is not a property of the non-Euclidean lines, only an artifice of the way they are being represented.