Elliptic geometry

From Wikipedia, the free encyclopedia
  (Redirected from Elliptical geometry)
Jump to: navigation, search

Elliptic 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 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°.



The distance between a pair of points is proportional to the angle between their absolute polars.[2]

As explained by H. S. M. Coxeter


Two dimensions[edit]

The spherical model[edit]


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 general 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 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[edit]





The Pythagorean theorem fails in elliptic geometry. In the 90°–90°–90° triangle described above, all three sides have the same length, and consequently do not satisfy a^2+b^2=c^2. The Pythagorean result is recovered in the limit of small triangles.


Elliptic space[edit]

The three-dimensional elliptic geometry makes use of the 3-sphere S3, and these points are well-accessed with the versors in the theory of quaternions. A versor is a quaternion of norm one, which must necessarily have the form

e^{ar} = \cos a + r \sin a , \quad r^2 = -1.

The origin corresponds to a = 0 and is the identity of the topological group consisting of versors. With r fixed, the versors

e^{ar}, \quad 0 \le a < \pi

form an elliptic line. The distance from e^{ar} to 1 is a. For an arbitrary versor u, the distance will be that θ for which cos θ = (u + u)/2 since this is the formula for the scalar part of any quaternion.

An elliptic motion is described by the quaternion mapping

q \mapsto u q v, where u and v are fixed versors.


Elliptic lines through versor u may be of the form

\lbrace u e^{ar} : 0 \le a < \pi \rbrace or \lbrace  e^{ar}u : 0 \le a < \pi \rbrace for a fixed r.

They are the right and left Clifford translations of u along an elliptic line through 1. The elliptic space is formed by identifying antipodal points on S3.[4]

Elliptic space has special structures called Clifford parallels and Clifford surfaces.

Higher-dimensional spaces[edit]

Hyperspherical model[edit]


Projective elliptic geometry[edit]



d(u, v) = \arccos \left(\frac{|u \cdot v|}{\|u\|\  \|v\|}\right);



Stereographic model[edit]


\delta(u, v)=\frac{2 \|u-v\|}{\sqrt{(1+\|u\|^2)(1+\|v\|^2)}}

where u and v are any two vectors in Rn and \|\cdot\| is the usual Euclidean norm. We also define

\delta(u, \infty)=\delta(\infty, u) = \frac{2}{\sqrt{1+\|u\|^2}}.


d(u, v) = 2 \arcsin\left(\frac{\delta(u,v)}{2}\right).




Tarski proved that elementary Euclidean geometry is complete: there is an algorithm which, for every proposition, can show it to be either true or false.[5] (This does not violate Gödel's theorem, because Euclidean geometry cannot describe a sufficient amount of arithmetic for the theorem to apply.[6]) It therefore follows that elementary elliptic geometry is also self-consistent and complete.

See also[edit]


  1. ^ Sommerville 1914 p 89
  2. ^ Sommerville 1914 p 101
  3. ^ Coxeter 1969 94
  4. ^ Coxeter 1950 Synopsis of Lemaitre
  5. ^ Tarski (1951)
  6. ^ Franzén 2005, pp. 25–26.