Elliptic geometry is an example of ageometry in which Euclid'sparallel postulate does not hold. Instead, as inspherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point (rather than two). Because of this, the elliptic geometry described in this article is sometimes referred to assingle elliptic geometry whereas spherical geometry is sometimes referred to asdouble elliptic geometry.
The appearance of this geometry in the nineteenth century stimulated the development ofnon-Euclidean geometry generally, includinghyperbolic geometry.
Elliptic geometry has a variety of properties that differ from those of classical Euclidean plane geometry. For example, the sum of the interiorangles of anytriangle is always greater than 180°.
Elliptic geometry may be derived fromspherical geometry by identifyingantipodal points of the sphere to a single elliptic point. The elliptic lines correspond togreat circles reduced by the identification of antipodal points. As any two great circles intersect, there are no parallel lines in elliptic geometry.
In elliptic geometry, two linesperpendicular to a given line must intersect. In fact, all perpendiculars to a given line intersect at a single point called theabsolute pole of that line.
Every point corresponds to anabsolute polar line of which it is the absolute pole. Any point on this polar line forms anabsolute conjugate pair with the pole. Such a pair of points isorthogonal, and the distance between them is aquadrant.[1]: 89
Thedistance between a pair of points is proportional to the angle between their absolute polars.[1]: 101
As explained byH. S. M. Coxeter:
The elliptic plane is thereal projective plane provided with ametric.Kepler andDesargues used thegnomonic projection to relate a plane σ to points on ahemisphere tangent to it. WithO the center of the hemisphere, a pointP in σ determines a lineOP intersecting the hemisphere, and any lineL ⊂ σ determines a planeOL which intersects the hemisphere in half of agreat circle. The hemisphere is bounded by a plane through O and parallel to σ. No ordinary line ofσ corresponds to this plane; instead aline at infinity is appended toσ. As any line in this extension of σ corresponds to a plane throughO, and since any pair of such planes intersects in a line throughO, one can conclude that any pair of lines in the extension intersect: the point of intersection lies where the plane intersection meets σ or the line at infinity. Thus the axiom of projective geometry, requiring all pairs of lines in a plane to intersect, is confirmed.[3]
GivenP andQ inσ, theelliptic distance between them is the measure of the anglePOQ, usually taken in radians.Arthur Cayley initiated the study of elliptic geometry when he wrote "On the definition of distance".[4]: 82 This venture into abstraction in geometry was followed byFelix Klein andBernhard Riemann leading tonon-Euclidean geometry andRiemannian geometry.
In Euclidean geometry, a figure can be scaled up or scaled down indefinitely, and the resulting figures are similar, i.e., they have the same angles and the same internal proportions. In elliptic geometry, this is not the case. For example, in the spherical model we can see that the distance between any two points must be strictly less than half the circumference of the sphere (because antipodal points are identified). A line segment therefore cannot be scaled up indefinitely.
A great deal of Euclidean geometry carries over directly to elliptic geometry. 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. Postulate 3, that one can construct a circle with any given center and radius, fails if "any radius" is taken to mean "any real number", but holds if it is taken to mean "the length of any given line segment". Therefore any result in Euclidean geometry that follows from these three postulates will hold in elliptic geometry, such as proposition 1 from book I of theElements, which states that given any line segment, an equilateral triangle can be constructed with the segment as its base.
Elliptic geometry is also like Euclidean geometry in that space is continuous, homogeneous, isotropic, and without boundaries. Isotropy is guaranteed by the fourth postulate, that all right angles are equal. For an example of homogeneity, note that Euclid's proposition I.1 implies that the same equilateral triangle can be constructed at any location, not just in locations that are special in some way. The lack of boundaries follows from the second postulate, extensibility of a line segment.
One way in which elliptic geometry differs from Euclidean geometry is that the sum of the interior angles of a triangle is greater than 180 degrees. In the spherical model, for example, a triangle can be constructed with vertices at the locations where the three positive Cartesian coordinate axes intersect the sphere, and all three of its internal angles are 90 degrees, summing to 270 degrees. For sufficiently small triangles, the excess over 180 degrees can be made arbitrarily small.
ThePythagorean 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. The Pythagorean result is recovered in the limit of small triangles.
The ratio of a circle's circumference to its area is smaller than in Euclidean geometry. In general, area and volume do not scale as the second and third powers of linear dimensions.
Note: This section uses the term "elliptic space" to refer specifically to 3-dimensional elliptic geometry. This is in contrast to the previous section, which was about 2-dimensional elliptic geometry. The quaternions are used to elucidate this space.
Elliptic space can be constructed in a way similar to the construction of three-dimensional vector space: withequivalence classes. One uses directed arcs on great circles of the sphere. As directed line segments areequipollent when they are parallel, of the same length, and similarly oriented, so directed arcs found on great circles are equipollent when they are of the same length, orientation, and great circle. These relations of equipollence produce 3D vector space and elliptic space, respectively.
Access to elliptic space structure is provided through the vector algebra ofWilliam Rowan Hamilton: he envisioned a sphere as a domain of square roots of minus one. ThenEuler's formula (wherer is on the sphere) represents thegreat circle in the plane containing 1 andr. Opposite pointsr and –r correspond to oppositely directed circles. An arc between θ and φ is equipollent with one between 0 and φ – θ. In elliptic space, arc length is less than π, so arcs may be parametrized with θ in [0, π) or (–π/2, π/2].[5]
For It is said that the modulus or norm ofz is one (Hamilton called it the tensor of z). But sincer ranges over a sphere in 3-space, exp(θ r) ranges over a sphere in 4-space, now called the3-sphere, as its surface has three dimensions. Hamilton called his algebraquaternions and it quickly became a useful and celebrated tool of mathematics. Its space of four dimensions is evolved in polar co-ordinates witht in thepositive real numbers.
When doing trigonometry on Earth or thecelestial sphere, the sides of the triangles are great circle arcs. The first success of quaternions was a rendering ofspherical trigonometry to algebra.[6] Hamilton called a quaternion of norm one aversor, and these are the points of elliptic space.
Withr fixed, the versors
form anelliptic line. The distance from to 1 isa. For an arbitrary versor u, the distance will be that θ for whichcos θ = (u +u∗)/2 since this is the formula for the scalar part of any quaternion.
Anelliptic motion is described by the quaternion mapping
Distances between points are the same as between image points of an elliptic motion. In the case thatu andv are quaternion conjugates of one another, the motion is aspatial rotation, and their vector part is the axis of rotation. In the caseu = 1 the elliptic motion is called arightClifford translation, or aparataxy. The casev = 1 corresponds to left Clifford translation.
Elliptic lines through versor u may be of the form
They are the right and left Clifford translations of u along an elliptic line through 1.Theelliptic space is formed fromS3 by identifying antipodal points.[7]
Elliptic space has special structures calledClifford parallels andClifford surfaces.
The versor points of elliptic space are mapped by theCayley transform to for an alternative representation of the space.
The hyperspherical model is the generalization of the spherical model to higher dimensions. The points ofn-dimensional elliptic space are the pairs of unit vectors(x, −x) inRn+1, that is, pairs ofantipodal points on the surface of the unit ball in(n + 1)-dimensional space (then-dimensional hypersphere). Lines in this model aregreat circles, i.e., intersections of the hypersphere with flat hypersurfaces of dimensionn passing through the origin.
In the projective model of elliptic geometry, the points ofn-dimensionalreal projective space are used as points of the model. This models an abstract elliptic geometry that is also known asprojective geometry.
The points ofn-dimensional projective space can be identified with lines through the origin in(n + 1)-dimensional space, and can be represented non-uniquely by nonzero vectors inRn+1, with the understanding thatu andλu, for any non-zero scalar λ, represent the same point. Distance is defined using the metric
that is, the distance between two points is the angle between their corresponding lines inRn+1. The distance formula is homogeneous in each variable, withd(λu, μv) =d(u, v) ifλ andμ are non-zero scalars, so it does define a distance on the points of projective space.
A notable property of the projective elliptic geometry is that for even dimensions, such as the plane, the geometry is non-orientable. It erases the distinction between clockwise and counterclockwise rotation by identifying them.
A model representing the same space as the hyperspherical model can be obtained by means ofstereographic projection. LetEn representRn ∪ {∞}, that is,n-dimensional real space extended by a single point at infinity. We may define a metric, thechordal metric, onEn by
whereu andv are any two vectors inRn and is the usual Euclidean norm. We also define
The result is a metric space onEn, which represents the distance along a chord of the corresponding points on the hyperspherical model, to which it maps bijectively by stereographic projection. We obtain a model of spherical geometry if we use the metric
Elliptic geometry is obtained from this by identifying theantipodal pointsu and−u / ‖u‖2, and taking the distance fromv to this pair to be the minimum of the distances fromv to each of these two points.
Because spherical elliptic geometry can be modeled as, for example, a spherical subspace of a Euclidean space, it follows that if Euclidean geometry is self-consistent, so is spherical elliptic geometry. Therefore it is not possible to prove the parallel postulate based on the other four postulates of Euclidean geometry.
Tarski proved that elementary Euclidean geometry iscomplete: there is an algorithm which, for every proposition, can show it to be either true or false.[8] (This does not violateGödel's theorem, because Euclidean geometry cannot describe a sufficient amount ofarithmetic for the theorem to apply.[9]) It therefore follows that elementary elliptic geometry is also self-consistent and complete.
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