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Elliptic geometry is an example of a
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
in which Euclid's parallel postulate does not hold. Instead, as in
spherical geometry 300px, A sphere with a spherical triangle on it. Spherical geometry is the geometry of the two-dimensional surface of a sphere. In this context the word "sphere" refers only to the 2-dimensional surface and other terms like "ball" or "solid sp ...
, 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 as ''single elliptic geometry'' whereas spherical geometry is sometimes referred to as ''double elliptic geometry''. The appearance of this geometry in the nineteenth century stimulated the development of non-Euclidean geometry generally, including
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P ...
. Elliptic geometry has a variety of properties that differ from those of classical Euclidean plane geometry. For example, the sum of the interior
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
s of any
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...
is always greater than 180°.


Definitions

In elliptic geometry, two lines
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...
to a given line must intersect. In fact, the perpendiculars on one side all intersect at a single point called the ''absolute pole'' of that line. The perpendiculars on the other side also intersect at a point. However, unlike in spherical geometry, the poles on either side are the same. This is because there are no
antipodal points In mathematics, antipodal points of a sphere are those diametrically opposite to each other (the specific qualities of such a definition are that a line drawn from the one to the other passes through the center of the sphere so forms a true ...
in elliptic geometry. For example, this is achieved in the hyperspherical model (described below) by making the "points" in our geometry actually be pairs of opposite points on a sphere. The reason for doing this is that it allows elliptic geometry to satisfy the axiom that there is a unique line passing through any two points. Every point corresponds to an ''absolute polar line'' of which it is the absolute pole. Any point on this polar line forms an ''absolute conjugate pair'' with the pole. Such a pair of points is ''orthogonal'', and the distance between them is a ''quadrant''. Duncan Sommerville (1914) ''The Elements of Non-Euclidean Geometry'', chapter 3 Elliptic geometry, pp 88 to 122,
George Bell & Sons George Bell & Sons was a book publishing house located in London, United Kingdom, from 1839 to 1986. History George Bell & Sons was founded by George Bell as an educational bookseller, with the intention of selling the output of London un ...
The distance between a pair of points is proportional to the angle between their absolute polars. As explained by
H. S. M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...
: :The name "elliptic" is possibly misleading. It does not imply any direct connection with the curve called an ellipse, but only a rather far-fetched analogy. A central conic is called an ellipse or a hyperbola according as it has no asymptote or two
asymptote In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related context ...
s. Analogously, a non-Euclidean plane is said to be elliptic or hyperbolic according as each of its
line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Art ...
s contains no
point at infinity In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line. In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. ...
or two points at infinity.


Two dimensions


Elliptic plane

The elliptic plane is the real projective plane provided with a
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathe ...
:
Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws o ...
and
Desargues Girard Desargues (; 21 February 1591 – September 1661) was a French mathematician and engineer, who is considered one of the founders of projective geometry. Desargues' theorem, the Desargues graph, and the crater Desargues on the Moo ...
used the
gnomonic projection A gnomonic map projection is a map projection which displays all great circles as straight lines, resulting in any straight line segment on a gnomonic map showing a geodesic, the shortest route between the segment's two endpoints. This is achi ...
to relate a plane σ to points on a
hemisphere Hemisphere refers to: * A half of a sphere As half of the Earth * A hemisphere of Earth ** Northern Hemisphere ** Southern Hemisphere ** Eastern Hemisphere ** Western Hemisphere ** Land and water hemispheres * A half of the (geocentric) celes ...
tangent to it. With ''O'' the center of the hemisphere, a point ''P'' in σ determines a line ''OP'' intersecting the hemisphere, and any line ''L'' ⊂ σ determines a plane ''OL'' which intersects the hemisphere in half of a
great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geome ...
. The hemisphere is bounded by a plane through O and parallel to σ. No ordinary line of ''σ'' corresponds to this plane; instead a line at infinity is appended to ''σ''. As any line in this extension of σ corresponds to a plane through ''O'', and since any pair of such planes intersects in a line through ''O'', 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. Given ''P'' and ''Q'' in ''σ'', the elliptic distance between them is the measure of the angle ''POQ'', usually taken in radians.
Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics. As a child, Cayley enjoyed solving complex maths problem ...
initiated the study of elliptic geometry when he wrote "On the definition of distance". This venture into abstraction in geometry was followed by Felix Klein and
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ...
leading to non-Euclidean geometry and Riemannian geometry.


Comparison with Euclidean 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 geometer measuring the geometrical properties of the space he or she inhabits can detect, via measurements, that there is a certain distance scale that is a property of the space. On scales much smaller than this one, the space is approximately flat, geometry is approximately Euclidean, and figures can be scaled up and down while remaining approximately similar. 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 the ''Elements'', 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. 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. 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.


Elliptic space (the 3D case)

''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: with
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es. One uses directed arcs on great circles of the sphere. As directed line segments are equipollent 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 of
William Rowan Hamilton Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the ...
: he envisioned a sphere as a domain of square roots of minus one. Then
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that ...
\exp(\theta r) = \cos \theta + r \sin \theta (where ''r'' is on the sphere) represents the
great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geome ...
in the plane containing 1 and ''r''. Opposite points ''r'' 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 , π) or (–π/2, π/2 For z = \exp(\theta r), \ z^* = \exp(-\theta r) \implies z z^* = 1 . It is said that the modulus or norm of ''z'' is one (Hamilton called it the tensor of z). But since ''r'' ranges over a sphere in 3-space, exp(θ r) ranges over a sphere in 4-space, now called the 3-sphere, as its surface has three dimensions. Hamilton called his algebra
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
s and it quickly became a useful and celebrated tool of mathematics. Its space of four dimensions is evolved in polar co-ordinates t \exp(\theta r), with ''t'' in the
positive real numbers In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used f ...
. When doing trigonometry on Earth or the
celestial sphere In astronomy and navigation, the celestial sphere is an abstract sphere that has an arbitrarily large radius and is concentric to Earth. All objects in the sky can be conceived as being projected upon the inner surface of the celestial sphe ...
, the sides of the triangles are great circle arcs. The first success of quaternions was a rendering of spherical trigonometry to algebra. Hamilton called a quaternion of norm one a versor, and these are the points of elliptic space. With fixed, the versors :e^, \quad 0 \le a < \pi form an ''elliptic line''. The distance from e^ to 1 is . For an arbitrary versor , the distance will be that θ for which 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 and are fixed versors. Distances between points are the same as between image points of an elliptic motion. In the case that and are quaternion conjugates of one another, the motion is a spatial rotation, and their vector part is the axis of rotation. In the case the elliptic motion is called a
right Rights are legal, social, or ethical principles of freedom or entitlement; that is, rights are the fundamental normative rules about what is allowed of people or owed to people according to some legal system, social convention, or ethical ...
''Clifford translation'', or a ''parataxy''. The case corresponds to left Clifford translation. ''Elliptic lines'' through versor  may be of the form :\lbrace u e^ : 0 \le a < \pi \rbrace or \lbrace e^u : 0 \le a < \pi \rbrace for a fixed . They are the right and left Clifford translations of  along an elliptic line through 1. The ''elliptic space'' is formed from by identifying antipodal points. Elliptic space has special structures called
Clifford parallel In elliptic geometry, two lines are Clifford parallel or paratactic lines if the perpendicular distance between them is constant from point to point. The concept was first studied by William Kingdon Clifford in elliptic space and appears only in s ...
s and Clifford surfaces. The versor points of elliptic space are mapped by the
Cayley transform In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by , the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform i ...
to ℝ3 for an alternative representation of the space.


Higher-dimensional spaces


Hyperspherical model

The hyperspherical model is the generalization of the spherical model to higher dimensions. The points of ''n''-dimensional elliptic space are the pairs of unit vectors in R''n''+1, that is, pairs of
antipodal points In mathematics, antipodal points of a sphere are those diametrically opposite to each other (the specific qualities of such a definition are that a line drawn from the one to the other passes through the center of the sphere so forms a true ...
on the surface of the unit ball in -dimensional space (the ''n''-dimensional hypersphere). Lines in this model are
great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geome ...
s, i.e., intersections of the hypersphere with flat hypersurfaces of dimension ''n'' passing through the origin.


Projective elliptic geometry

In the projective model of elliptic geometry, the points of ''n''-dimensional real projective space are used as points of the model. This models an abstract elliptic geometry that is also known as projective geometry. The points of ''n''-dimensional projective space can be identified with lines through the origin in -dimensional space, and can be represented non-uniquely by nonzero vectors in R''n''+1, with the understanding that and , for any non-zero scalar , represent the same point. Distance is defined using the metric :d(u, v) = \arccos \left(\frac\right); that is, the distance between two points is the angle between their corresponding lines in R''n''+1. The distance formula is homogeneous in each variable, with if and 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.


Stereographic model

A model representing the same space as the hyperspherical model can be obtained by means of
stereographic projection In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (the ''projection plane'') perpendicular to the diameter thro ...
. Let E''n'' represent that is, -dimensional real space extended by a single point at infinity. We may define a metric, the ''chordal metric'', on E''n'' by :\delta(u, v)=\frac where and are any two vectors in R''n'' and \, \cdot\, is the usual Euclidean norm. We also define :\delta(u, \infty)=\delta(\infty, u) = \frac. The result is a metric space on E''n'', 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 :d(u, v) = 2 \arcsin\left(\frac\right). Elliptic geometry is obtained from this by identifying the
antipodal points In mathematics, antipodal points of a sphere are those diametrically opposite to each other (the specific qualities of such a definition are that a line drawn from the one to the other passes through the center of the sphere so forms a true ...
and , and taking the distance from to this pair to be the minimum of the distances from to each of these two points.


Self-consistency

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 is
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
: there is an algorithm which, for every proposition, can show it to be either true or false. (This does not violate Gödel's theorem, because Euclidean geometry cannot describe a sufficient amount of
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
for the theorem to apply.Franzén 2005, pp. 25–26.) It therefore follows that elementary elliptic geometry is also self-consistent and complete.


See also

*
Elliptic tiling In geometry, a (globally) projective polyhedron is a tessellation of the real projective plane. These are projective analogs of spherical polyhedra – tessellations of the sphere – and toroidal polyhedra – tessellations of the toroids. Proje ...
* Spherical tiling


Notes


References

* Alan F. Beardon,
The Geometry of Discrete Groups
', Springer-Verlag, 1983 *
H. S. M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...
(1942) ''Non-Euclidean Geometry'', chapters 5, 6, & 7: Elliptic geometry in 1, 2, & 3 dimensions,
University of Toronto Press The University of Toronto Press is a Canadian university press founded in 1901. Although it was founded in 1901, the press did not actually publish any books until 1911. The press originally printed only examination books and the university cale ...
, reissued 1998 by
Mathematical Association of America The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure a ...
, . * H.S.M. Coxeter (1969) ''Introduction to Geometry'', §6.9 The Elliptic Plane, pp. 92–95.
John Wiley & Sons John Wiley & Sons, Inc., commonly known as Wiley (), is an American multinational publishing company founded in 1807 that focuses on academic publishing and instructional materials. The company produces books, journals, and encyclopedias, i ...
. * * Felix Klein (1871) "On the so-called noneuclidean geometry"
Mathematische Annalen ''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, ...
4:573–625, translated and introduced in John Stillwell (1996) ''Sources of Hyperbolic Geometry'',
American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meeting ...
. * Boris Odehna
"On isotropic congruences of lines in elliptic three-space"
* Eduard Study (1913) D.H. Delphenich translator
"Foundations and goals of analytical kinematics"
page 20. * Alfred Tarski (1951) ''A Decision Method for Elementary Algebra and Geometry''. Univ. of California Press. * *
Alfred North Whitehead Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He is best known as the defining figure of the philosophical school known as process philosophy, which today has found applica ...
(1898
Universal Algebra
, Book VI Chapter 2: Elliptic Geometry, pp 371–98.


External links

* {{Authority control Classical geometry Non-Euclidean geometry Metric geometry