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e angle between their absolute polars 1 101 as explained by h s m coxeter 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 asymptotes analogously a non euclidean plane is said to be elliptic or hyperbolic according as each of its lines contains no point at infinity or two points at infinity 2 two dimensions edit elliptic plane edit the elliptic plane is the real projective plane provided with a metric kepler and desargues used the gnomonic projection to relate a plane σ to points on a hemisphere 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 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 3 given p and q in σ the elliptic distance between them is the measure of the angle poq 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 by felix klein and bernhard riemann leading to non euclidean geometry and riemannian geometry comparison with euclidean geometry edit comparison of elliptic euclidean and hyperbolic geometries in two dimensions 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 displaystyle 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 edit 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 classes 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 he envisioned a sphere as a domain of square roots of minus one then euler s formula exp θ r cos θ r sin θ displaystyle exp theta r cos theta r sin theta where r is on the sphere represents the great circle 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 0 π or π 2 π 2 5 for z exp θ r z exp θ r z z 1 displaystyle z exp theta r z exp theta r implies zz 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 quaternions and it quickly became a useful and celebrated tool of mathematics its space of four dimensions is evolved in polar co ordinates t exp θ r displaystyle t exp theta r with t in the positive real numbers when doing trigonometry on earth or the celestial sphere the sides of the triangles are great circle arcs the first success of quaternions was a rendering of spherical trigonometry to algebra 6 hamilton called a quaternion of norm one a versor and these are the points of elliptic space with r fixed the versors e a r 0 a π displaystyle e ar quad 0 leq a pi form an elliptic line the distance from e a r displaystyle 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 u q v displaystyle q mapsto uqv where u and v are fixed versors distances between points are the same as between image points of an elliptic motion in the case that u and v are quaternion conjugates of one another the motion is a spatial rotation and their vector part is the axis of rotation in the case u 1 the elliptic motion is called a right clifford translation or a parataxy the case v 1 corresponds to left clifford translation elliptic lines through versor u may be of the form u e a r 0 a π displaystyle lbrace ue ar 0 leq a pi rbrace or e a r u 0 a π displaystyle lbrace e ar u 0 leq 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 from s 3 by identifying antipodal points 7 elliptic space has special structures called clifford parallels and clifford surfaces the versor points of elliptic space are mapped by the cayley transform to ℝ 3 for an alternative representation of the space higher dimensional spaces edit hyperspherical model edit 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 x x in r n 1 that is pairs of opposite points on the surface of the unit ball in n 1 dimensional space the n dimensional hypersphere lines in this model are great circles i e intersections of the hypersphere with flat hypersurfaces of dimension n passing through the origin projective elliptic geometry edit 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 n 1 dimensional space and can be represented non uniquely by nonzero vectors in r n 1 with the understanding that u and λ u for any non zero scalar λ represent the same point distance is defined using the metric d u v arccos u v u v displaystyle d u v arccos left frac u cdot v u v 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 d λ u μ v d u v if λ and μ 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 edit a model representing the same space as the hyperspherical model can be obtained by means of stereographic projection let e n represent r n that is n dimensional real space extended by a single point at infinity we may define a metric the chordal metric on e n by δ u v 2 u v 1 u 2 1 v 2 displaystyle delta u v frac 2 u v sqrt 1 u 2 1 v 2 where u and v are any two vectors in r n and displaystyle cdot is the usual euclidean norm we also define δ u δ u 2 1 u 2 displaystyle delta u infty delta infty u frac 2 sqrt 1 u 2 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 δ u v 2 displaystyle d u v 2 arcsin left frac delta u v 2 right elliptic geometry is obtained from this by identifying the points u and u and taking the distance from v to this pair to be the minimum of the distances from v to each of these two points self consistency edit 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 there is an algorithm which for every proposition can show it to be either true or false 8 this does not violate gödel s theorem because euclidean geometry cannot describe a sufficient amount of arithmetic for the theorem to apply 9 it therefore follows that elementary elliptic geometry is also self consistent and complete see also edit elliptic tiling spherical tiling notes edit a b duncan sommerville 1914 the elements of non euclidean geometry chapter 3 elliptic geometry pp 88 to 122 george bell sons coxeter 1969 94 h s m coxeter 1965 introduction to geometry page 92 cayley arthur 1859 a sixth memoir upon quantics pdf philosophical transactions of the royal society of london 149 61 90 doi 10 1098 rstl 1859 0004 issn 0080 4614 jstor 108690 rafael artzy 1965 linear geometry chapter 3 8 quaternions and elliptic three space pp 186 94 addison wesley w r hamilton 1844 to 1850 on quaternions or a new system of imaginaries in algebra philosophical magazine link to david r wilkins collection at trinity college dublin lemaître georges quaternions et espace elliptique pontificia academia scientiarum acta 12 57 78 issn 0370 2138 tarski 1951 franzén 2005 pp 25 26 references edit alan f beardon the geometry of discrete groups springer verlag 1983 h s m coxeter 1942 non euclidean geometry chapters 5 6 7 elliptic geometry in 1 2 3 dimensions university of toronto press reissued 1998 by mathematical association of america isbn 0 88385 522 4 h s m coxeter 1969 introduction to geometry 6 9 the elliptic plane pp 92 95 john wiley sons elliptic geometry encyclopedia of mathematics ems press 2001 1994 felix klein 1871 on the so called noneuclidean geometry mathematische annalen 4 573 625 translated and introduced in john stillwell 1996 sources of hyperbolic geometry american mathematical society isbn 0 8218 0529 0 boris odehnal 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 franzén torkel 2005 gödel s theorem an incomplete guide to its use and abuse ak peters isbn 1 56881 238 8 alfred north whitehead 1898 universal algebra archived 2014 09 03 at the wayback machine book vi chapter 2 elliptic geometry pp 371 98 external links edit media related to elliptic geometry at wikimedia commons authority control national libraries germany retrieved 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