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pheres surfaces with constant negative gaussian curvature can be embedded in 3 dimensional space under the standard euclidean metric and so can be made into tangible physical models of these the tractoid often called the pseudosphere is the best known using the tractoid as a model of the hyperbolic plane is analogous to using a cone or cylinder as a model of the euclidean plane however the entire hyperbolic plane cannot be embedded into euclidean space in this way and various other models are more convenient for abstractly exploring hyperbolic geometry there are four models commonly used for hyperbolic geometry the klein model the poincaré disk model the poincaré half plane model and the lorentz or hyperboloid model these models define a hyperbolic plane which satisfies the axioms of a hyperbolic geometry despite their names the first three mentioned above were introduced as models of hyperbolic space by beltrami not by poincaré or klein all these models are extendable to more dimensions the beltrami klein model edit main article beltrami klein model the beltrami klein model also known as the projective disk model klein disk model and klein model is named after eugenio beltrami and felix klein for the two dimensions this model uses the interior of the unit circle for the complete hyperbolic plane and the chords of this circle are the hyperbolic lines for higher dimensions this model uses the interior of the unit ball and the chords of this n ball are the hyperbolic lines this model has the advantage that lines are straight but the disadvantage that angles are distorted the mapping is not conformal and also circles are not represented as circles the distance in this model is half the logarithm of the cross ratio which was introduced by arthur cayley in projective geometry the poincaré disk model edit poincaré disk model with truncated triheptagonal tiling main article poincaré disk model the poincaré disk model also known as the conformal disk model also employs the interior of the unit circle but lines are represented by arcs of circles that are orthogonal to the boundary circle plus diameters of the boundary circle this model preserves angles and is thereby conformal all isometries within this model are therefore möbius transformations circles entirely within the disk remain circles although the euclidean center of the circle is closer to the center of the disk than is the hyperbolic center of the circle horocycles are circles within the disk which are tangent to the boundary circle minus the point of contact hypercycles are open ended chords and circular arcs within the disc that terminate on the boundary circle at non orthogonal angles the poincaré half plane model edit main article poincaré half plane model the poincaré half plane model takes one half of the euclidean plane bounded by a line b of the plane to be a model of the hyperbolic plane the line b is not included in the model the euclidean plane may be taken to be a plane with the cartesian coordinate system and the x axis is taken as line b and the half plane is the upper half y 0 of this plane hyperbolic lines are then either half circles orthogonal to b or rays perpendicular to b the length of an interval on a ray is given by logarithmic measure so it is invariant under a homothetic transformation x y λ x λ y λ 0 displaystyle x y mapsto lambda x lambda y quad lambda 0 like the poincaré disk model this model preserves angles and is thus conformal all isometries within this model are therefore möbius transformations of the plane the half plane model is the limit of the poincaré disk model whose boundary is tangent to b at the same point while the radius of the disk model goes to infinity the hyperboloid model edit main article hyperboloid model the hyperboloid model or lorentz model employs a 2 dimensional hyperboloid of revolution of two sheets but using one embedded in 3 dimensional minkowski space this model is generally credited to poincaré but reynolds 33 says that wilhelm killing used this model in 1885 this model has direct application to special relativity as minkowski 3 space is a model for spacetime suppressing one spatial dimension one can take the hyperboloid to represent the events positions in spacetime that various inertially moving observers starting from a common event will reach in a fixed proper time the hyperbolic distance between two points on the hyperboloid can then be identified with the relative rapidity between the two corresponding observers the model generalizes directly to an additional dimension a hyperbolic 3 space three dimensional hyperbolic geometry relates to minkowski 4 space the hemisphere model edit the hemisphere model is not often used as model by itself but it functions as a useful tool for visualising transformations between the other models the hemisphere model uses the upper half of the unit sphere x 2 y 2 z 2 1 z 0 displaystyle x 2 y 2 z 2 1 z 0 the hyperbolic lines are half circles orthogonal to the boundary of the hemisphere the hemisphere model is part of a riemann sphere and different projections give different models of the hyperbolic plane stereographic projection from 0 0 1 displaystyle 0 0 1 onto the plane z 0 displaystyle z 0 projects corresponding points on the poincaré disk model stereographic projection from 0 0 1 displaystyle 0 0 1 onto the surface x 2 y 2 z 2 1 z 0 displaystyle x 2 y 2 z 2 1 z 0 projects corresponding points on the hyperboloid model stereographic projection from 1 0 0 displaystyle 1 0 0 onto the plane x 1 displaystyle x 1 projects corresponding points on the poincaré half plane model orthographic projection onto a plane z c displaystyle z c projects corresponding points on the beltrami klein model central projection from the centre of the sphere onto the plane z 1 displaystyle z 1 projects corresponding points on the gans model see further connection between the models below the gans model edit in 1966 david gans proposed a flattened hyperboloid model in the journal american mathematical monthly 34 it is an orthographic projection of the hyperboloid model onto the xy plane this model is not as widely used as other models but nevertheless is quite useful in the understanding of hyperbolic geometry unlike the klein or the poincaré models this model utilizes the entire euclidean plane the lines in this model are represented as branches of a hyperbola 35 the band model edit main article band model the band model employs a portion of the euclidean plane between two parallel lines 36 distance is preserved along one line through the middle of the band assuming the band is given by z c im z π 2 displaystyle z in mathbb c operatorname im z pi 2 the metric is given by d z sec im z displaystyle dz sec operatorname im z connection between the models edit poincaré disk hemispherical and hyperboloid models are related by stereographic projection from 1 beltrami klein model is orthographic projection from hemispherical model poincaré half plane model here projected from the hemispherical model by rays from left end of poincaré disk model all models essentially describe the same structure the difference between them is that they represent different coordinate charts laid down on the same metric space namely the hyperbolic plane the characteristic feature of the hyperbolic plane itself is that it has a constant negative gaussian curvature which is indifferent to the coordinate chart used the geodesics are similarly invariant that is geodesics map to geodesics under coordinate transformation hyperbolic geometry generally is introduced in terms of the geodesics and their intersections on the hyperbolic plane 37 once we choose a coordinate chart one of the models we can always embed it in a euclidean space of same dimension but the embedding is clearly not isometric since the curvature of euclidean space is 0 the hyperbolic space can be represented by infinitely many different charts but the embeddings in euclidean space due to these four specific charts show some interesting characteristics since the four models describe the same metric space each can be transformed into the other see for example the beltrami klein model s relation to the hyperboloid model the beltrami klein model s relation to the poincaré disk model and the poincaré disk model s relation to the hyperboloid model isometries of the hyperbolic plane edit see also hyperbolic motion and transformation geometry every isometry transformation or motion of the hyperbolic plane to itself can be realized as the composition of at most three reflections in n dimensional hyperbolic space up to n 1 reflections might be required these are also true for euclidean and spherical geometries but the classification below is different all the isometries of the hyperbolic plane can be classified into these classes orientation preserving the identity isometry nothing moves zero reflections zero degrees of freedom inversion through a point half turn two reflections through mutually perpendicular lines passing through the given point i e a rotation of 180 degrees around the point two degrees of freedom rotation around a normal point two reflections through lines passing through the given point includes inversion as a special case points move on circles around the center three degrees of freedom rotation around an ideal point horolation two reflections through lines leading to the ideal point points move along horocycles centered on the ideal point two degrees of freedom translation along a straight line two reflections through lines perpendicular to the given line points off the given line move along hypercycles three degrees of freedom orientation reversing reflection through a line one reflection two degrees of freedom combined reflection through a line and translation along the same line the reflection and translation commute three reflections required three degrees of freedom citation needed hyperbolic geometry in art edit m c escher s famous prints circle limit iii and circle limit iv illustrate the conformal disc model poincaré disk model quite well the white lines in iii are not quite geodesics they are hypercycles but are close to them it is also possible to see quite plainly the negative curvature of the hyperbolic plane through its effect on the sum of angles in triangles and squares for example in circle limit iii every vertex belongs to three triangles and three squares in the euclidean plane their angles would sum to 450 i e a circle and a quarter from this we see that the sum of angles of a triangle in the hyperbolic plane must be smaller than 180 another visible property is exponential growth in circle limit iii for example one can see that the number of fishes within a distance of n from the center rises exponentially the fishes have an equal hyperbolic area so the area of a ball of radius n must rise exponentially in n the art of crochet has been used to demonstrate hyperbolic planes pictured above with the first being made by daina taimiņa 28 whose book crocheting adventures with hyperbolic planes won the 2009 bookseller diagram prize for oddest title of the year 38 hyperrogue is a roguelike game set on various tilings of the hyperbolic plane higher dimensions edit main article hyperbolic space hyperbolic geometry is not limited to 2 dimensions a hyperbolic geometry exists for every higher number of dimensions homogeneous structure edit hyperbolic space of dimension n is a special case of a riemannian symmetric space of noncompact type as it is isomorphic to the quotient o 1 n o 1 o n displaystyle mathrm o 1 n mathrm o 1 times mathrm o n the orthogonal group o 1 n acts by norm preserving transformations on minkowski space r 1 n and it acts transitively on the two sheet hyperboloid of norm 1 vectors timelike lines i e those with positive norm tangents through the origin pass through antipodal points in the hyperboloid so the space of such lines yields a model of hyperbolic n space the stabilizer of any particular line is isomorphic to the product of the orthogonal groups o n and o 1 where o n acts on the tangent space of a point in the hyperboloid and o 1 reflects the line through the origin many of the elementary concepts in hyperbolic geometry can be described in linear algebraic terms geodesic paths are described by intersections with planes through the origin dihedral angles between hyperplanes can be described by inner products of normal vectors and hyperbolic reflection groups can be given explicit matrix realizations in small dimensions there are exceptional isomorphisms of lie groups that yield additional ways to consider symmetries of hyperbolic spaces for example in dimension 2 the isomorphisms so 1 2 psl 2 r psu 1 1 allow one to interpret the upper half plane model as the quotient sl 2 r so 2 and the poincaré disc model as the quotient su 1 1 u 1 in both cases the symmetry groups act by fractional linear transformations since both groups are the orientation preserving stabilizers in pgl 2 c of the respective subspaces of the riemann sphere the cayley transformation not only takes one model of the hyperbolic plane to the other but realizes the isomorphism of symmetry groups as conjugation in a larger group in dimension 3 the fractional linear action of pgl 2 c on the riemann sphere is identified with the action on the conformal boundary of hyperbolic 3 space induced by the isomorphism o 1 3 pgl 2 c this allows one to study isometries of hyperbolic 3 space by considering spectral properties of representative complex matrices for example parabolic transformations are conjugate to rigid translations in the upper half space model and they are exactly those transformations that can be represented by unipotent upper triangular matrices see also edit constructions in hyperbolic geometry hyperbolic 3 manifold hyperbolic manifold hyperbolic set hjelmslev transformation hyperbolic tree kleinian group lambert quadrilateral open universe poincaré metric saccheri quadrilateral systolic geometry uniform tilings in hyperbolic plane δ hyperbolic space band model notes edit a b curvature of curves on the hyperbolic plane math stackexchange retrieved 24 september 2017 hyde s t ramsden s 2003 some novel three dimensional euclidean crystalline networks derived from two dimensional hyperbolic tilings the european physical journal b 31 2 273 284 bibcode 2003epjb 31 273h citeseerx 10 1 1 720 5527 doi 10 1140 epjb e2003 00032 8 s2cid 41146796 a b sommerville d m y 2005 the elements of non euclidean geometry unabr and unaltered republ ed mineola n y dover publications p 58 isbn 0 486 44222 5 ramsay arlan richtmyer robert d 1995 introduction to hyperbolic geometry new york springer verlag pp 97 103 isbn 0387943390 see for instance omar khayyam 1048 1131 archived from the original on 2007 09 28 retrieved 2008 01 05 non euclidean geometry seminar math columbia edu retrieved 21 january 2018 boris a rosenfeld and adolf p youschkevitch 1996 geometry in ro...
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