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ith for any given line r and point p not on r in the plane containing both line r and point p there are at least two distinct lines through p that do not intersect r compare the above with playfair s axiom the modern version of euclid s parallel postulate hyperbolic plane geometry is also the geometry of pseudospherical surfaces surfaces with a constant negative gaussian curvature saddle surfaces have negative gaussian curvature in at least some regions where they locally resemble the hyperbolic plane a modern use of hyperbolic geometry is in the theory of special relativity particularly the minkowski model when geometers first realised they were working with something other than the standard euclidean geometry they described their geometry under many different names felix klein finally gave the subject the name hyperbolic geometry to include it in the now rarely used sequence elliptic geometry spherical geometry parabolic geometry euclidean geometry and hyperbolic geometry in the former soviet union it is commonly called lobachevskian geometry named after one of its discoverers the russian geometer nikolai lobachevsky this page is mainly about the 2 dimensional planar hyperbolic geometry and the differences and similarities between euclidean and hyperbolic geometry see hyperbolic space for more information on hyperbolic geometry extended to three and more dimensions contents 1 properties 1 1 relation to euclidean geometry 1 2 lines 1 2 1 non intersecting parallel lines 1 3 circles and disks 1 4 hypercycles and horocycles 1 5 triangles 1 6 regular apeirogon 1 7 tessellations 2 standardized gaussian curvature 2 1 cartesian like coordinate systems 2 2 distance 3 history 3 1 19th century developments 3 2 philosophical consequences 3 3 geometry of the universe spatial dimensions only 3 4 geometry of the universe special relativity 4 physical realizations of the hyperbolic plane 5 models of the hyperbolic plane 5 1 the beltrami klein model 5 2 the poincaré disk model 5 3 the poincaré half plane model 5 4 the hyperboloid model 5 5 the hemisphere model 5 6 the gans model 5 7 the band model 5 8 connection between the models 6 isometries of the hyperbolic plane 7 hyperbolic geometry in art 8 higher dimensions 9 homogeneous structure 10 see also 11 notes 12 references 13 external links properties edit relation to euclidean geometry edit comparison of elliptic euclidean and hyperbolic geometries in two dimensions hyperbolic geometry is more closely related to euclidean geometry than it seems the only axiomatic difference is the parallel postulate when the parallel postulate is removed from euclidean geometry the resulting geometry is absolute geometry there are two kinds of absolute geometry euclidean and hyperbolic all theorems of absolute geometry including the first 28 propositions of book one of euclid s elements are valid in euclidean and hyperbolic geometry propositions 27 and 28 of book one of euclid s elements prove the existence of parallel non intersecting lines this difference also has many consequences concepts that are equivalent in euclidean geometry are not equivalent in hyperbolic geometry new concepts need to be introduced further because of the angle of parallelism hyperbolic geometry has an absolute scale a relation between distance and angle measurements lines edit single lines in hyperbolic geometry have exactly the same properties as single straight lines in euclidean geometry for example two points uniquely define a line and line segments can be infinitely extended two intersecting lines have the same properties as two intersecting lines in euclidean geometry for example two distinct lines can intersect in no more than one point intersecting lines form equal opposite angles and adjacent angles of intersecting lines are supplementary when a third line is introduced then there can be properties of intersecting lines that differ from intersecting lines in euclidean geometry for example given two intersecting lines there are infinitely many lines that do not intersect either of the given lines these properties are all independent of the model used even if the lines may look radically different non intersecting parallel lines edit lines through a given point p and asymptotic to line r non intersecting lines in hyperbolic geometry also have properties that differ from non intersecting lines in euclidean geometry for any line r and any point p which does not lie on r in the plane containing line r and point p there are at least two distinct lines through p that do not intersect r this implies that there are through p an infinite number of coplanar lines that do not intersect r these non intersecting lines are divided into two classes two of the lines x and y in the diagram are limiting parallels sometimes called critically parallel horoparallel or just parallel there is one in the direction of each of the ideal points at the ends of r asymptotically approaching r always getting closer to r but never meeting it all other non intersecting lines have a point of minimum distance and diverge from both sides of that point and are called ultraparallel diverging parallel or sometimes non intersecting some geometers simply use the phrase parallel lines to mean limiting parallel lines with ultraparallel lines meaning just non intersecting these limiting parallels make an angle θ with pb this angle depends only on the gaussian curvature of the plane and the distance pb and is called the angle of parallelism for ultraparallel lines the ultraparallel theorem states that there is a unique line in the hyperbolic plane that is perpendicular to each pair of ultraparallel lines circles and disks edit in hyperbolic geometry the circumference of a circle of radius r is greater than 2 π r displaystyle 2 pi r let r 1 k displaystyle r frac 1 sqrt k where k displaystyle k is the gaussian curvature of the plane in hyperbolic geometry k displaystyle k is negative so the square root is of a positive number then the circumference of a circle of radius r is equal to 2 π r sinh r r displaystyle 2 pi r sinh frac r r and the area of the enclosed disk is 4 π r 2 sinh 2 r 2 r 2 π r 2 cosh r r 1 displaystyle 4 pi r 2 sinh 2 frac r 2r 2 pi r 2 left cosh frac r r 1 right therefore in hyperbolic geometry the ratio of a circle s circumference to its radius is always strictly greater than 2 π displaystyle 2 pi though it can be made arbitrarily close by selecting a small enough circle if the gaussian curvature of the plane is 1 then the geodesic curvature of a circle of radius r is 1 tanh r displaystyle frac 1 tanh r 1 hypercycles and horocycles edit hypercycle and pseudogon in the poincare disk model main articles hypercycle hyperbolic geometry and horocycle in hyperbolic geometry there is no line all of whose points are equidistant from another line instead the points that all have the same orthogonal distance from a given line lie on a curve called a hypercycle another special curve is the horocycle a curve whose normal radii perpendicular lines are all limiting parallel to each other all converge asymptotically in one direction to the same ideal point the centre of the horocycle through every pair of points there are two horocycles the centres of the horocycles are the ideal points of the perpendicular bisector of the line segment between them given any three distinct points they all lie on either a line hypercycle horocycle or circle the length of the line segment is the shortest length between two points the arc length of a hypercycle connecting two points is longer than that of the line segment and shorter than that of a horocycle connecting the same two points the arclength of both horocycles connecting two points are equal the arc length of a circle between two points is larger than the arc length of a horocycle connecting two points if the gaussian curvature of the plane is 1 then the geodesic curvature of a horocycle is 1 and of a hypercycle is between 0 and 1 1 triangles edit main article hyperbolic triangle unlike euclidean triangles where the angles always add up to π radians 180 a straight angle in hyperbolic geometry the sum of the angles of a hyperbolic triangle is always strictly less than π radians 180 a straight angle the difference is referred to as the defect the area of a hyperbolic triangle is given by its defect in radians multiplied by r 2 as a consequence all hyperbolic triangles have an area that is less than or equal to r 2 π the area of a hyperbolic ideal triangle in which all three angles are 0 is equal to this maximum as in euclidean geometry each hyperbolic triangle has an incircle in hyperbolic geometry if all three of its vertices lie on a horocycle or hypercycle then the triangle has no circumscribed circle as in spherical and elliptical geometry in hyperbolic geometry if two triangles are similar they must be congruent regular apeirogon edit an apeirogon and circumscribed horocycle in the poincare disk model main article apeirogon hyperbolic geometry a special polygon in hyperbolic geometry is the regular apeirogon a uniform polygon with an infinite number of sides in euclidean geometry the only way to construct such a polygon is to make the side lengths tend to zero and the apeirogon is indistinguishable from a circle or make the interior angles tend to 180 degrees and the apeirogon approaches a straight line however in hyperbolic geometry a regular apeirogon has sides of any length i e it remains a polygon the side and angle bisectors will depending on the side length and the angle between the sides be limiting or diverging parallel see lines above if the bisectors are limiting parallel the apeirogon can be inscribed and circumscribed by concentric horocycles if the bisectors are diverging parallel then a pseudogon distinctly different from an apeirogon can be inscribed in hypercycles all vertices are the same distance of a line the axis also the midpoint of the side segments are all equidistant to the same axis tessellations edit main article uniform tilings in hyperbolic plane see also regular hyperbolic tiling rhombitriheptagonal tiling of the hyperbolic plane seen in the poincaré disk model like the euclidean plane it is also possible to tessellate the hyperbolic plane with regular polygons as faces there are an infinite number of uniform tilings based on the schwarz triangles p q r where 1 p 1 q 1 r 1 where p q r are each orders of reflection symmetry at three points of the fundamental domain triangle the symmetry group is a hyperbolic triangle group there are also infinitely many uniform tilings that cannot be generated from schwarz triangles some for example requiring quadrilaterals as fundamental domains 2 standardized gaussian curvature edit though hyperbolic geometry applies for any surface with a constant negative gaussian curvature it is usual to assume a scale in which the curvature k is 1 this results in some formulas becoming simpler some examples are the area of a triangle is equal to its angle defect in radians the area of a horocyclic sector is equal to the length of its horocyclic arc an arc of a horocycle so that a line that is tangent at one endpoint is limiting parallel to the radius through the other endpoint has a length of 1 3 the ratio of the arc lengths between two radii of two concentric horocycles where the horocycles are a distance 1 apart is e 1 3 cartesian like coordinate systems edit main article coordinate systems for the hyperbolic plane in hyperbolic geometry the sum of the angles of a quadrilateral is always less than 360 degrees and hyperbolic rectangles differ greatly from euclidean rectangles since there are no equidistant lines so a proper euclidean rectangle would need to be enclosed by two lines and two hypercycles these all complicate coordinate systems there are however different coordinate systems for hyperbolic plane geometry all are based around choosing a point the origin on a chosen directed line the x axis and after that many choices exist the lobachevski coordinates x and y are found by dropping a perpendicular onto the x axis x will be the label of the foot of the perpendicular y will be the distance along the perpendicular of the given point from its foot positive on one side and negative on the other another coordinate system measures the distance from the point to the horocycle through the origin centered around 0 displaystyle 0 infty and the length along this horocycle 4 other coordinate systems use the klein model or the poincare disk model described below and take the euclidean coordinates as hyperbolic distance edit a cartesian like citation needed coordinate system x y on the oriented hyperbolic plane is constructed as follows choose a line in the hyperbolic plane together with an orientation and an origin o on this line then the x coordinate of a point is the signed distance of its projection onto the line the foot of the perpendicular segment to the line from that point to the origin the y coordinate is the signed distance from the point to the line with the sign according to whether the point is on the positive or negative side of the oriented line the distance between two points represented by x_i y_i i 1 2 in this coordinate system is citation needed dist x 1 y 1 x 2 y 2 arcosh cosh y 1 cosh x 2 x 1 cosh y 2 sinh y 1 sinh y 2 displaystyle operatorname dist langle x_ 1 y_ 1 rangle langle x_ 2 y_ 2 rangle operatorname arcosh left cosh y_ 1 cosh x_ 2 x_ 1 cosh y_ 2 sinh y_ 1 sinh y_ 2 right this formula can be derived from the formulas about hyperbolic triangles the corresponding metric tensor field is d s 2 cosh 2 y d x 2 d y 2 displaystyle mathrm d s 2 cosh 2 y mathrm d x 2 mathrm d y 2 in this coordinate system straight lines take one of these forms x y is a point on the line x 0 y 0 a and α are parameters ultraparallel to the x axis tanh y tanh y 0 cosh x x 0 displaystyle tanh y tanh y_ 0 cosh x x_ 0 asymptotically parallel on the negative side tanh y a exp x displaystyle tanh y a exp x asymptotically parallel on the positive side tanh y a exp x displaystyle tanh y a exp x intersecting perpendicularly x x 0 displaystyle x x_ 0 intersecting at an angle α tanh y tan α sinh x x 0 displaystyle tanh y tan alpha sinh x x_ 0 generally these equations will only hold in a bounded domain of x values at the edge of that domain the value of y blows up to infinity see also coordinate systems for the hyperbolic plane polar coordinate system history edit see also non euclidean geometry history since the publication of euclid s elements circa 300 bce many geometers made attempts to prove the parallel postulate some tried to prove it by assuming its negation and trying to derive a contradiction foremost among these were proclus ibn al haytham alhacen omar khayyám 5 nasīr al dīn al tūsī witelo gersonides alfonso and later giovanni gerolamo saccheri john wallis johann heinrich lambert and legendre 6 their attempts were doomed to failure as we now know the paral...
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