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on 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 parallel postulate is not provable from the other postulates but their efforts led to the discovery of hyperbolic geometry the theorems of alhacen khayyam and al tūsī on quadrilaterals including the ibn al haytham lambert quadrilateral and khayyam saccheri quadrilateral were the first theorems on hyperbolic geometry their works on hyperbolic geometry had a considerable influence on its development among later european geometers including witelo gersonides alfonso john wallis and saccheri 7 in the 18th century johann heinrich lambert introduced the hyperbolic functions 8 and computed the area of a hyperbolic triangle 9 19th century developments edit in the 19th century hyperbolic geometry was explored extensively by nikolai ivanovich lobachevsky jános bolyai carl friedrich gauss and franz taurinus unlike their predecessors who just wanted to eliminate the parallel postulate from the axioms of euclidean geometry these authors realized they had discovered a new geometry 10 11 gauss wrote in an 1824 letter to franz taurinus that he had constructed it but gauss did not publish his work gauss called it non euclidean geometry 12 causing several modern authors to continue to consider non euclidean geometry and hyperbolic geometry to be synonyms taurinus published results on hyperbolic trigonometry in 1826 argued that hyperbolic geometry is self consistent but still believed in the special role of euclidean geometry the complete system of hyperbolic geometry was published by lobachevsky in 1829 1830 while bolyai discovered it independently and published in 1832 in 1868 eugenio beltrami provided models see below of hyperbolic geometry and used this to prove that hyperbolic geometry was consistent if and only if euclidean geometry was the term hyperbolic geometry was introduced by felix klein in 1871 13 klein followed an initiative of arthur cayley to use the transformations of projective geometry to produce isometries the idea used a conic section or quadric to define a region and used cross ratio to define a metric the projective transformations that leave the conic section or quadric stable are the isometries klein showed that if the cayley absolute is a real curve then the part of the projective plane in its interior is isometric to the hyperbolic plane 14 for more history see article on non euclidean geometry and the references coxeter 15 and milnor 16 philosophical consequences edit the discovery of hyperbolic geometry had important philosophical consequences before its discovery many philosophers for example hobbes and spinoza viewed philosophical rigour in terms of the geometrical method referring to the method of reasoning used in euclid s elements kant in the critique of pure reason came to the conclusion that space in euclidean geometry and time are not discovered by humans as objective features of the world but are part of an unavoidable systematic framework for organizing our experiences 17 it is said that gauss did not publish anything about hyperbolic geometry out of fear of the uproar of the boeotians which would ruin his status as princeps mathematicorum latin the prince of mathematicians 18 the uproar of the boeotians came and went and gave an impetus to great improvements in mathematical rigour analytical philosophy and logic hyperbolic geometry was finally proved consistent and is therefore another valid geometry geometry of the universe spatial dimensions only edit main article philosophy of space and time see also shape of the universe curvature of the universe because euclidean hyperbolic and elliptic geometry are all consistent the question arises which is the real geometry of space and if it is hyperbolic or elliptic what is its curvature lobachevsky had already tried to measure the curvature of the universe by measuring the parallax of sirius and treating sirius as the ideal point of an angle of parallelism he realised that his measurements were not precise enough to give a definite answer but he did reach the conclusion that if the geometry of the universe is hyperbolic then the absolute length is at least one million times the diameter of the earth s orbit 2 000 000 au 10 parsec 19 some argu...
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