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constructions in hyperbolic geometry wikipedia 10 captures 19 apr 2022 17 jul 2026 aug sep oct 07 2021 2022 2023 success fail about this capture timestamps the wayback machine http web archive org web 20220907121244 https en wikipedia org wiki constructions_in_hyperbolic_geometry constructions in hyperbolic geometry from wikipedia the free encyclopedia jump to navigation jump to search hyperbolic geometry is a non euclidean geometry where the first four axioms of euclidean geometry are kept but the fifth axiom the parallel postulate is changed the fifth axiom of hyperbolic geometry says that given a line l and a point p not on that line there are at least two lines passing through p that are parallel to l 1 as in euclidean geometry where ancient greek mathematicians used a compass and idealized ruler for constructions of lengths angles and other geometric figures constructions can also be made in hyperbolic geometry pseudosphere contents 1 models of hyperbolic geometry 2 tools 3 simple constructions 3 1 angle bisector 3 2 common parallel line to two lines 3 3 line perpendicular to another line at a point 3 4 midpoint of a line segment 4 definitions for complex constructions 4 1 cyclic quadrilaterals 4 2 pseudoaltitudes 4 3 pseudolengths 4 4 homothety 5 three dunce cap theorem 6 applying to spherical geometry 7 references models of hyperbolic geometry edit there are a couple of models for hyperbolic geometry which can make performing and visualizing constructions easier portions of the hyperbolic plane can be placed onto a pseudosphere and maintain angles and hyperbolic distances as well as be bent around the pseudosphere and still keep its properties 2 however not the entire hyperbolic plane can be placed onto the pseudosphere as a model only a portion of the hyperbolic plane 2 poincare disc with hyperbolic parallel lines the entire hyperbolic plane can also be placed on a poincaré disk and maintain its angles however the lines will turn into circular arcs which warps them 2 tools edit in hyperbolic geometry one can use the standard ruler and compass that is often used in euclidean plane geometry however there are a variety of compasses and rulers developed for hyperbolic constructions a hypercompass can be used to construct a hypercycle given the central line and radius 3 a horocompass can be used to construct a horocycle through a specific point if the diameter and direction are also provided both of these also require a straight edge like the standard ruler 3 when doing constructions in hyperbolic geometry as long as you are using the proper ruler for the construction the three compasses meaning the horocompass hypercompass and the standard compass can all perform the same constructions 3 a parallel ruler can be used to draw a line through a given point a and parallel to a given ray a 3 for any two lines a hyperbolic ruler can be used to construct a line that is parallel to the first line and perpendicular to the second 3 a few notes on the uses of rulers are a parallel ruler can be used to construct anything that a standard ruler and the three rulers can also construct 3 a parallel ruler can act as a ruler in euclidean geometry 3 a hyperbolic ruler cannot perform euclidean geometry constructions 3 in hyperbolic geometry constructions that can be done using any one of the three compasses listed above and the parallel ruler can also be done using the hyperbolic ruler 3 simple constructions edit angle bisector edit consider a given angle ᗉ iai π 2 radians whose angle bisector is sought this results in two different cases either ᗉ iai π 2 radians or ᗉ iai π 2 radians 3 for both cases a hyperbolic ruler is needed to construct a line bi where bi is perpendicular to ai and parallel to ai also construct a line b i where b i is perpendicular to ai and parallel to ai 3 case 1 ᗉ iai π 2 radians let c be the intersection of bi and b i the result of this is that the line ac bisects ᗉ iai 3 case 2 ᗉ iai π 2 radians this case is further broken down into three sub cases case 2a ib intersects i b let a be the intersection of ib and i b then aa is the angle bisector of ᗉ iai 3 case 2b ib is parallel to i b construct the line segment bb and using a hyperbolic ruler construct the line oi such that oi is perpendicular to bb and parallel to b i then line oa is the angle bisector for ᗉ iai 3 case 2c ib is ultraparallel to i b using the ultraparallel theorem construct the common perpendicular of ib and i b cc let the intersection of cb and bc be d as a result ad will be the angle bisector of ᗉ bdb we then find that the line through od is also the angle bisector of ᗉ iai 3 common parallel line to two lines edit we consider the problem of finding a line parallel to two given lines a and a there are three cases a and a intersect at a point o a and a are parallel to each other and a and a are ultraparallel to each other 3 case 1 a and a intersect at a point o bisect one of the angles made by these two lines and name the angle bisector b using a hyperbolic ruler construct a line c such that c is perpendicular to b and parallel to a as a result c is also parallel to a making c the common parallel to lines a and a 3 case 2 a and a are parallel to each other using a hyperbolic ruler construct ai such that ai is parallel to a and perpendicular to a construct another line a i such that a i is parallel to a and perpendicular to a let the intersection of ai and a i be b because ᗉ ibi π 2 radians the case now plays out like case 1 allowing construction of the common parallel to bi and bi 3 case 3 a and a are ultraparallel to each other using a hyperbolic ruler construct bi such that bi is perpendicular to a and parallel to a and construct a line b i such that b i is perpendicular to a and parallel to a in a way that puts bi and b i on the same side of the common perpendicular to a and a which can be found using the ultraparallel theorem let the intersection of bi and b i be c then ᗉ ici π 2 radians which allows you to finish the construction like the other two cases 3 line perpendicular to another line at a point edit suppose you have a line a and a point a on that line and you want to construct a line perpendicular to a and through a then let a be a line through a where a and a are two distinct lines then you will have one of two cases 3 case 1 a is perpendicular to a in this case we already have the line perpendicular to a through a 3 case 2 a and a are not perpendicular to each other using a hyperbolic ruler construct a line bi such that bi is perpendicular to a and parallel to a also construct a line ci such that ci is perpendicular to a and parallel to a but in the opposite direction of bi now draw a line ii so that ii is the common parallel to bi and i c the ultraparallel theorem now allows us to create the common perpendicular to ii and a because these two lines are ultraparallel this common perpendicular is now the line perpendicular to a and through a 3 midpoint of a line segment edit suppose you are trying to find the midpoint of a line segment ab then construct the line ai such that ai is through a and perpendicular to ab also construct a line bi such that bi intersects ab at b and is perpendicular to ab now construct the line ii such that ii is the common parallel to ai and bi 3 construct the common perpendicular to ii and ab which can be done using the ultraparallel theorem because ii and ab are ultraparallel to each other name this line cc c now ends up being the midpoint of ab 3 definitions for complex constructions edit when giving angles a positive or negative sign angle ᗉ xyz will be positive if the direction of the shortest path from xy to yz is counterclockwise for the purposes of the following definitions the following assumptions will be made which usually cannot be made in hyperbolic geometry three distinct points create a unique circle 4 given any two lines they meet at a unique point 4 normally this would contradict the parallel axiom of hyperbolic geometry since there can be many different lines parallel to the same line 1 angle measures have signs here they will be defined in the following way consider a triangle xyz the sign of angle ᗉ xyz is positive if and only if the direction of the path along the shortest arc from side xy to side yz is counterclockwise the picture of the triangle on the right describes this to make a comparison when working with the unit circle the angle measure is positive when going counterclockwise and negative when going clockwise 4 cyclic quadrilaterals edit a quadrilateral is cyclic if the two opposite vertices add up to pi radians or 180 degrees 4 also if a quadrilateral is inscribed in a circle in a way that all of its vertexes lie on the circle it is cyclic 5 pseudoaltitudes edit consider triangle abc where the points are labeled in a clockwise manner so all angles are positive let x be a point moving along bc from b to c as x moves closer to c angle ᗉaxb will decrease and angle ᗉ axc will increase when x is close enough to b ᗉ axb ᗉ axc when x is close enough to c ᗉ axb ᗉ axc this means that at some point x will be in a position where ᗉ axb ᗉ axc when x is in this position it is defined as the foot of the pseudoaltitude from vertex a 4 the pseudoaltitude would then be the line segment ax 4 here examples of pseudoaltitudes would be a 1 h 1 a 2 h 2 and a 3 h 3 pseudolengths edit let d e a b denote the pseudolength for a given hyperbolic line segment ab let a transformation move a to the center of a poincaré disk with a radius equalling 1 the pseudolength d e a b is the length of this segment in euclidean geometry 4 homothety edit given a point p a point a where a is the center of the homothety and a number k that represents the ratio of the homothety a homothety is a transformation that will move p to a point p where p is on the ray ap and d e a p k d e a p 4 three dunce cap theorem edit consider the three circles ω 1 ω 2 and ω 3 in a common plane let p 1 be the intersection of the two external tangent lines of ω 2 and ω 3 let p 2 and p 3 be found the same way the three dunce caps theorem then says that p 1 p 2 and p 3 all lie on the same line 4 proof construct a sphere on top of each circle and then construct a plane tangent to these three spheres the plane intersects the plane that the circles lies on at a straight line containing p 1 p 2 and p 3 these points are also the centers of homothety for the circles that they were derived from 4 applying to spherical geometry edit algebraically hyperbolic and spherical geometry have the same structure 4 this allows us to apply concepts and theorems to one geometry to the other 4 applying hyperbolic geometry to spherical geometry can make it easier to understand because spheres are much more concrete which then makes spherical geometry easier to conceptualize references edit a b cannon james w floyd william j kenyon richard perry walter r 1997 hyperbolic geometry pdf library msri org retrieved 2018 12 13 a b c rothe franz 2006 09 07 hyperbolic geometry and the pseudo sphere pdf math2 uncc edu archived from the original pdf on 2018 01 09 retrieved 2018 12 13 a b c d e f g h i j k l m n o p q r s t u v w x al dhahir m w 1962 an instrument in hyperbolic geometry proceedings of the american mathematical society 13 2 298 304 doi 10 1090 s0002 9939 1962 0138036 7 jstor 2034487 a b c d e f g h i j k l akopyan arseniy v 2011 05 11 on some classical constructions extended to hyperbolic geometry arxiv 1105 2153 math mg leonard i ed lewis j e liu a c f tokarsky g w 2014 06 04 classical geometry euclidean transformational inversive and projective hoboken nj isbn 9781118839430 oclc 861966488 retrieved from https en wikipedia org w index php title constructions_in_hyperbolic_geometry oldid 1068634813 categories hyperbolic geometry navigation menu personal tools not logged in talk contributions create account log in namespaces article talk english views read edit view history more search 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