Let the word \"point\" mean ordered pair (x,y) of real numbers withx^2+y^2<1, th
ID: 2938516 • Letter: L
Question
Let the word "point" mean ordered pair (x,y) of real numbers withx^2+y^2<1, the word line a set which is the portion of a circleorthogonal to x^2+y^2=1 inside the set {(x,y) s.t x^2+y^2<1} orthe intersection of th eset of the form {(x,y) s.t ax+by=0} wherea^2+b^2=1 with the disk.Problem: Verify the remaining step to show that this isan incidence geometry
Incidence axioms IA1: Given any two distinct points there exists a unique lineto which both belong. IA2: Each line has at least two points that belong toit. IA3: There exists at least three distinct points. IA4: Not all points belong to any one line.
Apparently the way I worked out in 7 hours was notcorrect so I am interested to see what the real verification wouldbe.
Problem: Verify the remaining step to show that this isan incidence geometry
Incidence axioms IA1: Given any two distinct points there exists a unique lineto which both belong. IA2: Each line has at least two points that belong toit. IA3: There exists at least three distinct points. IA4: Not all points belong to any one line.
Apparently the way I worked out in 7 hours was notcorrect so I am interested to see what the real verification wouldbe.
Explanation / Answer
What you are working with is known as the Poincare disk model forhyperbolic geometry. The only axiom of the four mentionedthat is not obviously true is the first. Here is an outlineof a proof from wikipedia, where they give the formula of theunique circle that goes through two points that are not on adiameter: http://en.wikipedia.org/wiki/Poincar%C3%A9_disk_model#Analytic_geometry_constructions_in_the_hyperbolic_plane There should be lots written on the Poincare model, in books and onthe internet, if this proves insufficient.
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