I know that parallel axiom in Hyperbolic geometry is that there at least two parallel lines to line $a$ through a given point $A$. But as I know it can be proven that there are infinitely many of them, what is the proof ?
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I swear the proof is given under one of Euclid's propositions... but I can't seem to remember it myself! – Mr Pie Apr 07 '18 at 10:03
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What are your other axioms? – Hagen von Eitzen Apr 07 '18 at 10:12
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1@user477343 By Euclid's propositions you mean one of his theorems in neutral geometry ? – Юрій Ярош Apr 07 '18 at 10:12
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Euclid's fifth postulate says exactly not this. – Steve Kangas Apr 07 '18 at 10:12
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@SteveKangas I'm talking about Hyperbolic geometry which is non-euclidean. – Юрій Ярош Apr 07 '18 at 10:13
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@HagenvonEitzen Let it be Hilbert's axioms with the parallel axiom of Hyperbolic geometry instead of Euclid's – Юрій Ярош Apr 07 '18 at 10:14
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1@Юрій Ярош I was responding to user477343, who was, maybe joking? – Steve Kangas Apr 07 '18 at 10:17
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@SteveKangas Oh, sorry. – Юрій Ярош Apr 07 '18 at 10:21
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2You can literally see the infinitely many parallel lines in the Poincare disk model: https://en.wikipedia.org/wiki/Poincar%C3%A9_disk_model#/media/File:Poincare_disc_hyperbolic_parallel_lines.svg – Steve Kangas Apr 07 '18 at 10:26
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Once you know it's true for one point, you can move all the parallel lines to any other point using isometries. But probably you're looking for a proof derived from axioms. – Steve Kangas Apr 07 '18 at 10:36
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@SteveKangas I know that I can see infinitely many of them in models of Hyperbolic geometry, but I was interested in a proof. Does this proof at least exists ? – Юрій Ярош Apr 07 '18 at 10:44
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It's possible to prove this given a particular set of axioms. Like Hilbert's. – Steve Kangas Apr 07 '18 at 10:56
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@SteveKangas sorry I got confused, hahah. – Mr Pie Apr 08 '18 at 06:35
1 Answers
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Suppose you have the two lines through $A$, that converge on the two ends of $a$.
A point $B$ that lies between these convergent lines, will create a line $AB$, that crosses the two convergent lines at $A$, and so will always be opposite one of the convergent lines to $a$. Thus it never crosses $a$.
wendy.krieger
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I don't know this terminology, so sorry for the additional questions, but what convergent line means in hyperbolic geometry ? – Юрій Ярош Apr 07 '18 at 10:51
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When they say there are two parallels, these are the ones that meet at infinity, such at the edge of a Poincare disk. These two cross at A, and make an X-shape. If the line a is at the bottom of the X, and B is placed in the left or right side of the X, the line AB will pass from the left-side into the right side. – wendy.krieger Apr 07 '18 at 10:58
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The underlying idea here is that the hyperbolic plane is still orientable, and the line $AB$ can not cross $a$ because it's above the two lines that meet at infinity. – wendy.krieger Apr 07 '18 at 10:59
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Suppose the line $a$ has the ends at $L$ and $R$. Then the lines through $A$ run $AL$ and $AR$. If $B$ is in the X-shape, on the right hand side, a line to the left of $AB$ crosses $BR$, $BA$, $BL$ and $a$ in that order, because $AB$ crosses $BL$ at $B$ only. On the right hand side, the order is $BL$, $BA$, $BR$ and $a$, meaning that $BA$ never crosses $BR$ and thus can not approach $a$. Therefore $AB$ never crosses $a$. – wendy.krieger Apr 07 '18 at 11:11
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Sorry but what is the "end of a line" ? Line doesn't have ends as far as I know. – Юрій Ярош Apr 07 '18 at 16:06