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 Hyperbolic geometry
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the_hill1962
Advanced Member

USA
1468 Posts

Posted - 11/02/2012 :  13:48:30  Show Profile  Reply with Quote
Can a right angle be drawn in hyperbolic geometry?
I read a passage in a textbook that stated "there are no right triangles on a hyperbolic plane".
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royhaas
Moderator

USA
3059 Posts

Posted - 11/05/2012 :  12:39:29  Show Profile  Reply with Quote
The (interior) angles of all triagles in hyperbolic geometric sum to less than 180. The plane is really a saddle surface, so you can draw a right angle, but any "right" triangle would be "defective" in the sense that its angles sum less than 180.
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the_hill1962
Advanced Member

USA
1468 Posts

Posted - 11/12/2012 :  13:18:39  Show Profile  Reply with Quote
"defective" is a very interesting description.
I am not convinced that means it is not a right triangle. It is still a "triangle" with a "90 degree angle".
I am just going to "agree to disagree" with the textbook.
Again, your use of "defective" is interesting.
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royhaas
Moderator

USA
3059 Posts

Posted - 11/13/2012 :  10:26:42  Show Profile  Reply with Quote
I didn't invent the term "defective" in this context; I don't know who did, but it seems to be in common usage. In the case of Riemannian geometry, I believe the term "excess" is used, since the sum of interior triangle angles exceeds 180. So on a sphere, you can have an equilateral triangle with a total of 270.
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the_hill1962
Advanced Member

USA
1468 Posts

Posted - 11/14/2012 :  08:56:32  Show Profile  Reply with Quote
Ahh, thanks! I was wondering about that term.
I still disagree with the text that I read in a lesson somewhere that stated something like "right triangles do not exist in hyperbolic geometry".
It seems like a better statement would be "normal right triangles do not exist in hyperbolic geometry". That would open up the discussion of the terms "defective" and "excess".
This could just be me.
BTW, if "excess" means a triangle with sum of interior angels exceeding 180, then what is the term for the triangles in hyperbolic geometry?
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