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Topic: Euclidean, Hyperbolic and Elliptic Geometry
Replies: 4   Last Post: Mar 12, 2009 9:24 PM

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K. E. Pledger

Posts: 6
Registered: 3/12/09
Re: Euclidean, Hyperbolic and Elliptic Geometry
Posted: Mar 12, 2009 9:24 PM
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> Can someone tell me some universal rules that apply
> all the time no matter what type of geometry?

Euclidean, hyperbolic and elliptic geometry have quite a lot in common. For example, the Euclidean criteria for congruent triangles also apply in the other two geometries, and from those you can prove many other things.

To give a more historical answer, Euclid I.1-15 apply to all three geometries. Euclid I.16 and various later propositions fail in the elliptic plane. Euclid I.29 and various later propositions fail in the hyperbolic plane.

> My friend Brian asks: Is there a specific ratio
> between hyperbolic sines and euclidean sines?

Are you willing to use complex numbers? If so, then
sinh(iz) = i.sin(z),
cosh(iz) = cos(z).

Ken Pledger.

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