Question

In hyperbolic geometry. Playfair's Postulate (i.e., the Euclidean parallel postulate) is replaced by the following statement: If l is a line and if P is a point not on l, then there exists AT LEAST 2 lines through P that are parallel to l.
1) This statement contradicts the Euclidean parallel postulate. Does this mean that none of the theorems from Euclidean geometry are valid in hyperbolic geometry?
My answer: surely not. Other Euclidean postulates are applied on hyperbolic geometry.
My problem: I don't know what hyperbolic geometry is. I am making a research about it but didn't (cnt. in c

Answer 1

But I didn't get a good site yet. It means I no nothing about it. Please give me a good site or good explanation about it.
2) The hyperbolic parallel postulate stated above implies that there are AT LEAST 2 lines through P that are parallel to l. Is it possible that there are "exactly 2 lines through P that are parallel to l"
My answer: Yes, it is. in some special case, the problem is arranged that only 2 lines pass through P that are // to l.

Answer 2

@Halmos

Answer 3

ok i'll give you a model that would make you understand all hyperbolic gimme a sec to post :D
PS:- your answer is correct since both Euclid's and hyperbolic are two different type of geometry which are also independent

Answer 4

here are examples for models ,the Klein model, the Poincare Disk model, and the Poincare Half-Plane model.

i'll describe the Poincare Disk model since its the easiest one to explain.

let a unit circle be \(\lambda\), the circle points itself are not in hyperbolic geometry we call them ideal points, or points at infinity.

undefined terms :-
Points: in the hyperbolic plane are interpreted to be the set of all interior points
of \(\lambda\).

Lines:- in the hyperbolic plane are of two types:
1. The open diameters, that is diameters without endpoints.
2. The open arcs, that are portions inside \(\lambda\) of circles orthogonal to
. These lines are called Poincare lines. (or open chords)


other stuff can beinterpreted as the Eclids sense

Answer 5

Hence, the line in hyperbolic geometry is defined as it is in Euclidean, right?

Answer 6

My prof defined that is the disk without boundary.