## Loser66 one year ago 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

1. Loser66

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.

2. Loser66

@Halmos

3. anonymous

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

4. anonymous

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

5. anonymous

|dw:1441574010939:dw|

6. anonymous

|dw:1441574129229:dw|

7. anonymous

|dw:1441574177174:dw|

8. Loser66

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

9. Loser66

My prof defined that is the disk without boundary.

10. anonymous

|dw:1441574255876:dw|

11. anonymous

Hence, the line in hyperbolic geometry is defined as it is in Euclidean, right? NO, EXTREMELY NO, in Euclid lines defines as between two points there is a line, but n hyperbolic see i've stated two different type of lines only |dw:1441574530904:dw|

12. anonymous

My prof defined that is the disk without boundary. exactly i also said without boundary :-D

13. anonymous

okay so far ? @Loser66 you can ask any question before i start parallel axiom

14. Loser66

|dw:1441574638141:dw|

15. anonymous

i see your point, ur saying diameters can be also line in Euclid's geometry, which also incorrect as there is no end point in lines of Euclid, you might call a segment but also wouldn't make sense as it has no end no start

16. anonymous

just think of hyperbolic as if it's something not related to Euclid's as long as you don't try to link them you would understand them better, okay ?

17. Loser66

Hold on. In Euclid, he defined a line is a breadthless length. It doesn't relate to end points.

18. Loser66

He gave the definition of a line while in hyperbolic geometry, they consider a line as an undefined term. Am I right?

19. Loser66

Ok, ignore me, go ahead. Study the very new stuff is always hard. However, Since the question is about comparison, I have to link them to see whether the fifth different postulates leads to all postulates are different or not. Sorry, friend.

20. anonymous

undefined terms means it has no definition like (line has endless points which is something wrong to say), undefined does not exactly means not defined both in Euclid's and hyperbolic they consider points,and lines are undefined term to AVOID physical explanation of them, like how much thin the line should be? or could we see the point.. or such stuff, got this now ?

21. anonymous

assume this circle in Euclid's define the line, segment, open segment |dw:1441575350298:dw|

22. Loser66

yes

23. Loser66

|dw:1441575468982:dw|

24. anonymous

see now you got the point :D what we might call open segment in Euclid is (sort of seen in the model as line) and lines in Euclid are not the same in hyperbolic segments are a cut off lines so this segment might not been defined in hyperbolic but here are things we might consider as segment in hyperbolic |dw:1441575652342:dw| |dw:1441575678689:dw|

25. anonymous

confused @Loser66 ? or that make sense ?

26. Loser66

|dw:1441575823244:dw|

27. anonymous

hahaha Two circles are called orthogonal to one another, if at the points of intersection their radii are perpendicular.

28. Loser66

You make me think of 3D, not 2 D any more.

29. Loser66

|dw:1441575963819:dw|

30. Loser66

|dw:1441576002403:dw|

31. anonymous

haha that might be new model as well :D why not.

32. Loser66

hahaha..... but I can't imagine how 2 circles perpendicular to each other in 2D, friend.

33. anonymous

i see only work on the definition "Two circles are called orthogonal to one another, if at the points of intersection their radii are perpendicular. " |dw:1441576158882:dw|

34. anonymous

|dw:1441576194227:dw|

35. Loser66

36. Loser66

|dw:1441576339793:dw|

37. Loser66

aAAAAAAh, I got you!!

38. Loser66

Hence, from point A , you can draw many "lines" perpendicular to the "line" of lambda. That is what the fifth postulate says in hyperbolic, right?

39. anonymous

so when there is orthogonal circle to the unit circle we have the chord that gets in the unit we call it open arcs which is the second type of lines. |dw:1441576427062:dw|

40. anonymous

yes exactly Hoa !

41. Loser66

|dw:1441576471993:dw|

42. anonymous

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. example |dw:1441576552016:dw|

43. Loser66

Hence if a triangle is formed from the "lines", it should look like |dw:1441576649907:dw|

44. anonymous

parallel means ( there is no common points ) so in the last model i have stated, m || l and n || l so there are two different lines at least that are parallel to l from a point p.

45. anonymous

yes you are correct, i think its less than 180 i'm i right ?

46. Loser66

Thanks a ton. I got it now. Just confirm the answer for my question 1, 2. You says!! hihihi.... I want to be a baby now to be indulged.

47. anonymous

haha okay. 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 answer is :- Hyperbolic geometry is, by definition, the geometry obtained by, assuming the axioms of neutral geometry and the negation of Hilbert's parallel axiom. and neutral geometry is valid in Euclid's. some of properties are the same in both hyperbolic and Euclid's but both are independent and not related to each other even we might understand the higher geometry more if we could compare them to Euclid's geometry. 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 is Universal Hyperbolic Theorem says there is at least two lines, so for each R in l , there corresponds a line through P that is parallel to l. Also, note that different points on l give rise to different parallel lines. so In hyperbolic geometry, for every line l and every point P not on l, there exist infinitely many lines through P and parallel to l.

48. Loser66

Got you. again, thank you so so so much. I love you, friend.

49. anonymous

here you can use my reference, my teacher (Aminah Afaneh) wrote it and made it so easy to read and understand.

50. anonymous

i love you too my friend <3.

51. Loser66

Thanks for the link. I am out now to read it. :)

52. anonymous

i also can explain Klein model if you want to or need it someday lol, anyway i'm here for you for any other question :)