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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

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  1. Loser66
    • one year ago
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    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
    • one year ago
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    @Halmos

  3. anonymous
    • one year ago
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    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
    • one year ago
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    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
    • one year ago
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    |dw:1441574010939:dw|

  6. anonymous
    • one year ago
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    |dw:1441574129229:dw|

  7. anonymous
    • one year ago
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    |dw:1441574177174:dw|

  8. Loser66
    • one year ago
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    Hence, the line in hyperbolic geometry is defined as it is in Euclidean, right?

  9. Loser66
    • one year ago
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    My prof defined that is the disk without boundary.

  10. anonymous
    • one year ago
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    |dw:1441574255876:dw|

  11. anonymous
    • one year ago
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    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
    • one year ago
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    My prof defined that is the disk without boundary. exactly i also said without boundary :-D

  13. anonymous
    • one year ago
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    okay so far ? @Loser66 you can ask any question before i start parallel axiom

  14. Loser66
    • one year ago
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    |dw:1441574638141:dw|

  15. anonymous
    • one year ago
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    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
    • one year ago
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    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
    • one year ago
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    Hold on. In Euclid, he defined a line is a breadthless length. It doesn't relate to end points.

  18. Loser66
    • one year ago
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    He gave the definition of a line while in hyperbolic geometry, they consider a line as an undefined term. Am I right?

  19. Loser66
    • one year ago
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    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
    • one year ago
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    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
    • one year ago
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    assume this circle in Euclid's define the line, segment, open segment |dw:1441575350298:dw|

  22. Loser66
    • one year ago
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    yes

  23. Loser66
    • one year ago
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    |dw:1441575468982:dw|

  24. anonymous
    • one year ago
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    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
    • one year ago
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    confused @Loser66 ? or that make sense ?

  26. Loser66
    • one year ago
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    |dw:1441575823244:dw|

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

  28. Loser66
    • one year ago
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    You make me think of 3D, not 2 D any more.

  29. Loser66
    • one year ago
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    |dw:1441575963819:dw|

  30. Loser66
    • one year ago
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    |dw:1441576002403:dw|

  31. anonymous
    • one year ago
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    haha that might be new model as well :D why not.

  32. Loser66
    • one year ago
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    hahaha..... but I can't imagine how 2 circles perpendicular to each other in 2D, friend.

  33. anonymous
    • one year ago
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    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
    • one year ago
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    |dw:1441576194227:dw|

  35. Loser66
    • one year ago
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    ok, go ahead.

  36. Loser66
    • one year ago
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    |dw:1441576339793:dw|

  37. Loser66
    • one year ago
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    aAAAAAAh, I got you!!

  38. Loser66
    • one year ago
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    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
    • one year ago
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    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
    • one year ago
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    yes exactly Hoa !

  41. Loser66
    • one year ago
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    |dw:1441576471993:dw|

  42. anonymous
    • one year ago
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    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
    • one year ago
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    Hence if a triangle is formed from the "lines", it should look like |dw:1441576649907:dw|

  44. anonymous
    • one year ago
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    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
    • one year ago
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    yes you are correct, i think its less than 180 i'm i right ?

  46. Loser66
    • one year ago
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    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
    • one year ago
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    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
    • one year ago
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    Got you. again, thank you so so so much. I love you, friend.

  49. anonymous
    • one year ago
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    here you can use my reference, my teacher (Aminah Afaneh) wrote it and made it so easy to read and understand.

  50. anonymous
    • one year ago
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    i love you too my friend <3.

  51. Loser66
    • one year ago
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    Thanks for the link. I am out now to read it. :)

  52. anonymous
    • one year ago
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    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 :)

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