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Affine Geometry

  1. May 8, 2009 #1
    1. The problem statement, all variables and given/known data

    For any subset ¬ of the affine plane R^2, let G(¬) denote the group of all affine transformations f of R^2 such that f(¬) = ¬.

    Exercise: Find all the elements of the group G = G(¬), where ¬ is the hyperbola xy = 1
    in R^2.




    3. The attempt at a solution

    Hey! I'm afraid I’m completely lost with this, any hints please on where to start would be very much appreciated. Thanks x
     
  2. jcsd
  3. May 8, 2009 #2
    Welcome to PF, bishbashbosh.

    Definitions. What is your definition of an affine transformation?

    Hopefully you can put that in the form

    x' = function of x and y
    y' = function of x and y

    Here (x',y') denotes the image of (x,y), not derivatives.

    Now use the fact that xy=1 must imply x'y'=1.
     
  4. May 9, 2009 #3
    Thank you Billy Bob, glad to be here!

    OK

    Definition:
    an affine transformation f of Rn is any transformation of the form
    f A,t : x -> Ax + t
    where A is an invertible n×n matrix; detA = 0, and t is an element of Rn.

    so ??
    x' = ax+by+t1
    y' = cx+dy+t2

    x'y'=acx2+(ad +bc)xy+bdy2+axt2+byt2+cxt1+dyt1+t2t1

    ...?

    Sorry, what then are a,b,c,d please?

    Thanks again x
     
    Last edited: May 9, 2009
  5. May 9, 2009 #4
    If x'y'=xy=1

    Comparing coefficients:

    ac=0
    bd=0
    ad+bc=1

    & as detA ≠ 0, ad-bc ≠ 0

    Either a=0 & d=0 or b=0 & c=0

    So elements of G:
    ( ax ) ,( by )
    ( dy ) ( cx )

    Correct?!

    Further please, let H be the subgroup of G preserving each of the two branches of ¬. Determine the index of H in G, whether H is normal in G, and whether H is abelian.

    Definitions:

    A subgroup H ≤ G of G is a subset H c_ G which is a group with respect to the same operation as G; equivalently, H is non-empty and is closed with respect to products and inverses.
    We say that H is a normal subgroup if gH = Hg for all g in G; these cosets gH
    then form a quotient group G/H with gH.g’H = (gg’)H.

    Let H be a subgroup of the group G and let a in G. Then aH = {ah | h in H} is called the left coset of H in G determined by a.
    Similarly, Ha = {ha | h in H} is called the right coset of H in G determined by a.

    Let H ≤ G and let G be written as the disjoint union of the left cosets of H. Then the number of the left cosets in this decomposition is called the index of H in G, written |G : H|.

    A group (G,*) is called abelian if the binary operation is commutative, i.e. if for all a, b in G, a * b = b * a.

    Any ideas on this one please?

    Thanks x
     
    Last edited: May 10, 2009
  6. May 9, 2009 #5
    Pretty good. For the original question, matrix A should have a=d=0 or b=c=0, but continue further to obtain bc=1 in the first case and ad=1 in the second. Did you mention t1=t2=0?

    If you write down some examples, like A=[2 0 | 0 1/2], then you have nice geometric meaning; in this example x-coordinate is doubled and y-coordinate is halved.

    Consider enough examples and you will easily that some of these preserve the branches of the hyperbola and others interchange them.
     
  7. May 10, 2009 #6
    Sorry, couple of typo's in there. I think i understand the original question now, many thanks to you!

    Ok so for 2nd, H= (ax|dy) for ad=1 ?!
    Do all elements of G preserve each of the two asymptotes of ¬ ?

    What extent will results apply to the groups G' = G(¬') preserving other hyperbolas ¬' not equal to ¬ in R2?

    x
     
  8. May 10, 2009 #7
    I almost made the same mistake I think you are making. But matrix A=[a 0 | 0 d] (ad=1) does not necessarily carry each branch to itself; it might switch the two. And matrix A=[0 b | c 0] (bc=1) does not necessarily switch the two. It might just reflect across the line y=x.

    Don't forget negative numbers!

    For different hyperbolas, I guess rotate the coordinate axes first and/or dilate so that you have xy=1.
     
  9. May 11, 2009 #8
    Ahhh Thank you once more!
     
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