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Math Help - Finite Reflection Groups in Two Dimensions - Grove & Benson Section 2.1

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    Super Member Bernhard's Avatar
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    Finite Reflection Groups in Two Dimensions - Grove & Benson Section 2.1

    I am seeking to understand Finite Reflection Groups and am reading Grove and Benson (G&B) : Finite Reflection Groups

    Grove and Benson, in Section 21 Orthogonal Transformations in Two Dimensions define T as a linear transformation belonging to the groups of all orthogonal transformations  O( {\mathbb{R}}^2 ).

    G&B point out that the vector  x_1 = (cos \ \theta /2 , sin \ \theta /2)  is an eigenvector having eigenvalue 1 for T and that similarly  x_2 = ( - sin \ \theta /2 , cos \ \theta /2 ) is an eigenvector with eigenvalue -1 and  x_1 \ \bot  \ x_2 [see attachment]

    G&B then state that if  x = {\lambda}_1 x_1 + {\lambda}_2 x_2 , then  Tx = {\lambda}_1 x_1 - {\lambda}_2 x_2 and T sends x to its mirror image with respect to the line L (see Figure 2.2(b) in attachement)

    The transformation T is called the refection through L or the reflection along  x_2 .

    G&B then say "observe that  Tx = x - 2( x , x_2) x_2 for all  x \in {\mathbb{R}}^2 "

    Can someone help me show (explicitly and formally) that  Tx = x - 2( x , x_2) x_2 for all  x \in {\mathbb{R}}^2 ?

    And further (and possibly more important) can someone help me get a geometric sense of what the formula above means? ie why do G&B highlight this particular relationship?

    Would very much appreciate such help

    Peter
    Last edited by Bernhard; April 26th 2012 at 05:41 AM.
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    Re: Finite Reflection Groups in Two Dimensions - Grove & Benson Section 2.1

    Tx = T(λ1x1 + λ2x2) = λ1T(x1) + λ2T(x2)

    = λ1x1 - λ2x2 = λ1x1 + λ2x2 - 2λ2x2

    = x - 2λ2x2 (this is just simple algebra, to this point).

    now (x,x2) = (λ1x1 + λ2x2,x2) = λ1(x1,x2) + λ2(x2,x2)

    and since x1,x2 are orthogonal, (x1,x2) = 0, therefore:

    (x,x2) = λ2(x2,x2) = λ2|x2|2.

    but x2 lies on the unit circle, therefore it has length (and thus length squared) of 1.

    so (x,x2) = λ2, and we have:

    Tx = x - 2λ2x2 = x - 2(x,x2)x2.

    so why?

    the quantity (x,x2)x2 is a little misleading (because we are dealing with UNIT vectors x1,x2).

    normally, this is written as: [(x,x2)/(x2,x2)]x2 which is known as:

    the projection of the vector x in the direction of the vector x2. geometrically this is: "the part of x that lies on the line generated by x2".

    normally, we project a vector onto the unit vectors e1 = (1,0) and e2 = (0,1).

    that is: if x = (x1,x2) = x1e1 + x2e2, then:

    the projection of x in the direction of e1 is [(x.e1)/(e1.e1)]e1

    = [(x1*1 + x2*0)/(1*1 + 0*0)]e1 = x1e1 = (x1,0).
    Thanks from Bernhard
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    Super Member Bernhard's Avatar
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    Re: Finite Reflection Groups in Two Dimensions - Grove & Benson Section 2.1

    Thanks ... that post was most helpful, particularly the bit about the misleading nature of the formula as stated ... that clarified a few issues for me.

    Geometrical interpretation is most helpful

    Peter
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