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Math Help - Isometries of R2 and R3 - unique expression

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    Super Member Bernhard's Avatar
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    Isometries of R2 and R3 - unique expression

    I am reading Papantonopoulou: Algebra Ch 14 Symmetries. I am seeking to fully understand Theorem 14.21 (see attached Papantonopoulou pp 462 -463)

    On page 462, Papantonopoulou defines translations, rotations and reflections for R2 and R3 (see attached). Note that the rotations are defined as about the origin and the reflections are about the X-axis or e_1 axis.

    Then on Page 463 he states Theorem 14.21 as follows:

    14.21 Theorem An isometry S of \mathbb{R}^2 or \mathbb{R}^3 can be uniquely expressed as S = t_b \circ  \rho_{\phi} \circ r^i where i = 0 or 1

    I have some questions related to the nature and sense of the above theorem.

    Since S = t_b \circ  \rho_{\phi} \circ r^i it seems that Papantonopoulou is defining all isometries in terms of (and in this order!) first, a reflection (about the X-axis or e_1 axis), followed by a rotation (about the origin) followed by a translation. Can all isometries be expressed in this way? How does this fit an isometry that maps L_1 onto line L_2? [See attached Figure 1] Presumably you would consider a transformation of the space to a situation where O' was the origin since Papantonopoulou defines rotations in terms of the origin and then rotate L_1 onto L_2?

    But alternatively, L_1 could be translated some distance along the Y axis so that it passed through O, then rotated until it was parallel to L_2, then reflected through a line equidistant between L_1 and L_2. But how would this process (ie order of transformations) fit with Theorem 14.21 above? And how do different ways of achieving an isometry fir with Theorem 14.21's idea of a 'unique expression" for isometries?

    Does Theorem 14.21 actually assert that despite many alternative isometries achieving a certain outcome (e.g. line L_1 mapped to line L_2, there is one of them that can be expressed in the form S = t_b \circ \rho_{\phi} \circ r^i [/TEX]

    Can anyone please help clarify the meaning of Papantonopoulou Theorem 14.21?

    Would appreciate some guidance

    Peter
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