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Thread: Isometries of R2 and R3 - unique expression

  1. #1
    Super Member Bernhard's Avatar
    Jan 2010
    Hobart, Tasmania, Australia

    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 $\displaystyle e_1 $ axis.

    Then on Page 463 he states Theorem 14.21 as follows:

    14.21 Theorem An isometry S of $\displaystyle \mathbb{R}^2$ or $\displaystyle \mathbb{R}^3$ can be uniquely expressed as $\displaystyle 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 $\displaystyle 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 $\displaystyle e_1$ 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 $\displaystyle L_1$ onto line $\displaystyle L_2$? [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 $\displaystyle L_1$ onto $\displaystyle L_2$?

    But alternatively, $\displaystyle L_1$ could be translated some distance along the Y axis so that it passed through O, then rotated until it was parallel to $\displaystyle L_2$, then reflected through a line equidistant between $\displaystyle L_1$ and $\displaystyle L_2$. 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 $\displaystyle L_1$ mapped to line $\displaystyle L_2$, 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

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