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Math Help - Isometries of R3

  1. #1
    Super Member Bernhard's Avatar
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    Isometries of R3

    I am reading Aigli Papantonopoulou's book Algebra: Pure and applied.

    Exercise 18 of Chapter 14:Symmetries reads as follows:

    ================================================== =====

    "Describe all the isometries of \mathbb{R}^3 that map the point (1,1,1) to (1,0,0) and express them in the form t_b \circ \rho_{\theta} \circ r^i as in Theorem 14.21, and in terms of S( \nu) = A \nu + b as in Corollary 14.22
    ================================================== =====

    I would really appreciate help with this problem.

    Peter



    Details from Papantonopoulou follow:

    On pages 462 - 463 we find:

    Possible isometries of \mathbb{R}^2 or \mathbb{R}^3 include:

    (1) Translations: These are of the form t_b(\nu) = \nu + b

    (2) Rotations: These are of the form T_A for some A \inSO(n). We write \rho_{\theta} for a rotation by an angle \theta (about some axis through the origin)

    (3) Reflections: These are given by T_A for some A \inO(n) with det A = -1, that is A not in SO(n). We write r for reflections given by the matrices

    \left(\begin{array}{cc}1&0\\0&-1\end{array}\right) \in O(2)

    \left(\begin{array}{ccc}1&0&0\\0&1&0\\0&0&-1\end{array}\right) \in O(3)

    Since SO(n) is a subgroup of O(n) of index 2, every reflection is of the form \rho_{\theta} \circr

    Theorem 14.21

    An isometry S of \mathbb{R}^2 or \mathbb{R}^3can be uniquely expressed as

    S = t_b \circ \rho_{\theta} \circ r^i where i = 0 or 1

    Corollary 14.22

    An isometry S of \mathbb{R}^2 or \mathbb{R}^3can be uniquely expressed as

    S( \nu) = A \nu + b where A \in O(n) and b \in \mathbb{R}^3
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  2. #2
    MHF Contributor

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    Re: Isometries of R3

    the only part of this worth investigating is the translation part, because any element of O(3) preserves the origin. so ask yourself: if x-->x+b takes (1,1,1) to (1,0,0), what must the 3-vector b be?
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