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Thread: 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 $\displaystyle \mathbb{R}^3$ that map the point (1,1,1) to (1,0,0) and express them in the form $\displaystyle t_b$$\displaystyle \circ$$\displaystyle \rho_{\theta}$$\displaystyle \circ$$\displaystyle r^i$ as in Theorem 14.21, and in terms of S($\displaystyle \nu$) = A$\displaystyle \nu$ + 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 $\displaystyle \mathbb{R}^2$ or $\displaystyle \mathbb{R}^3$ include:

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

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

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

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

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

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

    Theorem 14.21

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

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

    Corollary 14.22

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

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