# Isometries of R3

• Jan 3rd 2012, 03:51 PM
Bernhard
Isometries of R3
I am reading Aigli Papantonopoulou's book Algebra: Pure and applied.

Exercise 18 of Chapter 14:Symmetries reads as follows:

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"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
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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$
• Jan 4th 2012, 03:38 AM
Deveno
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?