I am reading Papantonopoulou: Algebra Ch 14 Symmetries. I am seeking to fully understand Theorem 14.21 (see attached Papantonopoulou pp 459 -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 TheoremAn 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

The proof (see attached) proceeds as follows:

Proof: To prove existence, let b = S(0). Then $\displaystyle t_{-b} \circ S$ fixes the origin and so $\displaystyle t_{-b} \circ S = T_A$ for some A $\displaystyle \in$ O(n) ... ... etc ... ( see attached for definition of $\displaystyle T_A$)

I need some help with this first line of the proof!

Restating the proof explicitly in $\displaystyle \mathbb{R}^2$ we have S(0,0) = ($\displaystyle b_1 , b_2$)

i.e. $\displaystyle S(0,0) = t_b \circ \rho_{\phi} \circ r^i (0,0) $ = ($\displaystyle b_1 , b_2$)

Thus $\displaystyle t_{-b}\circ S(0,0) = t_{-b} \circ t_b \circ \rho_{\phi} \circ r^i (0,0) = \rho_{\phi} \circ r^i (0,0) = (0,0) $

But ... why does $\displaystyle \rho_{\phi} \circ r^i (0,0) = (0,0)$???

Presumably this is because Papantonopoulou defines rotations as about the origin (0,0) and reflections are defined to be about the X-axis or $\displaystyle e_1$ axis. (see definitions of rotations and reflections at bottom of page 462 - attachment)

A further question is this ... what is Papantonopoulou actually trying to do when he starts his proof with "To prove existence, let b = S(0)"? What is his strategy here? What is he tryinh to do - I am somewhat lost regarding the overall point of this!

I would be really appreciative of some help here - or at least confirmation of my reasoning.

Peter