Rotation isometry on the complex plane derivation

[PcumP_Ravenclaw]

Dear All,
In the 2nd paragraph of the attachment can you please explain to me why we are trying to make $r(z) = z$

and what does "As r is not the identity..." mean??

and how did the line $L = 0.5*b + ρe^{θ/2}$ come about?

Danke...

 

[HallsofIvy]

The second sentence starts "If (ii) Is true" so everything said here depends upon "(ii)" being true but we don't know what "(ii)" is!

 

[PcumP_Ravenclaw]

Oops sorry!.

The missing scan is in this attachement.

danke.

 

[Fredrik]

"r is not the identity" means "r is not the identity map", which means "r is not the unique function f such that f(z)=z for all z". So there's at least one complex number z such that r(z)≠z.