Doubt about rotation isometry on the complex plane

[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.

 

[blue_raver22]

Hello,

I can provide some insight into your questions about rotation isometry on the complex plane. First, let's define what rotation isometry means. It refers to a transformation that preserves distances and angles between points on a plane, while also rotating the plane around a fixed point. In the context of the complex plane, this means that the transformation preserves the distance and angle between complex numbers while also rotating the plane.

Now, in the second paragraph of the attachment, the equation $r(z) = z$ is being used to represent the transformation of a complex number z under a rotation isometry. Essentially, this equation is saying that the transformed complex number (r(z)) is equal to the original complex number (z). This is important because we want to find a way to represent a rotation isometry in terms of complex numbers.

The phrase "As r is not the identity..." means that the transformation r is not simply the identity transformation, which would result in no change to the complex number. Instead, r is a rotation transformation, meaning it will change the complex number by rotating it around a fixed point.

The equation $L = 0.5*b + ρe^{θ/2}$ is used to represent the transformation matrix for a rotation isometry on the complex plane. This equation is derived from the fact that a rotation matrix in the complex plane can be represented as a combination of a scaling factor (ρ) and a rotation angle (θ), along with a translation represented by the complex number b. The specific values of 0.5 and θ/2 are used to ensure that the transformation is a rotation isometry, meaning it preserves distances and angles.

I hope this explanation helps clarify your doubts about rotation isometry on the complex plane. Let me know if you have any further questions. Danke.