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## Homework Statement

Let's call the axis of the ##z## complex plain ##x## and ##y##, so a general point can be written as ##z=x+iy##. Reflect the points of the complex plain so that the mirror line of the transformation is a line parallel to the vector ##v## and it passes trough the point ##u##. Give the function that transforms the point ##z=x+iy## to ##z'=x'+iy'## and show that the points on the mirror line are fix points of this transformation. (Sorry for the bad english.)

## Homework Equations

##u = 3+5i##

##v=8-5i##

## The Attempt at a Solution

First, I constructed the mirror line: ##5x+8y=55##. I checked it with a plotting program and it looks correct.

Then, the way we learnt reflection is the following:

- If the mirror line does not intersect the origin, we have to translate it so that it does. This can be done with any point of the line, let's call this point ##c##.

- Then, if the line does not lie on the x axis, we have to rotate it by an angle ##\beta##.

- Then we do the reflection with the complex conjugate.

-Then we do the inverse of the rotation and the translation.

##z_1 = z -c##

##z_2=z_1 e^{-i \beta}##

##z_3=z_2^{*}##

##z_4=z_3 e^{i \beta}##

##z_5=z_4+c##

So all together: ##z_5=z'=(z-c)^* e^{2i \beta} +c##

I substituted ##z=x+iy##, ##z'=x'+iy'##, ##c=a+bi## and made two functions for the real and the imaginary parts.

##x'=x \cdot cos{2 \beta} +y \cdot sin{2 \beta} -a \cdot cos{2 \beta} -b \cdot sin{2 \beta} +a##

##y'=x \cdot sin{2 \beta} -y \cdot cos{2 \beta} -a \cdot sin{2 \beta} +b \cdot cos{2 \beta} +b##

Now my problem is the last part which is about proving that the fix points are on the mirror line.

I guess I should substitute ##x'=x## and ##y'=y##, add the two equations together and express it as ##y=f(x)## and see if it matches my line equation from the beginning. But they are full of sines and cosines which won't have an exact value if I calculate ##\beta##, so I don't see how to prove this analytically. Numerically, it doesn't work yet, I'm still trying to figure out where did I mess it up.