I need to understand a proof of a lemma about the Illumination Problem in Euclidian Space; the proof's author is George Tokarsky (1991). See the images below. Can you help me understand?
- I know that $a \mod b$ is equal to the rest of the division $a/b$.
- I know that $a \equiv c \mod b$ is equivalent to say $a \mod b = c \mod b$
I have the next questions:
- What means "We measure all angles $\mod 2x$"?
Following $(1)$:
What represent here $a$?
There is $b$? What represent?
Why $2x$ and no $x$ or other coefficient?
- What means $\theta \equiv 90 \mod 2x$?
- What means $\theta \equiv 0 \mod x$?
Thank you!
In the triangle shown, all edges slopes are at multiples of $x$ relative to the base line, hence an input angle $\alpha$ is reflected into an even multiple of $x$ minus $\alpha$, hence at each reflection, $\pm\theta+2kx$ becomes $\mp\theta +2k'x$. As we start with $\theta+ 0x$, we will by induction always travel at some $\pm \theta +2kx$, i.e., at an angle that is $\equiv \pm\theta \pmod{2x}$