How to evaluate an angle

Question

Is it possibile to evaluate an angle in Tikz? I googled a lot but I can't find anything exhausting. My problem is the following: I have an arc intersected by a bundle of parallel lines, like shawn in the picture below

Once drawn the arc (path name=mirror) I used this code

\foreach \y in {-1.6,-0.8,...,1.6}
                {
                    \path [name path=ray] (-1,\y) -- ($ (-1,\y) + (10,0) $);
                    \path [name intersections={of=ray and mirror, by={P}}];
                    \draw [thick, postaction={on each segment={mid arrow=black}}] (-1,\y) -- (P);
                    \draw [thick, dashed, name path=refracted] (P) -- (F);

                    \begin{scope}[shift={(P)}]

                    \end{scope}                     
                }

to generate the bundle of lines. I would like to draw some lines like these in red

To do this, I thought to evaluate the angle between P--F--V and, after a shift of the coordinate system in P (which changes for each line), then draw a simply line using

\draw [thick] (\myAngle:2cm);

Answer 1

The computation you are asking for is simpler than the one in the previous answer since the rays are all horizontal. You just need to get the angle, alpha, at which they hit the mirror, and outgoing ray will have an angle of 180-2*alpha. This angle can be computed with the calc syntax

\path let \p1=($(<point>)-(<center>)$),
    \n1={180-atan2(\y1,\x1)} in ... ;

where <point> is the point at which the ray hits the mirror, i.e. the intersection point P in your code, and <center> is the center of the circle used to draw the mirror. The following self-contained code illustrates this.

\documentclass[tikz,border=3mm]{standalone}
\usetikzlibrary{decorations.markings,calc,angles,quotes}
\pgfdeclareradialshading{halo}{\pgfpointorigin}{
  color(0pt)=(pgftransparent!100); color(22.5bp)=(pgftransparent!100);
  color(25bp)=(pgftransparent!00); color(50bp)=(pgftransparent!00)}%
\pgfdeclarefading{halo}{\pgfuseshading{halo}}%
\tikzset{->-/.style={decoration={% https://tex.stackexchange.com/a/39282/194703
  markings,
  mark=at position #1 with {\arrow{>}}},postaction={decorate}},
  ->-/.default=0.5}
\begin{document}
\begin{tikzpicture}[>=stealth,every label/.append style={black},
    bullet/.style={circle,fill=black,inner sep=0.1em}]
\begin{scope}[xshift=6cm]
    \clip (75:-3)
    -- (75:-2.5) arc (75:-75:-2.5)
    -- (-75:-3) arc (-75:75:-3);
    \fill[path fading=halo] (0,0) circle[radius=3cm];
\end{scope}
\draw (-2,0) coordinate (L) -- (7,0) coordinate (R) 
  (3,0)  node[bullet,label=below left:$V$](V){}
  (6,0)  node[bullet,label=below:$C$](C){}
  (-1,0)  node[bullet,label=below:$O$](O){}
  foreach \X [count=\Y] in {-40,-20,20,40}
  {($(C)+(180+\X:3)$) node[bullet](P\Y){}};
  \foreach \Y in {1,...,4}
  {\draw[orange,->-,thick] (O|-P\Y) -- (P\Y);
   \draw[red,->-,thick] let \p1=($(P\Y)-(C)$),
    \n1={180-atan2(\y1,\x1)} in
    (P\Y) -- ++({180-2*\n1}:2);
   \draw[dashed, shorten >=-2cm] (C) -- (P\Y);}
\end{tikzpicture}
\end{document}