Can someone help me doing 2 points of intersection?

Question

i wanted to draw a cissoid curve using tikz. i managed to get this far:

however, i didn't manage to add visualizations of the 2 intersecting points between the big circle and the blue line and the graph of f(x) = tx.

I would love to add something like small crosses or filled black circles to visualize the points of intersection and call them R and P (i will remove the P above the blue line eventually).

can someone help me or show me an easy way of creating visual points of intersections between the circle and the other two lines?

this is my TeX-code:

\usetikzlibrary{calc,intersections}
\tikzset{>=latex}
\usetikzlibrary{datavisualization.formats.functions}
\usetikzlibrary{calc,patterns,angles,quotes}
\begin{document}

\begin{tikzpicture}[x=2cm, y=2cm]

\draw[->] (0,0) node[pos=0, left] {$O$} -- node[pos=0.8, below right, scale=0.5] {$x=2a$}(1.25,0); %% x axis
\draw[->] (0,-0.5) -- (0,1.25); %% y axis
\draw (0.5, 0) circle (0.5); % CIRCLE

%%%%%%%%%%%%%%%%%%%%

\draw(1,-0.5)[name path=line 2, pos=0, below left] coordinate(C) -- (1,1.25)coordinate (D); %% right vertical line parralel to y
\draw (0,0)[name path=line 1, pos=0, below left] coordinate(A) -- node[pos=1.025, right, scale=0.5] {$f(x) = tx$} (1.125,1) coordinate (B); %% f(x) = tx (graph)

%%%%%%%%%%%%%%%%%%%%

\draw[line width=1pt, color=blue] (0,0) coordinate(F) .. controls (0.5,0.025) ..node[pos=1, above, scale=0.65, color=black] {$P$} (0.75,0.75) coordinate(G); % curve through P
\node[below right, scale=0.65] at (intersection of  A--B and C--D){S}; % intersection point S

\end{tikzpicture}
\end{document}

Answer 1

\documentclass[border = 5pt, tikz]{standalone}

\usetikzlibrary{calc,intersections}
\tikzset{>=latex}
\usetikzlibrary{datavisualization.formats.functions}
\usetikzlibrary{calc,patterns,angles,quotes}
\begin{document}

\begin{tikzpicture}[x=2cm, y=2cm]

  \draw[->] (0,0) node[pos=0, left] {$O$} -- node[pos=0.8, below right, scale=0.5] {$x=2a$}(1.25,0); %% x axis
  \draw[->] (0,-0.5) -- (0,1.25); %% y axis
  \draw[name path = circle] (0.5, 0) circle (0.5); % CIRCLE

  %%%%%%%%%%%%%%%%%%%% 

  \draw(1,-0.5)[name path=line 2, pos=0, below left] coordinate(C) -- (1,1.25)coordinate (D); %% right vertical line parralel to y
  \draw (0,0)[name path=line 1, pos=0, below left] coordinate(A) -- node[pos=1.025, right, scale=0.5] {$f(x) = tx$} (1.125,1) coordinate (B); %% f(x) = tx (graph)

  %%%%%%%%%%%%%%%%%%%% 

  \draw[line width=1pt, color=blue, name path = blue line] (0,0) coordinate(F) .. controls (0.5,0.025) ..node[pos=1, above, scale=0.65, color=black] {$P$} (0.75,0.75) coordinate(G); % curve through P
  \node[below right, scale=0.65] at (intersection of  A--B and C--D){S}; % intersection point S

  %%%%%%%%%%%%%%%%%%%% <- added this
  \fill[red,name intersections={of=blue line and circle}]
  (intersection-1) circle (2pt) node[below right] {R};

  \fill[red,name intersections={of=line 1 and circle}]
  (intersection-1) circle (2pt) node[above left] {P};

\end{tikzpicture}
\end{document}