Marking a specific point on an arc

Question

I want to draw a line from one arc to other but I do not know if it is possible to mark a point on an arc and start drawing on that point. Is it possible to draw lines like below image?

\documentclass{article}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}
  \draw[thick] (90:3cm) arc (90:0:3cm);
  \draw[thick] (90:3cm) arc (180:90:3cm);
\end{tikzpicture}

\end{document}

Answer 1

There are a lot of possibilities with tikz.

First method

Below is one if you want to specify the points manually.

\documentclass{article}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}
  \draw[thick] (90:3cm) arc (90:0:3cm);
  \draw[thick] (90:3cm) arc (180:90:3cm);
  \coordinate (A) at (80:3cm);
  \path (90:3cm) arc (180:135:3cm) node [inner sep=0pt] (B) {};
  \draw [blue,thick] (A) -- (B);
  \foreach \X in {A,B}
    \fill [blue] (\X) circle (2pt);

  \coordinate (C) at (33:3cm);
  \path (90:3cm) arc (180:123:3cm) node [inner sep=0pt] (D) {};
  \draw [red,thick] (C) -- (D);
  \foreach \Y in {C,D}
        \fill [red] (\Y) circle (2pt);
\end{tikzpicture}

\end{document}

With decorations.markings library

And here is a solution using decorations.markings library.

%http://tex.stackexchange.com/questions/88053/marking-a-specific-point-on-an-arc/88060#88060
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{decorations.markings}

\begin{document}

\begin{tikzpicture}[
every node/.style={circle,inner sep=1.5pt,outer sep=0pt},
decoration={
markings,% switch on markings
mark=at position 1cm with {\node (A) [fill=blue] {};},
mark=at position 3cm with {\node (B) [fill=red] {};},
mark=at position 0.7 with {\node (C) [fill=blue] {};},
mark=at position 0.8 with {\node (D) [fill=red] {};}
}
]

\draw [thick,postaction={decorate}] (90:3cm) arc (90:0:3cm)(90:3cm) arc (180:90:3cm);
\draw [blue,thick] (A) -- (C);
\draw [red,thick] (B) -- (D);
\end{tikzpicture}
\end{document}

The advantage of the second method is that you can specify the placement of the points relatively (0.1, 0.23 and so on) between the first coordinate/node and the last coordinate/node of your path. So this may apply even to arbitrary paths.