Fill a section between two circles with TikZ

Question

I am trying to fill a pattern in a section between two circles with TikZ.

In the code below, C0 is a circle with radius=1, center=(0,0). C1 is a circle through Point z0 with center=z1.

Calculating the coordinates of the two circles' intersections is too complicated task, so I want to specify the arcs between the two intersections' label.

But I found a difficulty in specifying an arc with the start/end points and the center.

Is there any straightforward solution to fill this kind of section?

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc,patterns,through,intersections}
\begin{document}
\begin{tikzpicture}
 % x, y axis
 \draw[->] (-1.5,0) -- (3,0) node[right] {$x$};
 \draw[->] (0,-1.5) -- (0,3) node[above] {$y$};
 % Origin
 \coordinate (O) at (0,0);

 % Point z0
 \coordinate (z0) at (1,0);
 \node [below right] at (z0) {$z_0$};
 % Circle C0
 \draw [name path=C0] (O) circle [radius=1];
 \node [above left] at (170:1) {$C_0$};
 % Point z1
 \coordinate (z1) at (30:0.7);
 \fill (z1) circle [radius=1pt];
 \node [below] at (z1) {$z_1$};

 % Circle C1
 \node [draw, circle through=(z0), name path=C1] at (z1) {};
 \node [above right] at ($(30:0.7)+(20:0.5)$) {$C_1$};

 % Intersections
 \path [name intersections={of=C0 and C1}];

\draw [pattern=north west lines, pattern color=gray!60!white] (z0)  arc (?:?:?) --(intersection-2) arc (?:?:1) --(z0);
\end{tikzpicture}
\end{document}

Answer 1

Could you not just clip the filling pattern? (I say 'just' not because this is especially obvious, but because I find it much simpler than dealing with intersections!)

\documentclass[border=10pt,multi,tikz]{standalone}
\usetikzlibrary{calc,patterns,through}
\begin{document}
\begin{tikzpicture}
 % x, y axis
 \draw[->] (-1.5,0) -- (3,0) node[right] {$x$};
 \draw[->] (0,-1.5) -- (0,3) node[above] {$y$};
 % Origin
 \coordinate (O) at (0,0);
 % Point z0
 \coordinate (z0) at (1,0);
 \node [below right] at (z0) {$z_0$};
 % Circle C0
 \draw  (O) circle [radius=1];
 \node [above left] at (170:1) {$C_0$};
 % Point z1
 \coordinate (z1) at (30:0.7);
 \fill (z1) circle [radius=1pt];
 \node [below] at (z1) {$z_1$};
 % Circle C1
 \node [draw, circle through=(z0)] at (z1) {};
 \node [above right] at ($(30:0.7)+(20:0.5)$) {$C_1$};
 \begin{scope}
   \clip (3,0) -- (z0) arc (0:90:1) -- (0,3) -| cycle;
   \node [pattern=north west lines, draw, pattern color=gray!60!white, circle through=(z0)] at (z1) {};
  \end{scope}
\end{tikzpicture}
\end{document}

Answer 2

Probably the easiest way is after filling C1 to re-filldraw C0, and finaly to draw the axes. Here is another solution using clipping, that use the "inverse clip" idea from an old Jake's answer and path picture clipping.

\documentclass[tikz,border=7mm]{standalone}
\usetikzlibrary{calc,patterns,through,intersections}
\begin{document}
  \begin{tikzpicture}
    % x, y axis
    \draw[->] (-1.5,0) -- (3,0) node[right] {$x$};
    \draw[->] (0,-1.5) -- (0,3) node[above] {$y$};
    % Origin
    \coordinate (O) at (0,0);
    % Point z0
    \coordinate (z0) at (1,0);
    \node [below right] at (z0) {$z_0$};
    % Circle C0
    \draw [name path=C0] (O) circle [radius=1];
    \node [above left] at (170:1) {$C_0$};
    % Point z1
    \coordinate (z1) at (30:0.7);
    \fill (z1) circle [radius=1pt];
    \node [below] at (z1) {$z_1$};
    % Circle C1
    \node [draw, circle through=(z0), name path=C1,
      path picture={
        \clip (-20,-20) rectangle +(40,40) (O) circle [radius=1 cm +.5\pgflinewidth];
        \fill[pattern=crosshatch, pattern color=green] (-20,-20) rectangle +(40,40);
      }
    ] at (z1) {};
    \node [above right] at ($(30:0.7)+(20:0.5)$) {$C_1$};
  \end{tikzpicture}
\end{document}

Answer 3

I know this question has been dormant for a long time, but I thought of a novel solution using a couple of Metapost features. (Compile this with lualatex, or work out how to adapt it for GMP + pdflatex).

My idea was to try and capture the look of the hand-drawn green scribble in the OP.

\documentclass[border=5mm]{standalone}
\usepackage{luamplib}
\usepackage{luatex85}
\begin{document}
\mplibtextextlabel{enable}
\begin{mplibcode}
beginfig(1);
    numeric u; 
    u = 1cm;
    path C[], xx, yy;

    xx = (3 left -- 5 right) scaled u; yy = xx rotated 90;

    C0 = fullcircle scaled 4u;
    z0 = point 0 of C0;

    z1 = (1.2u, 0.7u);
    C1 = fullcircle scaled 2 abs(z1-z0) rotated angle (z0-z1) shifted z1;

    numeric n, s, t;
    (s, t) = C0 intersectiontimes C1;
    n = 16;

    draw point 0 of C0 for i=1 upto n: -- point s*i/n of C0 -- point t*i/n of C1 endfor 
         withpen pencircle scaled 1 
         withcolor 1/2 green;

    draw C0;
    draw C1;
    dotlabel.lrt("$z_0$", z0);
    dotlabel.bot("$z_1$", z1);
    drawarrow xx;
    drawarrow yy;

endfig;
\end{mplibcode}
\end{document}

Note that to make the trick work, I had to rotate the smaller circle so that its point 0 coincides with the point 0 of the larger circle; this makes s and t give the times of the other intersection.

Answer 4

You can first fill the whole circle C1 with the pattern you want, then fill the whole C0 with the background color; i.e., write something like this:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc,patterns,through,intersections}
\begin{document}
\begin{tikzpicture}
 % Coordinates
 \coordinate (O) at (0,0);
 \coordinate (z0) at (1,0);
 \coordinate (z1) at (30:0.7);

 \node [fill, circle through=(z0), pattern=north west lines, pattern color=gray!60!white] at (z1) {};
 \fill [color=white] (O) circle [radius=1];

 % Point z0
 \node [below right] at (z0) {$z_0$};
 % Circle C0
 \draw [name path=C0] (O) circle [radius=1];
 \node [above left] at (170:1) {$C_0$};
 % Point z1
 \fill (z1) circle [radius=1pt];
 \node [below] at (z1) {$z_1$};

 % Circle C1
 \node [draw, circle through=(z0), name path=C1] at (z1) {};
 \node [above right] at ($(30:0.7)+(20:0.5)$) {$C_1$};

 % x, y axis
 \draw[->] (-1.5,0) -- (3,0) node[right] {$x$};
 \draw[->] (0,-1.5) -- (0,3) node[above] {$y$};
\end{tikzpicture}
\end{document}

Pay attention that the background filling commands must come before anything that describes labels, lines etc, otherwise the latter would be hidden by the background filling. Personally, I find it more convenient to define the axes in the end of the TikZ picture code block rather than at the start — precisely for the filling reasons.