[Tex/LaTex] Problem filling intersection using TikZ

tikz-pgf

I was trying to use TikZ to draw a figure, I want to fill the common area of four circles of which centers are at four points of a square, and the radios for the arc is 0.9772 of the length of square.

I use the following code:

\begin{tikzpicture}[scale = 4]
\clip (-0.2, -0.2) rectangle (1.2,1.2);
\begin{scope}[even odd rule]
\clip (0.9772,0) arc (0:90:0.9772);
\clip (0.0228,1) arc (180:270:0.9772);
\fill [fill = green] (0,0) rectangle (1,1);
\end{scope}
\begin{scope}[even odd rule]
\clip (1,0.9772) arc (90:180:0.9772);
\clip (0,0.0228) arc (270:360:0.9772);
\fill [fill = green] (0,0) rectangle (1,1);
\end{scope}
\draw [very thick](0,0) rectangle (1,1);
\draw [very thin] (0.9772,0) arc (0:90:0.9772);
\draw [very thin] (1,0.9772) arc (90:180:0.9772);
\draw [very thin] (0.0228,1) arc (180:270:0.9772);
\draw [very thin] (0,0.0228) arc (270:360:0.9772);
\end{tikzpicture}

However, I got a weird drawing out like this:

Why this does not work? I used the similar code generated a similar graph but with a smaller radius, the drawing was generated successfully as this:

So can anyone help me out here?

Best Answer

Clipping, like filling, has to use a closed path. So when you give TikZ/PGF an open path and ask it to clip, or fill, it then it (sensibly) closes it first before trying to fill or clip against it. Since this is crucial to this problem, and it took me a few experiments to realise what was going on here, I'd like to over-emphasise this and formulate it as a rule:

When debugging clipping, render your paths using fills, not draws.

If we do that with your diagram then we can see what is happening: when TikZ/PGF closes up the arcs, as the radius increases then this cuts into the region that you want to describe. As the radius increases from 0.8 to 0.9 then this turns from not-a-problem into a problem as can be seen in the following progression: the light-grey areas are what is defined by the paths and the darker grey is their intersection which is what the clipping produces:

intersection progression

The solution is to ensure that your paths are big enough to contain the region that you want to render. Another take-home from this is that it is good to ensure that your paths are explicitly closed so that you don't get any unexpected surprises. Since you are already clipping against the bigger rectangle (so the bounding box is set), you could simply draw circles at each corner. An alternative would be to join the arcs to the corner. Either way, the same diagram as the above would be:

intersection progression with closed paths

which is what you want.

There are probably lots of ways to do this (and as I write, someone's posted one which I haven't looked at). Here's one which I would bet involves the minimum change to your code:

\documentclass{article}
%\url{http://tex.stackexchange.com/q/29745/86}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}[scale = 4]
\clip (-0.2, -0.2) rectangle (1.2,1.2);
\begin{scope}[even odd rule]
\clip (0,0) -- (0.9772,0) arc (0:90:0.9772) -- cycle;
\clip (1,1) -- (0.0228,1) arc (180:270:0.9772) -- cycle;
\fill [fill = green] (0,0) rectangle (1,1);
\end{scope}
\begin{scope}[even odd rule]
\clip (1,0) -- (1,0.9772) arc (90:180:0.9772) -- cycle;
\clip (0,1) -- (0,0.0228) arc (270:360:0.9772) -- cycle;
\fill [fill = green] (0,0) rectangle (1,1);
\end{scope}
\draw [very thick](0,0) rectangle (1,1);
\draw [very thin] (0.9772,0) arc (0:90:0.9772);
\draw [very thin] (1,0.9772) arc (90:180:0.9772);
\draw [very thin] (0.0228,1) arc (180:270:0.9772);
\draw [very thin] (0,0.0228) arc (270:360:0.9772);
\end{tikzpicture}
\end{document}

Personally, I would go for circles at the corners as it's easier to get right and to change the radius later. Here's that example:

\documentclass{standalone}
%\url{http://tex.stackexchange.com/q/29745/86}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}[scale = 4,every circle/.style={radius=0.9772}]
\useasboundingbox (-0.2, -0.2) rectangle (1.2,1.2);
\begin{scope}[even odd rule]
\clip (0,0) circle;
\clip (1,1) circle;
\fill [fill = green] (0,0) rectangle (1,1);
\end{scope}
\begin{scope}[even odd rule]
\clip (1,0) circle;
\clip (0,1) circle;
\fill [fill = green] (0,0) rectangle (1,1);
\end{scope}
\path [preaction={draw,very thick},clip] (0,0) rectangle (1,1);
\draw [very thin] (0,0) circle;
\draw [very thin] (1,1) circle;
\draw [very thin] (0,1) circle;
\draw [very thin] (1,0) circle;
\end{tikzpicture}
\end{document}

Note that the first clip is needed here to set the bounding box (in fact, I've converted it from a \clip to a \useasboundingbox) since otherwise the circles would be taken into account when computing the size of the final drawing (there are other ways to deal with this). One thing I like about this solution is that I can change the radius by changing only one parameter.

Both produce:

filled intersections

Which is what you want, I believe.

Related Solutions

[Tex/LaTex] TikZ: Cropping the Bounding Box

The code adds some completely useless invisible (or rather white) stuff. The lines

\clip(0pt,403pt) -- (389.957pt,403pt) -- (389.957pt,99.6166pt) -- (0pt,99.6166pt) -- (0pt,403pt);
\color[rgb]{1,1,1}
\fill(3.76406pt,399.236pt) -- (380.923pt,399.236pt) -- (380.923pt,253.19pt) -- (3.76406pt,253.19pt) -- (3.76406pt,399.236pt);
\fill(53.4497pt,394.719pt) -- (374.901pt,394.719pt) -- (374.901pt,289.325pt) -- (53.4497pt,289.325pt) -- (53.4497pt,394.719pt);

draw a white background that is larger than the actual picture. TikZ sees that and thinks it is part of the picture. Simply removing/uncommenting these lines removes most of the whitespace.

Near the end of the first scope,

\color[rgb]{1,1,1}
\fill(3.76406pt,249.426pt) -- (386.193pt,249.426pt) -- (386.193pt,103.381pt) -- (3.76406pt,103.381pt) -- (3.76406pt,249.426pt);

does the same. Additionally (near the end of the second scope),

\pgftext[center, base, at={\pgfpoint{220.95pt}{106.392pt}}]{\sffamily\fontsize{9}{0}\selectfont{\textbf{ }}}

adds a blank node below the picture, again enlarging the bounding box.

Removing all those lines gives a tight bounding box.

As far as I know, TikZ cannot do the cropping for you, as it can't know whether the white stuff is intentional or not (there might for example be a dark background behind the image so that white is visible).

[Tex/LaTex] TikZ: Drawing an arc from an intersection to an intersection

A solution which allows to draw intersection segments of any two intersections is available as tikz library fillbetween.

This library works as general purpose tikz library, but it is shipped with pgfplots and you need to load pgfplots in order to make it work:

\documentclass{standalone}

\usepackage{tikz}
\usepackage{pgfplots}
\usetikzlibrary{fillbetween}

\begin{document}

\begin{tikzpicture}

\draw [name path=red,red] (120:1.06) circle (1.9);
%\draw [name path=yellow,yellow] (0:1.06) circle (2.12);
\draw [name path=green,green!50!black] (0:0.77) circle (2.41);
\draw [name path=blue,blue] (0:0) circle (1.06);

% substitute this temp path by `\path` to make it invisible:
\draw[name path=temp1, intersection segments={of=red and blue,sequence=L1}];
\draw[red,-stealth,ultra thick, intersection segments={of=temp1 and green,sequence=L3}];

\end{tikzpicture}

\end{document}

enter image description here

The key intersection segments is described in all detail in the pgfplots reference manual section "5.6.6 Intersection Segment Recombination"; the key idea in this case is to

create a temporary path temp1 which is the first intersection segment of red and blue, more precisely, it is the first intersection segment in the Left argument in red and blue : red. This path is drawn as thin black path. Substitute its \draw statement by \path to make it invisible.
Compute the desired intersection segment by intersecting temp1 and green and use the correct intersection segment. By trial and error I figured that it is the third segment of path temp1 which is written as L3 (L = left argument in temp1 and green and 3 means third segment of that path).

The argument involves some trial and error because fillbetween is unaware of the fact that end and startpoint are connected -- and we as end users do not see start and end point.

Note that you can connect these path segments with other paths. If such an intersection segment should be the continuation of another path, use -- as before the first argument in sequence. This allows to fill paths segments:

\documentclass{standalone}

\usepackage{tikz}
\usepackage{pgfplots}
\usetikzlibrary{fillbetween}

\begin{document}

\begin{tikzpicture}

\draw [name path=red,red] (120:1.06) circle (1.9);
%\draw [name path=yellow,yellow] (0:1.06) circle (2.12);
\draw [name path=green,green!50!black] (0:0.77) circle (2.41);
\draw [name path=blue,blue] (0:0) circle (1.06);

% substitute this temp path by `\path` to make it invisible:
\draw[name path=temp1, intersection segments={of=red and blue,sequence=L1}];
\draw[red,fill=blue,-stealth,ultra thick, intersection segments={of=temp1 and green,sequence=L3}]  
    [intersection segments={of=temp1 and green, sequence={--R2}}]
;

\end{tikzpicture}

\end{document}

enter image description here

Best Answer

Related Solutions

[Tex/LaTex] TikZ: Cropping the Bounding Box

[Tex/LaTex] TikZ: Drawing an arc from an intersection to an intersection

Related Question