Difficult TikZ picture

Question

I need to draw an integration path on top of a circle (with TikZ because I assume it's the best tool for the job), but I don't have any experience with it, and I literally don't know how to start, so what could a skeleton be?

Here it is

Answer 1

You can let TikZ do the calculations:

\documentclass[border=10pt]{standalone}
\usepackage{tikz}
\usetikzlibrary{intersections, decorations.markings}
\newlength{\cx}
\newlength{\cy}
\NewDocumentCommand{\extractangle}{ m O{0pt} O{0pt} m }{
    \pgfextractx{\cx}{\pgfpointanchor{#4}{center}}
    \pgfextracty{\cy}{\pgfpointanchor{#4}{center}}
    \pgfmathparse{atan2(\cy-(#3),\cx-(#2))}
    \let#1=\pgfmathresult
}
\begin{document}
\begin{tikzpicture}[
        font=\scriptsize,
        dot/.style={
            node contents={},
            inner sep=0.5pt,
            circle,
            fill,
        },
        three arrow heads/.style={
            postaction={
                decorate, 
                decoration={
                    markings,
                    mark={
                        at position -3.885cm
                        with { \arrow{>}}
                    },
                    mark={
                        at position -2.25cm
                        with { \arrow{>}}
                    },
                    mark={
                        at position -0.635cm
                        with { \arrow{>}}
                    }
                }
            }
        }
    ]
% base line
\draw (0:3) -- (180:3);
% coordinates and calculations
\coordinate (A) at (120:2);
\coordinate (B) at (90:2);
\coordinate (C) at (60:2);
\path[name path=CircleA] (A) circle[radius=0.25];
\path[name path=CircleB] (B) circle[radius=0.25];
\path[name path=CircleC] (C) circle[radius=0.25];
\path[name path=Arc] (0:2) arc[start angle=0, end angle=180, radius=2];
\path[name intersections={of=CircleA and Arc, name=i}];
\path[name intersections={of=CircleB and Arc, name=j}];
\path[name intersections={of=CircleC and Arc, name=k}];
\extractangle{\angleA}[cos(120)2cm][sin(120)2cm]{i-1}
\extractangle{\angleB}{i-1}
\extractangle{\angleC}{j-1}
\extractangle{\angleD}[0pt][2cm]{j-1}
\extractangle{\angleE}[0pt][2cm]{j-2}
\extractangle{\angleF}{j-2}
\extractangle{\angleG}{k-1}
\extractangle{\angleH}[cos(60)2cm][sin(60)2cm]{k-1}
% squarish shape
\draw[three arrow heads]
    ([yshift=0.25cm]120:2) coordinate[label={left:{$B$}}]
    arc[start angle=90, end angle={\angleA}, radius=0.25]
        coordinate[label={below:{$C$}}]
    arc[start angle=\angleB, end angle=\angleC, radius=2]
        coordinate[label={below:{$D$}}]
    arc[start angle={360+\angleD}, end angle=\angleE, radius=0.25]
        coordinate[label={below:{$E$}}]
    arc[start angle=\angleF, end angle=\angleG, radius=2]
        coordinate[label={below:{$F$}}] 
    arc[start angle=\angleH, end angle=90, radius=0.25]
        coordinate[label={right:{$G$}}] 
    -- ++(0,1.25) coordinate[label={above right:{$H$}}] 
    -| coordinate[label={above left:{$A$}}] cycle;
% big arc
\draw 
    (0:2) 
        arc[start angle=0, end angle=60, radius=2] 
        node[dot, label={below:{$-\bar{e}$}}]
    (B) node[dot, label={below:{$i$}}]
    (A) node[dot, label={below:{$e$}}]
        arc[start angle=120, end angle=180, radius=2];
\end{tikzpicture}
\end{document}

Some explanations to help you understand the code:

• I mostly used polar coordinates. These are very helpful when drawing things that are based on circles or arcs. The syntax is (A:B) where A is the angle (0 degrees is to the east) and B is the distance from the origin.
• I used the intersections library to find intersections of different paths, mainly of circles and arcs. Below I added the drawing with some of the helper paths colored to make it more clear where I used intersections. See the PGF manual for an in-depth explanation of how this library works.
• I used a macro (\extractangle) for which I found inspiration here to help me calculate the angle of a given cartesian coordinate using the atan2() function provided by PGF. I added two optional arguments to shift the origin, which was needed to calculate all the needed angles.
• Knowing all the coordinates and angles, I was finally able to draw the shapes, and to add the labels and decorations.

Answer 2

A solution via math.

Values that are used more than once are stored in value-keys in the /tikz namespace which is why I'm using a short command name for \tvo.

The radius of the small arcs as well as their angles will be calculated and stored in small radius and small angle.

Code

\documentclass[tikz]{standalone}
\newcommand*\tvo[1]{\pgfkeysvalueof{/tikz/#1}}
\usetikzlibrary{arrows.meta}
\tikzset{
  pics/arrow/.style={/tikz/sloped, /tikz/allow upside down,
    code=\pgfarrowdraw{#1}}, pics/arrow/.default=>}
\begin{document}
\begin{tikzpicture}[
  > = {Straight Barb[angle=60:3pt 1.5]},
  declare function={chord(\a,\r)=2*\r*sin(\a/2);},
  missing angle/.initial=5,
  big radius/.initial=4,
  big angle/.initial=60,
  small radius/.initial/.evaluated={chord(\tvo{missing angle},\tvo{big radius})},
  small angle/.initial/.evaluated={\tvo{big angle}+\tvo{missing angle}/2},
  r-/.style={radius=\tvo{small radius}}, r+/.style={radius=\tvo{big radius}},
  top/.initial=6,
  node font=\scriptsize]
\draw (left:5) -- (right:5);
\draw[r+] (left:\tvo{big radius})
  arc[start angle=180, delta angle=-\tvo{big angle}] coordinate (e)
         (right:\tvo{big radius})
  arc[start angle=0,   delta angle= \tvo{big angle}] coordinate (e');
\coordinate (A) at (e|-0,\tvo{top})
 coordinate (B) at ([shift=(up:\tvo{small radius})] e)
 coordinate (G) at ([shift=(up:\tvo{small radius})] e')
 coordinate (H) at (e'|-0,\tvo{top})
 coordinate (i) at (up:\tvo{big radius});
\draw[r+] (A) -- pic{arrow} (B)
  arc[start angle=90, delta angle=-\tvo{small angle}, r-] coordinate (C)
  arc[start angle=180-\tvo{big angle}-\tvo{missing angle},
        end angle=90+\tvo{missing angle}]                 coordinate (D)
  arc[start angle=180+\tvo{missing angle},
        end angle=-\tvo{missing angle}, r-]               coordinate (E)
  arc[start angle=90-\tvo{missing angle},
        end angle=\tvo{big angle}+\tvo{missing angle}]    coordinate (F)
  arc[end angle=90, delta angle=-\tvo{small angle}, r-] % -- (G) % maybe
  -- pic{arrow} (H) -- pic{arrow} cycle;
\foreach \lab/\pos in {A/above left, B/left, C/below, D/below,
                       E/below, F/below, G/right, H/above right}
  \node[\pos] at (\lab) {$\lab$};
\foreach \pos/\lab in {e, i, e'/-\bar e}
  \fill (\pos) circle[radius=.7pt] node[below=3pt]{$\lab$};
\end{tikzpicture}
\end{document}

Output

Answer 3

Purely for comparison, here is a version in an alternative Latex-friendly language, called Metapost.

This is wrapped up in the luamplib package, so you need to compile it with lualatex.

\documentclass[border=5mm]{standalone}
\usepackage{luamplib}
\begin{document}
\mplibtextextlabel{enable}
\begin{mplibcode}
beginfig(1);
    path base, hemi, loop;
    base = (left -- right) scaled 200;
    hemi = halfcircle scaled 300;
% controlling parameters...
numeric r, c, h; r = 1.27324; c = 12; h = 100;
pair A, H;
A = point 4-r of hemi shifted (0, h);
H = point r   of hemi shifted (0, h);

loop = buildcycle(
    A -- point 4-r of hemi, 
    subpath (3, 0)  of halfcircle scaled 2c shifted point 4-r of hemi, 
    subpath (4-r,2) of hemi,
    subpath (5, -1) of fullcircle scaled 2c shifted point 2 of hemi, 
    subpath (2, r)  of hemi, 
    subpath (4, 1)  of halfcircle scaled 2c shifted point r of hemi,
    point r of hemi -- H -- A
);

% draw loop withcolor red;  % to see the points round the loop...
% for i=1 upto length loop: dotlabel.top(decimal i, point i of loop); endfor

interim ahangle := 30;
drawarrow subpath (-1.5, 0.5) of loop;
drawarrow subpath (0.5, 19.5) of loop;
drawarrow subpath (19.5, 20.5) of loop;

draw base;
draw subpath (0, r) of hemi;
draw subpath (4-r, 4) of hemi;

begingroup; interim labeloffset := 8; dotlabeldiam := 2;
    dotlabel.bot("$-\bar e$", point r of hemi);
    dotlabel.bot("$i$", point 2 of hemi);
    dotlabel.bot("$e$", point 4-r of hemi);
endgroup;

drawoptions(withcolor 2/3 red);
label.ulft("$\scriptstyle A$", A);
label.lft ("$\scriptstyle B$", point 1 of loop);
label.bot ("$\scriptstyle C$", point 4 of loop);
label.bot ("$\scriptstyle D$", point 6 of loop);
label.bot ("$\scriptstyle E$", point 14 of loop);
label.bot ("$\scriptstyle F$", point 16 of loop);
label.rt  ("$\scriptstyle G$", point 19 of loop);
label.urt ("$\scriptstyle H$", H);
drawoptions();


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

Notes

• The drawing is done with three separate paths: base is just a horizontal line through the origin; hemi uses the built-in path halfcircle to get a suitably scaled semi-circle, but only parts of it are drawn; and loop.

• loop is created with the buildcycle macro: this macro takes a list of paths and produces a closed path that consists of the overlapping parts of the paths. It works best if all the paths are running in the same direction.

• A complicated path built by buildcycle may end up with more control points than it needs; the extra points don't affect the shape but they do make it harder to use point x of y and subpath (a, b) of y, so the commented out code contains a loop that I used to work out the correct subpaths for the arrows and the lettered labels.

• The "controlling parameters" are: r that controls the position of the points "e" and "-e". r=0 would be on the base, r=2 would make "e" coincide with "i"; c is the radius of the circles at the three points; h is the height of "A" above "e".