Clipping more complicated shapes in TikZ

Question

I have tried to understand TikZ clipping better but I am stuck. I'd like to clip the area bounded by the blue contour in this picture:

Unfortunately, the MWE is rather lengthy, and based on Alain Matthes cool macros and some additional macros from the answer here.

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc,fadings,decorations.pathreplacing,shadings}
\usepackage{verbatim}

\newcommand\pgfmathsinandcos[3]{%
  \pgfmathsetmacro#1{sin(#3)}%
  \pgfmathsetmacro#2{cos(#3)}%
}
\newcommand\LongitudePlane[3][current plane]{%
  \pgfmathsinandcos\sinEl\cosEl{#2} % elevation
  \pgfmathsinandcos\sint\cost{#3} % azimuth
  \tikzset{#1/.style={cm={\cost,\sint*\sinEl,0,\cosEl,(0,0)}}}
}

\newcommand\LatitudePlane[3][current plane]{%
  \pgfmathsinandcos\sinEl\cosEl{#2} % elevation
  \pgfmathsinandcos\sint\cost{#3} % latitude
  \pgfmathsetmacro\yshift{\RadiusSphere*\cosEl*\sint}
  \tikzset{#1/.style={cm={\cost,0,0,\cost*\sinEl,(0,\yshift)}}} %
}
\newcommand\NewLatitudePlane[4][current plane]{%
  \pgfmathsinandcos\sinEl\cosEl{#3} % elevation
  \pgfmathsinandcos\sint\cost{#4} % latitude
  \pgfmathsetmacro\yshift{#2*\cosEl*\sint}
  \tikzset{#1/.style={cm={\cost,0,0,\cost*\sinEl,(0,\yshift)}}} %
}
\newcommand\DrawLongitudeCircle[2][1]{
  \LongitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
   % angle of "visibility"
  \pgfmathsetmacro\angVis{atan(sin(#2)*cos(\angEl)/sin(\angEl))} %
  \draw[current plane] (\angVis:1) arc (\angVis:\angVis+180:1);
  \draw[current plane,opacity=0.4] (\angVis-180:1) arc (\angVis-180:\angVis:1);
}
\newcommand\DrawLongitudeArc[4][black]{
  \LongitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=1}}
  \pgfmathsetmacro\angVis{atan(sin(#2)*cos(\angEl)/sin(\angEl))} %
  \pgfmathsetmacro\angA{mod(max(\angVis,#3),360)} %
  \pgfmathsetmacro\angB{mod(min(\angVis+180,#4),360} %
  \draw[current plane,#1,opacity=0.4] (#3:\RadiusSphere) arc (#3:#4:\RadiusSphere);
  \draw[current plane,#1]  (\angA:\RadiusSphere) arc (\angA:\angB:\RadiusSphere);
}%
\newcommand\DrawLatitudeCircle[2][1]{
  \LatitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
  \pgfmathsetmacro\sinVis{sin(#2)/cos(#2)*sin(\angEl)/cos(\angEl)}
  % angle of "visibility"
  \pgfmathsetmacro\angVis{asin(min(1,max(\sinVis,-1)))}
  \draw[current plane] (\angVis:1) arc (\angVis:-\angVis-180:1);
  \draw[current plane,opacity=0.4] (180-\angVis:1) arc (180-\angVis:\angVis:1);
}

\newcommand\DrawLatitudeArc[4][black]{
  \LatitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=1}}
  \pgfmathsetmacro\sinVis{sin(#2)/cos(#2)*sin(\angEl)/cos(\angEl)}
  % angle of "visibility"
  \pgfmathsetmacro\angVis{asin(min(1,max(\sinVis,-1)))}
  \pgfmathsetmacro\angA{max(min(\angVis,#3),-\angVis-180)} %
  \pgfmathsetmacro\angB{min(\angVis,#4)} %
  \draw[current plane,#1,opacity=0.4] (#3:\RadiusSphere) arc (#3:#4:\RadiusSphere);
  \draw[current plane,#1] (\angA:\RadiusSphere) arc (\angA:\angB:\RadiusSphere);
}

%% document-wide tikz options and styles

\tikzset{%
  >=latex, % option for nice arrows
  inner sep=0pt,%
  outer sep=2pt,%
  mark coordinate/.style={inner sep=0pt,outer sep=0pt,minimum size=3pt,
    fill=black,circle}%
}

\begin{document}

\begin{tikzpicture} % "THE GLOBE" showcase
\def\RadiusSphere{4} % sphere radius
\def\angEl{20} % elevation angle
\def\angAz{-20} % azimuth angle

\shade[ball color = gray!40, opacity = 0.5] (0,0) circle (\RadiusSphere);

\tikzset{
    every path/.style={
        color=black
    }
}

\DrawLatitudeArc[blue]{40}{-140}{-30}
\DrawLongitudeArc[blue]{-140}{-30}{40}
\DrawLatitudeArc[blue]{-30}{-140}{-30}
\DrawLongitudeArc[blue]{-30}{-30}{40} 

\end{tikzpicture}
\end{document}

The problem is that, according to what I read, clip needs to be the only option of \draw, but this clashes with the way the arcs are drawn here. So I am wondering if there is a way to have both.

EDIT Motivated by the great progress by John Kormylo, I tried (and to some extent succeeded) to find alternative ways of shading the area. They are based on this post, and require the spath package (run pdflatex spath.dtx).

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{shadings}
\usepackage{spath} % from https://tex.stackexchange.com/a/26664/121799
\usetikzlibrary{calc,fadings,decorations.pathreplacing,shadings}
\usepackage{verbatim}

\newcommand\pgfmathsinandcos[3]{%
  \pgfmathsetmacro#1{sin(#3)}%
  \pgfmathsetmacro#2{cos(#3)}%
}
\newcommand\LongitudePlane[3][current plane]{%
  \pgfmathsinandcos\sinEl\cosEl{#2} % elevation
  \pgfmathsinandcos\sint\cost{#3} % azimuth
  \tikzset{#1/.style={cm={\cost,\sint*\sinEl,0,\cosEl,(0,0)}}}
}

\newcommand\LatitudePlane[3][current plane]{%
  \pgfmathsinandcos\sinEl\cosEl{#2} % elevation
  \pgfmathsinandcos\sint\cost{#3} % latitude
  \pgfmathsetmacro\yshift{\RadiusSphere*\cosEl*\sint}
  \tikzset{#1/.style={cm={\cost,0,0,\cost*\sinEl,(0,\yshift)}}} %
}
\newcommand\NewLatitudePlane[4][current plane]{%
  \pgfmathsinandcos\sinEl\cosEl{#3} % elevation
  \pgfmathsinandcos\sint\cost{#4} % latitude
  \pgfmathsetmacro\yshift{#2*\cosEl*\sint}
  \tikzset{#1/.style={cm={\cost,0,0,\cost*\sinEl,(0,\yshift)}}} %
}
\newcommand\DrawLongitudeCircle[2][1]{
  \LongitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
   % angle of "visibility"
  \pgfmathsetmacro\angVis{atan(sin(#2)*cos(\angEl)/sin(\angEl))} %
  \draw[current plane] (\angVis:1) arc (\angVis:\angVis+180:1);
  \draw[current plane,opacity=0.4] (\angVis-180:1) arc (\angVis-180:\angVis:1);
}
\newcommand\DrawLongitudeArc[4][black]{
  \LongitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=1}}
  \pgfmathsetmacro\angVis{atan(sin(#2)*cos(\angEl)/sin(\angEl))} %
  \pgfmathsetmacro\angA{mod(max(\angVis,#3),360)} %
  \pgfmathsetmacro\angB{mod(min(\angVis+180,#4),360} %
  \draw[current plane,#1,opacity=0.4] (#3:\RadiusSphere) arc (#3:#4:\RadiusSphere);
  \draw[current plane,#1]  (\angA:\RadiusSphere) arc (\angA:\angB:\RadiusSphere);
}%

\newcommand\ClipLongitudeArc[4][black]{
  \LongitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=1}}
  \pgfmathsetmacro\angVis{atan(sin(#2)*cos(\angEl)/sin(\angEl))} %
  \pgfmathsetmacro\angA{mod(max(\angVis,#3),360)} %
  \pgfmathsetmacro\angB{mod(min(\angVis+180,#4),360} %
  \path[save path=\tmppathI,current plane,#1,opacity=0.4] (#3:\RadiusSphere) arc (#3:#4:\RadiusSphere);
  \path[save path=\tmppathII,current plane,#1]  (\angA:\RadiusSphere) arc (\angA:\angB:\RadiusSphere);
}%

\newcommand\ClipLongitudeArcReverse[4][black]{
  \LongitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=1}}
  \pgfmathsetmacro\angVis{atan(sin(#2)*cos(\angEl)/sin(\angEl))} %
  \pgfmathsetmacro\angA{mod(max(\angVis,#3),360)} %
  \pgfmathsetmacro\angB{mod(min(\angVis+180,#4),360} %
  \path[save path=\tmppathI,current plane,#1,opacity=0.4] (#4:\RadiusSphere) arc (#4:#3:\RadiusSphere);
  \path[save path=\tmppathII,current plane,#1]  (\angB:\RadiusSphere) arc (\angB:\angA:\RadiusSphere);
}%

\newcommand\DrawLatitudeCircle[2][1]{
  \LatitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
  \pgfmathsetmacro\sinVis{sin(#2)/cos(#2)*sin(\angEl)/cos(\angEl)}
  % angle of "visibility"
  \pgfmathsetmacro\angVis{asin(min(1,max(\sinVis,-1)))}
  \draw[current plane] (\angVis:1) arc (\angVis:-\angVis-180:1);
  \draw[current plane,opacity=0.4] (180-\angVis:1) arc (180-\angVis:\angVis:1);
}

\newcommand\DrawLatitudeArc[4][black]{
  \LatitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=1}}
  \pgfmathsetmacro\sinVis{sin(#2)/cos(#2)*sin(\angEl)/cos(\angEl)}
  % angle of "visibility"
  \pgfmathsetmacro\angVis{asin(min(1,max(\sinVis,-1)))}
  \pgfmathsetmacro\angA{max(min(\angVis,#3),-\angVis-180)} %
  \pgfmathsetmacro\angB{min(\angVis,#4)} %
  \draw[current plane,#1,opacity=0.4] (#3:\RadiusSphere) arc (#3:#4:\RadiusSphere);
  \draw[current plane,#1] (\angA:\RadiusSphere) arc (\angA:\angB:\RadiusSphere);
}

\newcommand\ClipLatitudeArc[4][black]{
  \LatitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=1}}
  \pgfmathsetmacro\sinVis{sin(#2)/cos(#2)*sin(\angEl)/cos(\angEl)}
  % angle of "visibility"
  \pgfmathsetmacro\angVis{asin(min(1,max(\sinVis,-1)))}
  \pgfmathsetmacro\angA{max(min(\angVis,#3),-\angVis-180)} %
  \pgfmathsetmacro\angB{min(\angVis,#4)} %
  \path[save path=\tmppathI,current plane,#1,opacity=0.4] (#3:\RadiusSphere) arc (#3:#4:\RadiusSphere);
  \path[save path=\tmppathII,current plane,#1] (\angA:\RadiusSphere) arc (\angA:\angB:\RadiusSphere);
}

\newcommand\ClipLatitudeArcReverse[4][black]{
  \LatitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=1}}
  \pgfmathsetmacro\sinVis{sin(#2)/cos(#2)*sin(\angEl)/cos(\angEl)}
  % angle of "visibility"
  \pgfmathsetmacro\angVis{asin(min(1,max(\sinVis,-1)))}
  \pgfmathsetmacro\angA{max(min(\angVis,#3),-\angVis-180)} %
  \pgfmathsetmacro\angB{min(\angVis,#4)} %
  \path[save path=\tmppathI,current plane,#1,opacity=0.4] (#4:\RadiusSphere) arc (#4:#3:\RadiusSphere);
  \path[save path=\tmppathII,current plane,#1] (\angB:\RadiusSphere) arc (\angB:\angA:\RadiusSphere);
}


%% document-wide tikz options and styles

\tikzset{%
  >=latex, % option for nice arrows
  inner sep=0pt,%
  outer sep=2pt,%
  mark coordinate/.style={inner sep=0pt,outer sep=0pt,minimum size=3pt,
    fill=black,circle}%
}

\begin{document}

\begin{tikzpicture} % "THE GLOBE" showcase
\def\RadiusSphere{4} % sphere radius
\def\angEl{20} % elevation angle
\def\angAz{-20} % azimuth angle

\shade[ball color = gray!40, opacity = 0.5] (0,0) circle (\RadiusSphere);

\tikzset{
    every path/.style={
        color=black
    }
}

\ClipLatitudeArc[blue]{40}{-140}{-30}
\pgfoonew \patha=new spath(\tmppathI)
\pgfoonew \pathb=new spath(\tmppathII)
\ClipLongitudeArc[blue]{-30}{40}{-40} 
\pgfoonew \pathc=new spath(\tmppathI)
\pgfoonew \pathd=new spath(\tmppathII)
\ClipLatitudeArc[blue]{-40}{-30}{-140}
\pgfoonew \pathe=new spath(\tmppathI)
\pgfoonew \pathf=new spath(\tmppathII)
\ClipLongitudeArc[blue]{-140}{-30}{40}
\pgfoonew \pathg=new spath(\tmppathI)
\pgfoonew \pathh=new spath(\tmppathII)

\patha.concatenate with lineto(,\pathc)
\patha.concatenate with lineto(,\pathe)
\patha.concatenate with lineto(,\pathg)
\patha.close()

\patha.use path with tikz(line width=1pt,draw=black,fill=blue,path fading=south)

\end{tikzpicture}
\end{document}

Although this comes somewhat closer to what I wish to achieve, I am still puzzled that these paths can not be used for clippings. (This also means that John Kormylo's approach is much cleaner and better.) And I am wondering if there is a cleaner or at least alternative way that does not rely on the inofficial spath package.

Okay, the main problem is that the clip has to be one continuous path, and you have it subdivided. That plus the whole [current plane] style which simply complicates matters. Tell Alan Mathes that his macros are a pain. — John Kormylo, Commented Dec 31, 2017 at 15:42
@JohnKormylo I see what you mean but IMHO your code here is also quite sophisticated. Anyway, thanks a lot for your input and Happy New Year! — user121799, Commented Dec 31, 2017 at 23:56
I worked out the math and have done some trial runs, but am having trouble converting my longitudes to yours. I assume 0 longitude is at the center, in which case -140 is on the opposite side of the globe. — John Kormylo, Commented Jan 12, 2018 at 19:05
@JohnKormylo Oh WOW! These are not my longitudes, but they have been invented by Alain Matthes here. Yet I do not really understand what you mean by 0 longitude is at the center. But most likely you only need to play with the azimuth angle \angAz in order to get coordinates that you like better (but I never did that). — user121799, Commented Jan 12, 2018 at 19:11
It looks like 0 longitude is defined as right edge of the sphere, which is closer to arc angles. — John Kormylo, Commented Jan 12, 2018 at 19:15

John Kormylo · Accepted Answer · 2018-01-17 17:19:57Z

To clip, one must complete the border in a single path. The easiest way to do this is to do everything using screen coordinates.

Unfortunately, rotate seems to have no effect on arc. One can however use \pgfpatharcto instead.

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc,fadings,decorations.pathreplacing,shadings}
\usepackage{verbatim}

\newcommand{\location}[3][\empty]{% #1=label (optional) ,#2=latitude, #3=longitude
  \pgfmathsetmacro{\Xloc}{\RadiusSphere*cos(#2)*sin(#3-\Clong)}%
  \pgfmathsetmacro{\Yloc}{-\RadiusSphere*cos(#2)*sin(\Clat)*cos(#3-\Clong) + \RadiusSphere*sin(#2)*cos(\Clat)}%
  \ifx\empty#1\else
    \expandafter\let\csname Xloc#1\endcsname=\Xloc
    \expandafter\let\csname Yloc#1\endcsname=\Yloc
  \fi
}
% compute ellipse xradius=\RX, yradius=\RY, yshift=\CY, edge angles = \ArcStart, \ArcEnd
\newcommand{\latitude}[2][\empty]{% #1=label (optional), #2=latitude
  \pgfmathsetmacro{\RX}{\RadiusSphere*cos(#2)}%
  \pgfmathsetmacro{\RY}{\RX*sin(\Clat)}%
  \pgfmathsetmacro{\CY}{\RadiusSphere*sin(#2)*cos(\Clat)}%
  \pgfmathparse{tan(#2)*tan(\Clat)}%
  \pgfmathparse{ifthenelse(\pgfmathresult<1,\pgfmathresult,1)}%
  \pgfmathparse{ifthenelse(\pgfmathresult>-1,\pgfmathresult,-1)}%
  \pgfmathsetmacro{\ArcStart}{asin(\pgfmathresult)}%
  \pgfmathsetmacro{\ArcEnd}{-180-\ArcStart}%
  \ifx\empty#1\else
    \expandafter\let\csname RX#1\endcsname=\RX
    \expandafter\let\csname RY#1\endcsname=\RY
    \expandafter\let\csname CY#1\endcsname=\CY
    \expandafter\let\csname ArcStart#1\endcsname=\ArcStart
    \expandafter\let\csname ArcEnd#1\endcsname=\ArcEnd
  \fi
}
% compute ellipse rotation=\ROT, xradius=\RX, arc angle at equator=\LAT
\newcommand{\longitude}[2][\empty]{% #1=label (optional), #2=longitude
  \pgfmathsetmacro{\ROT}{atan2(sin(\Clat)*sin(#2-\Clong),cos(#2-\Clong))}%
  \pgfmathsetmacro{\LAT}{asin(cos(\Clat)*cos(\ROT))}% north pole
  \pgfmathsetmacro{\RX}{\RadiusSphere*tan(\LAT)*tan(\ROT)}%
  \pgfmathsetmacro{\LAT}{\LAT-90}%
  \ifx\empty#1\else
    \expandafter\let\csname ROT#1\endcsname=\ROT
    \expandafter\let\csname RX#1\endcsname=\RX
    \expandafter\let\csname LAT#1\endcsname=\LAT
  \fi
}

\begin{document}

\begin{tikzpicture}% "THE GLOBE" showcase
\def\RadiusSphere{4}% sphere radius
\def\Clat{20}% point of view latitude
\def\Clong{-90}% point of view longitude

\latitude[A]{40}%
\latitude[B]{-30}%
\longitude[C]{-140}%
\longitude[D]{-30}%
\location[AC]{40}{-140}%
\location[AD]{40}{-30}%
\location[BC]{-30}{-140}%
\location[BD]{-30}{-30}%

\begin{scope}
  \pgfpathcircle{\pgfpointorigin}{\RadiusSphere cm}%
  \pgfpathmoveto{\pgfpointxy{\XlocAC}{\YlocAC}}%
  \pgfpatharcto{\RXA cm}{\RYA cm}{0}{0}{1}{\pgfpointxy{\XlocAD}{\YlocAD}}%
  \pgfpatharcto{\RXD cm}{\RadiusSphere cm}{\ROTD}{0}{0}{\pgfpointxy{\XlocBD}{\YlocBD}}%
  \pgfpatharcto{\RXB cm}{\RYB cm}{0}{0}{0}{\pgfpointxy{\XlocBC}{\YlocBC}}%
  \pgfpatharcto{\RXC cm}{\RadiusSphere cm}{\ROTC}{0}{0}{\pgfpointxy{\XlocAC}{\YlocAC}}%
  \pgfpathclose
  \pgfusepath{clip}%
  \shade[ball color = gray!40, opacity = 0.5] (0,0) circle (\RadiusSphere);
\end{scope}

\draw[blue] (0,\CYA) circle[x radius=\RXA, y radius=\RYA];
%\location{40}{\ArcStartA}%
%\draw[red] (\Xloc,\Yloc) arc[x radius=\RXA, y radius=\RYA, start angle=\ArcStartA, end angle=\ArcEndA];
\draw[blue] (0,\CYB) circle[x radius=\RXB, y radius=\RYB];
\draw[blue] (0,0) circle[x radius=\RXC, y radius=\RadiusSphere, rotate=\ROTC];
\draw[blue] (0,0) circle[x radius=\RXD, y radius=\RadiusSphere, rotate=\ROTD];

\end{tikzpicture}
\end{document}

This code fragment performs the same function using \pgfpatharcaxes. I didn't use this originally as it does not explicitly use rotate.

\begin{scope}
  \pgfpathcircle{\pgfpointorigin}{\RadiusSphere cm}%
  \pgfpathmoveto{\pgfpointxy{\XlocAC}{\YlocAC}}%
  \pgfpatharcaxes{-140}{-30}{\pgfpointxy{\RXA}{0}}{\pgfpointxy{0}{\RYA}}%
  \pgfpatharcaxes{40+\LATD}{-30+\LATD}{\pgfpointpolarxy{\ROTD}{\RXD}}%
    {\pgfpointpolarxy{\ROTD+90}{\RadiusSphere}}%
  \pgfpatharcaxes{-30}{-140}{\pgfpointxy{\RXB}{0}}{\pgfpointxy{0}{\RYB}}%
  \pgfpatharcaxes{-30+\LATC}{40+\LATC}{\pgfpointpolarxy{\ROTC}{\RXC}}%
    {\pgfpointpolarxy{\ROTC+90}{\RadiusSphere}}%
  \pgfpathclose
  \pgfusepath{clip}%
  \shade[ball color = gray!40, opacity = 0.5] (0,0) circle (\RadiusSphere);
\end{scope}

The following code fragment uses normal TikZ \clip. It is based on this question & answer.

\begin{scope}
  \clip (0,0) circle[radius=\RadiusSphere] (\XlocAC,\YlocAC)
    arc[x radius=\RXA, y radius=\RYA, start angle=-140, end angle=-30]
    {[rotate=\ROTD] arc[x radius=\RXD, y radius=\RadiusSphere,
      start angle={40+\LATD}, end angle={-30+\LATD}]}
    arc[x radius=\RXB, y radius=\RYB, start angle=-30, end angle=-140]
    {[rotate=\ROTC] arc[x radius=\RXC, y radius=\RadiusSphere,
      start angle={-30+\LATC}, end angle={40+\LATC}]}
    -- cycle;
  \shade[ball color = gray!40, opacity = 0.5] (0,0) circle (\RadiusSphere);
\end{scope}

This is great! Your parametrizations seem so much simpler than the above ones! — user121799, Commented Jan 14, 2018 at 22:41
I also updated my post by an alternative method that partially successful. I'd like to ask you whether you recommend to post this as a separate question. — user121799, Commented Jan 14, 2018 at 23:30

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