[Tex/LaTex] Drawing a perspective ellipse with TikZ

tikz-3dtikz-3dplottikz-pgf

I made the following ellipse in Inkscape

But I would like to draw it within TikZ environment.My final goal is to put this ellipse inside a coordinate system. Using the image from Inkscape, here is my try

\documentclass[tikz,border=3.14mm]{standalone}
\usepackage{tikz-3dplot}
\usetikzlibrary{3d,shapes.geometric,shadows.blur}
\usepackage{graphicx}
% small fix for canvas is xy plane at z % https://tex.stackexchange.com/a/48776/121799
\makeatletter
\tikzoption{canvas is xy plane at z}[]{%
    \def\tikz@plane@origin{\pgfpointxyz{0}{0}{#1}}%
    \def\tikz@plane@x{\pgfpointxyz{1}{0}{#1}}%
    \def\tikz@plane@y{\pgfpointxyz{0}{1}{#1}}%
    \tikz@canvas@is@plane}
\makeatother
\begin{document}
\tdplotsetmaincoords{60}{130}
\begin{tikzpicture}[scale=3.2,tdplot_main_coords,>=latex,line join=bevel]
    \coordinate (O) at (0,0,0);
    \draw[thick,->] (O) -- (1.2,0,0) node[anchor=north east]{$x$};
    \draw[thick,->] (O) -- (0,1.2,0) node[anchor=north west]{$y$};
    \draw[thick,->] (O) -- (0,0,1.2) node[anchor=south]{$z$};
    \draw[dashed] (O) -- (-1.2,0,0);

    \pgfmathsetmacro{\rvec}{1.5}
    \pgfmathsetmacro{\thetavec}{40}
    \pgfmathsetmacro{\phivec}{60}
    \tdplotsetcoord{P}{\rvec}{\thetavec}{\phivec}
   \node[anchor=south west,color=red] at (P) {$B$};
   \draw[-stealth,color=red,very thick] (O) -- (P);

   \begin{scope}[canvas is yz plane at x=0]
   \node[] (elliL) at (0,0) {\includegraphics[width=.1\textwidth]{image.eps}}; 
   \end{scope}  
\end{tikzpicture}
\end{document}

Using Tikz my best attempt was

 \documentclass[tikz,border=3.14mm]{standalone}
\usepackage{tikz-3dplot}
\usetikzlibrary{3d,shapes.geometric,shadows.blur}
\usepackage{graphicx}
% small fix for canvas is xy plane at z % https://tex.stackexchange.com/a/48776/121799
\makeatletter
\tikzoption{canvas is xy plane at z}[]{%
    \def\tikz@plane@origin{\pgfpointxyz{0}{0}{#1}}%
    \def\tikz@plane@x{\pgfpointxyz{1}{0}{#1}}%
    \def\tikz@plane@y{\pgfpointxyz{0}{1}{#1}}%
    \tikz@canvas@is@plane}
\makeatother
\begin{document}
\tdplotsetmaincoords{60}{130}
\begin{tikzpicture}[scale=3.2,tdplot_main_coords,>=latex,line join=bevel]
    \coordinate (O) at (0,0,0);
    \draw[thick,->] (O) -- (1.2,0,0) node[anchor=north east]{$x$};
    \draw[thick,->] (O) -- (0,1.2,0) node[anchor=north west]{$y$};
    \draw[thick,->] (O) -- (0,0,1.2) node[anchor=south]{$z$};
    \draw[dashed] (O) -- (-1.2,0,0);

    \pgfmathsetmacro{\rvec}{1.5}
    \pgfmathsetmacro{\thetavec}{40}
    \pgfmathsetmacro{\phivec}{60}
    \tdplotsetcoord{P}{\rvec}{\thetavec}{\phivec}
   \node[anchor=south west,color=red] at (P) {};
   \draw[-stealth,color=red,very thick] (O) -- (P);

   \begin{scope}[canvas is xz plane at y=0]
   \node[ellipse,fill=gray,fill opacity=0.3,draw,minimum width=2cm,minimum height=1cm,rotate=90] (elliL) at (0,0) {};
   \end{scope}  
\end{tikzpicture}
\end{document}

Best Answer

This problem is actually less innocent than it might seem. The visible part of the ellipse is not obtained by just drawing an ellipse of the dimensions of the ellipsoid in the screen coordinates or, say, the xy plane. The problem as AFAIK only been used for the sphere, see e.g. the nice macros by Alain Matthes provided for a sphere and, in particular, this great answer by Fritz. Let me start by providing a brute force way to shade the relevant area.

\documentclass[tikz,border=3.14mm]{standalone}
\usepackage{tikz-3dplot}
\begin{document}
\tdplotsetmaincoords{60}{130}
\begin{tikzpicture}[scale=3.2,tdplot_main_coords,>=latex,line join=bevel]
    \coordinate (O) at (0,0,0);
    \draw[thick,->] (O) -- (1.2,0,0) node[anchor=north east]{$x$};
    \draw[thick,->] (O) -- (0,1.2,0) node[anchor=north west]{$y$};
    \draw[thick,->] (O) -- (0,0,1.2) node[anchor=south]{$z$};
    \draw[dashed] (O) -- (-1.2,0,0);
    \pgfmathsetmacro{\mya}{0.4}
    \pgfmathsetmacro{\myb}{0.8}
    % lines in the background
    \draw[gray,dashed] plot[variable=\x,domain=-70:-250,smooth,samples=51]({\mya*cos(\x)},{0},{\myb*sin(\x)});
    \draw[gray,dashed] plot[variable=\x,domain=-70:-250,smooth,samples=51]({0},{\mya*cos(\x)},{\myb*sin(\x)});
    \draw[gray,dashed] plot[variable=\x,domain=\tdplotmainphi:\tdplotmainphi+180,smooth,samples=51]({\mya*cos(\x)},{\mya*sin(\x)},0);   
    % fill
    \pgfmathtruncatemacro{\Xstart}{\tdplotmainphi-180}
    \pgfmathtruncatemacro{\DeltaX}{10}
    \pgfmathtruncatemacro{\Xnext}{\Xstart+\DeltaX}
    \pgfmathtruncatemacro{\Xend}{\tdplotmainphi+180}
    \begin{scope}[transparency group,opacity=0.5]
    \foreach \X in {\Xstart,\Xnext,...,\Xend}
    {\tdplotsetrotatedcoords{0}{0}{\X}
    \begin{scope}[tdplot_rotated_coords]
    \path[fill=gray!40] plot[variable=\x,domain=-90:90,smooth,samples=51]({\mya*cos(\x)},{0},{{\myb*sin(\x)}});
    \end{scope}}
    \end{scope}
    % lines in the foreground
    \draw[gray] plot[variable=\x,domain=-70:110,smooth,samples=51]({\mya*cos(\x)},{0},{\myb*sin(\x)});
    \draw[gray] plot[variable=\x,domain=-70:110,smooth,samples=51]({0},{\mya*cos(\x)},{\myb*sin(\x)});
    \draw[gray] plot[variable=\x,domain=\tdplotmainphi-180:\tdplotmainphi,smooth,samples=51]({\mya*cos(\x)},{\mya*sin(\x)},0);  
    % redraw "visible" part of the axes
    \draw[thick,->] (\mya,0,0) -- (1.2,0,0);
    \draw[thick,->] (0,\mya,0) -- (0,1.2,0);
    \draw[thick,->] (0,0,\myb) -- (0,0,1.2);   
    \pgfmathsetmacro{\rvec}{1.5}
    \pgfmathsetmacro{\thetavec}{40}
    \pgfmathsetmacro{\phivec}{60}
    \tdplotsetcoord{P}{\rvec}{\thetavec}{\phivec}
   \node[anchor=south west,color=red] at (P) {};
   \draw[-stealth,color=red,very thick] (O) -- (P);
\end{tikzpicture}
\end{document}

A somewhat more analytic variant thereof is

\documentclass[tikz,border=3.14mm]{standalone}
\usepackage{tikz-3dplot}
\usetikzlibrary{intersections,backgrounds}
\makeatletter
%from https://tex.stackexchange.com/a/375604/121799
%along x axis
\define@key{x sphericalkeys}{radius}{\def\myradius{#1}}
\define@key{x sphericalkeys}{theta}{\def\mytheta{#1}}
\define@key{x sphericalkeys}{phi}{\def\myphi{#1}}
\tikzdeclarecoordinatesystem{x spherical}{% %%%rotation around x
    \setkeys{x sphericalkeys}{#1}%
    \pgfpointxyz{\myradius*cos(\mytheta)}{\myradius*sin(\mytheta)*cos(\myphi)}{\myradius*sin(\mytheta)*sin(\myphi)}}

%along y axis
\define@key{y sphericalkeys}{radius}{\def\myradius{#1}}
\define@key{y sphericalkeys}{theta}{\def\mytheta{#1}}
\define@key{y sphericalkeys}{phi}{\def\myphi{#1}}
\tikzdeclarecoordinatesystem{y spherical}{% %%%rotation around x
    \setkeys{y sphericalkeys}{#1}%
    \pgfpointxyz{\myradius*sin(\mytheta)*cos(\myphi)}{\myradius*cos(\mytheta)}{\myradius*sin(\mytheta)*sin(\myphi)}}

%along z axis
\define@key{z sphericalkeys}{radius}{\def\myradius{#1}}
\define@key{z sphericalkeys}{theta}{\def\mytheta{#1}}
\define@key{z sphericalkeys}{phi}{\def\myphi{#1}}
\tikzdeclarecoordinatesystem{z spherical}{% %%%rotation around x
    \setkeys{z sphericalkeys}{#1}%
    \pgfpointxyz{\myradius*sin(\mytheta)*cos(\myphi)}{\myradius*sin(\mytheta)*sin(\myphi)}{\myradius*cos(\mytheta)}}


\makeatother % https://tex.stackexchange.com/a/438695/121799

% definitions to make your life easier
\tikzset{rotate axes about y axis/.code={
\path (y spherical cs:radius=1,theta=90,phi=0+#1) coordinate(xpp)
(y spherical cs:radius=1,theta=00,phi=90+#1) coordinate(ypp) 
(y spherical cs:radius=1,theta=90,phi=90+#1) coordinate(zpp);
},rotate axes about x axis/.code={
\path (x spherical cs:radius=1,theta=00,phi=90+#1) coordinate(xpp)
(x spherical cs:radius=1,theta=90,phi=00+#1) coordinate(ypp) 
(x spherical cs:radius=1,theta=90,phi=90+#1) coordinate(zpp);
},
pitch/.style={rotate axes about y axis=#1,x={(xpp)},y={(ypp)},z={(zpp)}},
roll/.style={rotate axes about x axis=#1,x={(xpp)},y={(ypp)},z={(zpp)}}
}
\begin{document}
\tdplotsetmaincoords{60}{130}
\begin{tikzpicture}[scale=3.2,tdplot_main_coords,>=latex,line join=bevel]
    \coordinate (O) at (0,0,0);
    \draw[thick,->] (O) -- (1.2,0,0) node[anchor=north east]{$x$};
    \draw[thick,->] (O) -- (0,1.2,0) node[anchor=north west]{$y$};
    \draw[thick,->] (O) -- (0,0,1.2) node[anchor=south]{$z$};
    \draw[dashed] (O) -- (-1.2,0,0);
    \pgfmathsetmacro{\mya}{0.4}
    \pgfmathsetmacro{\myb}{0.8}
    % lines in the background
    \draw[gray,dashed] plot[variable=\x,domain=-70:-250,smooth,samples=51]({\mya*cos(\x)},{0},{\myb*sin(\x)});
    \draw[gray,dashed] plot[variable=\x,domain=-70:-250,smooth,samples=51]({0},{\mya*cos(\x)},{\myb*sin(\x)});
    \draw[gray,dashed] plot[variable=\x,domain=\tdplotmainphi:\tdplotmainphi+180,smooth,samples=51]({\mya*cos(\x)},{\mya*sin(\x)},0);   
    % fill
    \tdplotsetrotatedcoords{0}{0}{\tdplotmainphi}   
    \begin{scope}[tdplot_rotated_coords]
    \begin{scope}[roll=-5]
    \fill[gray!40,opacity=0.6] plot[variable=\x,domain=0:360,smooth,samples=51]({\mya*cos(\x)},{0},{{\myb*sin(\x)}});
    \end{scope}
    \end{scope}
    % lines in the foreground
    \draw[gray] plot[variable=\x,domain=-70:110,smooth,samples=51]({\mya*cos(\x)},{0},{\myb*sin(\x)});
    \draw[gray] plot[variable=\x,domain=-70:110,smooth,samples=51]({0},{\mya*cos(\x)},{\myb*sin(\x)});
    \draw[gray] plot[variable=\x,domain=\tdplotmainphi-180:\tdplotmainphi,smooth,samples=51]({\mya*cos(\x)},{\mya*sin(\x)},0);  
    % redraw "visible" part of the axes
    \draw[thick,->] (\mya,0,0) -- (1.2,0,0);
    \draw[thick,->] (0,\mya,0) -- (0,1.2,0);
    \draw[thick,->] (0,0,\myb) -- (0,0,1.2);   
    \pgfmathsetmacro{\rvec}{1.5}
    \pgfmathsetmacro{\thetavec}{40}
    \pgfmathsetmacro{\phivec}{60}
    \tdplotsetrotatedcoords{0}{0}{\phivec}  
    \begin{scope}[tdplot_rotated_coords]
    \path[name path=elli] plot[variable=\x,domain=0:360,smooth,samples=51]({\mya*cos(\x)},{0},{{\myb*sin(\x)}});
    \end{scope}
    \tdplotsetcoord{P}{\rvec}{\thetavec}{\phivec}
    \node[anchor=south west,color=red] at (P) {P};
    \begin{scope}[on background layer]
    \draw[-stealth,color=red,very thick,name path global=P] (O) -- (P);
    \end{scope}
    \draw[-stealth,color=red,very thick,name intersections={of=P and elli}] 
    (intersection-1) -- (P);
\end{tikzpicture}
\end{document}

This is an ellipsoid in perspective, see e.g.

to note that you view on the ellipsoid from the top, as dictated by the angle theta=60 in \tdplotsetmaincoords{60}{130}.

Related Solutions

[Tex/LaTex] Rotate a node but not its content: the case of the ellipse decoration

The source of the difficulty is that ellipses are constructed in a particular way in TikZ. They are paths that start from the x-axis and proceed counter-clockwise around their centre. The vast majority of the time, the exact parametrisation doesn't matter. You appear to have found the one situation where it does!

In the actual question, you only want to be able to mirror the ellipse, and so draw it starting from the negative x-axis (the title of the question suggests a more flexible approach). That's actually not too hard since we can exploit the symmetry of the ellipse. The key is to provide it with a negative x-radius, since then it will start from the negative x-axis (and proceed clockwise, but we could correct for that by negating the y-radius as well). To do this, we interrupt the call from the node shape to the drawing command and flip the sign of the x-radius. The simplest way to do this is to redefine the \pgfpathellipse macro to do the negation and then call the original macro. The following code does this.

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{decorations,shapes,decorations.markings}
\makeatletter
\let\origpgfpathellipse=\pgfpathellipse
\def\revpgfpathellipse#1#2#3{%
  #2%
  \pgf@xa=-\pgf@x
  \origpgfpathellipse{#1}{\pgfqpoint{\pgf@xa}{0pt}}{#3}}
\makeatother

\tikzset{
  reversed ellipse/.style={
    ellipse,
    reverse the ellipse%
  },
  reverse the ellipse/.code={
    \let\pgfpathellipse=\revpgfpathellipse
  }
}
\begin{document}
\begin{tikzpicture}
\node[ellipse,
  draw,
  postaction={
    decorate,
    decoration={
      markings,
      mark=at position 1 with {
        \arrow[line width=5pt,blue]{>}
      }
    }
  }
] at (0,0) {hello world};

\node[reversed ellipse,
  draw,
  postaction={
    decorate,
    decoration={
      markings,
      mark=at position 1 with {
        \arrow[line width=5pt,blue]{>}
      }
    }
  }
] at (0,-2) {hello world};
\end{tikzpicture}

\end{document}

Here's the result:

rotated ellipse

(the arrow got clipped, but you can see where it lies)

[Tex/LaTex] How to define the default vertical distance between nodes

I don't really get the question so I hope this is what you wanted. If you include a full document (such that we copy paste and see the problem on our systems) things are much more easier.

Here, you can change the default setting within a scope but your block style had a node distance which was resetting every time it is issued. I've made it 2mm such that we can see the difference easier.

\documentclass[tikz]{standalone}
\usetikzlibrary{arrows,shapes.geometric,positioning}

\begin{document}
\begin{tikzpicture}[decision/.style={diamond, draw, text width=4.5em, text badly centered, node distance=3.5cm, inner sep=0pt},
                    block/.style   ={rectangle, draw, text width=6em, text centered, rounded corners, minimum height=4em, minimum height=2em},
                    cloud/.style   ={draw, ellipse, minimum height=2em},
                    line/.style    ={draw,-latex'},
                    node distance = 1cm, 
                    auto]
    \node [block] (1st) {1st};
    \node [block, right= of 1st]  (2nd1) {2nd1};
\begin{scope}[node distance=2mm and 10mm]%Here we change it for everything inside this scope
    \node [block, above= of 2nd1] (2nd2) {2nd2};
    \node [block, below= of 2nd1] (2nd3) {2nd3};
    \node [block, right= of 2nd1] (3rd1) {3rd1};
    \node [block, above= of 3rd1] (3rd2) {3rd2};
    \node [block, above= of 3rd2] (3rd3) {3rd3};
\end{scope}
    \node [block, below= of 3rd1] (3rd4) {3rd4};
    \node [block, below= of 3rd4] (3rd5) {3rd5};

    \path [line] (1st) -- (2nd1);
    \path [line] (2nd1) -- (2nd2);
    \path [line] (2nd1) -- (2nd3);
    \path [line] (2nd2) -- (3rd3);
    \path [line] (2nd1) -- (3rd1);
    \path [line] (1st) -- (2nd1);
\end{tikzpicture}
\end{document}

enter image description here

Best Answer

Related Solutions

[Tex/LaTex] Rotate a node but not its content: the case of the ellipse decoration

[Tex/LaTex] How to define the default vertical distance between nodes

Related Question