[Tex/LaTex] Tikz: placing a node at the intersection of two circles along a hyperbola

intersectionsnodestikz-pgf

I know one can use the intersection package to find the point of intersection but I don't understand how to do it yet.

I would like to label the increasing order of the intersections F^*_2, F^*_1, F^*_m. F^*_1 (but over set with a ~), and F^*_2 with the same over set.

Note: I am ignoring the lower and upper most intersections.

The code is:

\documentclass[tikz,convert=false]{standalone}
\usetikzlibrary{through,calc}
\makeatletter
\tikzset{circle through extra radius/.code={% unorthodox addon for the through library
                                            % needs to be used after 'circle through'!
                                            % this can be avoided by slightly changing the source
  \tikz@addoption{%
    \pgfmathsetlengthmacro\pgf@tempa{\pgfkeysvalueof{/pgf/minimum width}+2*(#1)}%
    \pgfset{/pgf/minimum width/.expanded=\pgf@tempa}%
  }%
}}
\tikzset{
  special style/.code={%
    \if#1\tikz@nonactiveexlmark
      \pgfkeysalso{@special style}%
    \else
      \pgfkeysalso{style/.expanded=#1}%
    \fi
  },
  @special style/.style={draw=none,fill=none}
}
\makeatother
\begin{document}
\begin{tikzpicture}[
  every label/.append style={font=\small},
  dot/.style={fill,outer sep=+0pt,inner sep=+0pt,minimum size=2pt,shape=circle,draw=none,label={#1}},
  dot/.default={}
]
\begin{scope}[rotate around ={-20.5560452:(1, -.75)}]
      \node[scale = .75, dot = {right: \(P_1\)}] (P1) at (3cm, 0) {};
      \node[scale = .75, dot = {\(P_2\)}] (P2) at (-1, 0) {};
      \node[scale = .75, dot = {below: \(F\)}] (F) at (1cm, -.75cm) {};
      \path[blue] (F) edge (P1) edge (P2) (P1) edge (P2);
      \path ($(P1)!.75!(P2)$) coordinate (half-center);
      \foreach \sStyle/\xFocus in {{draw, thick}/half}
        \foreach \cPoint in {1, 2}
          \foreach \sStyleR/\dDeltaRadius[count = \cRadius from 0] in
          {/.0cm, /.5cm, /1cm, /1.5cm}
          {
            \node[style/.expanded = \sStyle,
            special style/.expanded = {\sStyleR}] at (P\cPoint.center)
            ({\xFocus:\cPoint:\cRadius}) [circle through/.expanded =
            {(\xFocus-center)}, circle through extra radius = \dDeltaRadius] {};
            \global\let\cRadius\cRadius
         }

      \clip(-4, -5) rectangle (8, 5);
      \draw [samples = 50, domain = -0.99:0.99, xshift = 1cm, red,
      thick] plot ({(-1 - (\x)^2) / (1 - (\x)^2)},
      {1.73 * (-2) * (\x) / (1 - (\x)^2)});
    \end{scope}
  \end{tikzpicture}
  \end{document}

enter image description here

From my post Tikz: drawing concentric circles with specified origins and the arc that connects the intersections
Qrrbrbirlbel found the intersections with

\foreach \cSolution in {1,2}
    \foreach \cRadius in {1,...,3}
    \coordinate (i-\cRadius-\cSolution) at (intersection cs: first node={F:1:\cRadius}, second node={F:2:\cRadius}, solution=\cSolution);

but I am not clear on how to adapt the solution for my problem.

Edit 1:

I don't know if there is a smooth way to automate the process, but I have the middle point defined as:

\node[scale = .75, dot = {right:\(\overset{\mbox{~}}{F}_m^*\)}] (Fm)
      at ($(P1)!.75!(P2)$) {};

How can I make the the dot bigger and the overset ~ wont appear. I have tried \~ and ~ but nothing works.

Edit 2:

Close up of the output from Qrr.. answer:

The \tilde isn't overset and the node placement isn't similar.

enter image description here

Edit 3:

I was able to adjust some of the nodes but for the nodes along the hyperbola, if I adjust them more, the top nodes will be out of place.

enter image description here

\begin{tikzpicture}[scale = .7,
    every label/.append style = {font = \small},
    dot/.style = {fill, outer sep = +0pt, inner sep = +0pt,
      shape = circle, draw = none, label = {#1}},
    dot/.default =,
    small dot/.style = {minimum size = 2pt, dot = {#1}},
    small dot/.default =,
    big dot/.style = {minimum size = 3pt, dot = {#1}},
    big dot/.default =
  ]
    \begin{scope}[rotate around ={-23.9625:(.75, -1)}]
      \begin{scope}
        \clip(-2, -4) rectangle (0, 4);
        \draw [samples = 50, domain = -0.99:0.99, xshift = 1cm, red, thick]
        plot ({(-1 - (\x)^2) / (1 - (\x)^2)},
        {1.73 * (-2) * (\x) / (1 - (\x)^2)});
      \end{scope}
      \node[scale = .75, small dot = {right: \(P_1\)}] (P1) at (3, 0) {};
      \node[scale = .75, small dot = {below, left = 3.5pt: \(P_2\)}] (P2) at
      (-1, 0) {};
      \node[scale = .75, small dot = {below, left = 3.5pt: \(F\)}] (F)
      at (.75, -1) {};
      \path[blue] (F) edge (P1) edge (P2) (P1) edge (P2);
      \path ($(P1)!.75!(P2)$) coordinate (Fm) node[big dot =
      {below = 10pt, right = 3pt: \(F_m^*\)}] {};
      \foreach \cPoint in {1, 2}
        \foreach \sStyleR/\dDeltaRadius[count = \cRadius from 0] in
        {.0cm, .5cm, 1cm}
          \node[draw, thick] at (P\cPoint.center) (\cPoint: \cRadius)
          [circle through = (Fm), circle through extra radius = \dDeltaRadius]
          {};
      \foreach \cSolution in {1, 2}
        \foreach \cRadius in {1, 2}
          \node[black, big dot = {right = 2pt, below = 2pt:
            $\ifnum\cSolution = 2\expandafter\tilde\fi F^*_\cRadius$}]
          (n-\cRadius-\cSolution) at
          (intersection cs: first node = {1: \cRadius},
                            second node = {2: \cRadius},
                            solution = \cSolution) {};
    \end{scope}
  \end{tikzpicture}

Best Answer

You can use the same \foreach loop for the labeling:

\foreach \cSolution in {1,2}
  \foreach \cRadius in {1,...,2}
    \node[green,dot={right:$\ifnum\cSolution=2\expandafter\tilde\fi F^*_\cRadius$}]
         (n-\cRadius-\cSolution)
      at (intersection cs: first node={1:\cRadius},
                          second node={2:\cRadius},
                            solution=\cSolution) {};

The math behind the intersection cs seems to be a little bit un-precise, so I repeated the task with the intersections library.

(I also have cleaned up a little bit.)

Code (`intersection cs`)

\documentclass[tikz,convert=false]{standalone}
\usetikzlibrary{through,calc}
\makeatletter
\tikzset{circle through extra radius/.code={% unorthodox addon for the through library
                                            % needs to be used after 'circle through'!
                                            % this can be avoided by slightly changing the source
  \tikz@addoption{%
    \pgfmathsetlengthmacro\pgf@tempa{\pgfkeysvalueof{/pgf/minimum width}+2*(#1)}%
    \pgfset{/pgf/minimum width/.expanded=\pgf@tempa}%
  }%
}}
\makeatother
\begin{document}
\begin{tikzpicture}[
  every label/.append style={font=\small},
  dot/.style={fill,outer sep=+0pt,inner sep=+0pt,shape=circle,draw=none,label={#1}},
  dot/.default=,
  small dot/.style={minimum size=2pt,dot={#1}},
  small dot/.default=,
  big dot/.style={minimum size=3pt,dot={#1}},
  big dot/.default=
]
\begin{scope}[rotate around ={-20.5560452:(1, -.75)}]
      \node[small dot = {right: \(P_1\)}] (P1) at (3cm, 0) {};
      \node[small dot = {\(P_2\)}]        (P2) at (-1, 0) {};
      \node[small dot = {below: \(F\)}]   (F)  at (1cm, -.75cm) {};
      \path[blue] (F) edge (P1) edge (P2) (P1) edge (P2);
      \path ($(P1)!.75!(P2)$) coordinate (Fm) node[dot={right:$F_m^*$}] {};
      \foreach \cPoint in {1, 2}
        \foreach \sStyleR/\dDeltaRadius[count = \cRadius from 0] in {.0cm, .5cm, 1cm, 1.5cm}
            \node[draw] at (P\cPoint.center) (\cPoint:\cRadius) [circle through=(Fm), circle through extra radius = \dDeltaRadius] {};
      \foreach \cSolution in {1,2}
        \foreach \cRadius in {1,2}
          \node[green,big dot={right:$\ifnum\cSolution=2\expandafter\tilde\fi F^*_\cRadius$}]
            (n-\cRadius-\cSolution) at (intersection cs: first node={1:\cRadius},
                                                        second node={2:\cRadius},
                                                           solution=\cSolution) {};

      \clip(-4, -5) rectangle (8, 5);
      \draw [samples = 50, domain = -0.99:0.99, xshift = 1cm, red,
      thick] plot ({(-1 - (\x)^2) / (1 - (\x)^2)},
      {1.73 * (-2) * (\x) / (1 - (\x)^2)});
\end{scope}
\end{tikzpicture}
\end{document}

Output (`intersection cs`)

enter image description here

Code (`intersections` library)

\documentclass[tikz,convert=false]{standalone}
\usetikzlibrary{through,calc,intersections}
\makeatletter
\tikzset{circle through extra radius/.code={% unorthodox addon for the through library
                                            % needs to be used after 'circle through'!
                                            % this can be avoided by slightly changing the source
  \tikz@addoption{%
    \pgfmathsetlengthmacro\pgf@tempa{\pgfkeysvalueof{/pgf/minimum width}+2*(#1)}%
    \pgfset{/pgf/minimum width/.expanded=\pgf@tempa}%
  }%
}}
\makeatother
\begin{document}
\begin{tikzpicture}[
  every label/.append style={font=\small},
  dot/.style={fill,outer sep=+0pt,inner sep=+0pt,shape=circle,draw=none,label={#1}},
  dot/.default=,
  small dot/.style={minimum size=2pt,dot={#1}},
  small dot/.default=,
  big dot/.style={minimum size=3pt,dot={#1}},
  big dot/.default=
]
\begin{scope}[rotate around ={-20.5560452:(1, -.75)}]
      \node[small dot = {right: \(P_1\)}] (P1) at (3cm, 0) {};
      \node[small dot = {\(P_2\)}]        (P2) at (-1, 0) {};
      \node[small dot = {below: \(F\)}]   (F)  at (1cm, -.75cm) {};
      \path[blue] (F) edge (P1) edge (P2) (P1) edge (P2);
      \path ($(P1)!.75!(P2)$) coordinate (Fm) node[dot={right:$F_m^*$}] {};
      \foreach \cPoint in {1, 2}
        \foreach \dDeltaRadius[count = \cRadius from 0] in {.0cm, .5cm, 1cm, 1.5cm}
            \node[draw, name path global/.expanded=\cPoint:\cRadius] at (P\cPoint.center) (\cPoint:\cRadius) [circle through=(Fm), circle through extra radius = \dDeltaRadius] {};
      \foreach \cRadius in {1,2} {
        \tikzset{name intersections={of/.expanded={1:\cRadius} and 2:\cRadius,name/.expanded=n'-\cRadius}}
        \foreach \cSolution in {1,2}
          \node[green,big dot={right:$\ifnum\cSolution=2\expandafter\tilde\fi F^*_\cRadius$}] (n-\cRadius-\cSolution) at (n'-\cRadius-\cSolution){};
      }

      \clip (-4, -5) rectangle (8, 5);
      \draw [samples = 50, domain = -0.99:0.99, xshift = 1cm, red,
      thick] plot ({(-1 - (\x)^2) / (1 - (\x)^2)},
      {1.73 * (-2) * (\x) / (1 - (\x)^2)});
\end{scope}
\end{tikzpicture}
\end{document}

Output (`intersections` library)

enter image description here

Code

\documentclass[tikz]{standalone}
\begin{document}
\begin{tikzpicture}[
   every node/.append style={circle, draw=blue!80, inner sep=0pt, minimum size=12pt}]
\node                 (1) at (0,0) {1};
\node[thick]          (2) at (1,0) {2};
\node[line width=1mm] (3) at (2,0) {3};
\end{tikzpicture}
\end{document}

Output

enter image description here

[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

Code (intersection cs)

Output (intersection cs)

Code (intersections library)

Output (intersections library)

Related Solutions

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

Related Question

Code (`intersection cs`)

Output (`intersection cs`)

Code (`intersections` library)

Output (`intersections` library)