Tikz: drawing concentric circles with specified origins and the arc that connects the intersections

Question

I would like to draw a few concentric circles with origins of P1 and P2 where all the circles are greater than the minimum distances. The min distances is where the circles will touch at one point only. I would also like to have that circle added to the diagram as well.

How can I find the min circles where this would occur? For the other circles greater than this distance, how can I add them in with a for loop or something of that nature?

Then I want to draw the arc that connects all the intersecting points.

\documentclass{article}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}[scale = .75]
  \node[scale = .75] (P1) at (3, 0) {\(P_1\)};
  \node[scale = .75] (P2) at (-1.5, 2) {\(P_2\)};
  \node[scale = .75] (F) at (0, 0) {\(F\)};
  \draw (F) -- (P1);
  \draw (F) -- (P2);
  \draw (P1) -- (P2);
\end{tikzpicture}
\end{document}

Here is my horrible example drawing, just remember those are circles:

Here is another extremely poor example:

Answer 1

I hope this helps you with the first two parts of your question.

The calc library provides the ($(p1)!<magic>!(p2)$) syntax that evaluates to various coordinates.
If <magic> is a ratio (say, .1, .5 or even -.1) the resulting coordinate lies between (p1) (ratio = 0) and (p2) (ratio = 1). If <magic> is a coordinate (like (F)), this coordinate is projected orthogonal onto the line between (p1) and (p2) (in the example below, the resulting circles a gray).

The very small library through provides only one option: circle through. This option accepts one coordinate through which the circle (this is a node of the shape circle) goes.

The option circle through extra radius which is to be used after circle through adds its argument to the circle’s radius.

The coordinate which will produce equal-radius circles will be saved under the name (half-center), the coordinate where (F) is projected onto the line will be stored unter the name (F-center) (please use better names in your project ;)). One could also use calc’s ($<stuff>$) syntax for circle through but this way, we can reference these coordinates later without the need to let TikZ re-calculate the coordinate over and over again (and it is easier to maintain).

The—apparently un-documented—intersection cs can be used to find the intersection of line/line, lines/circle and circle/circle. This works only if the circle is a node!

There is also the intersections library which can find any intersection between arbitrary paths (refer to Gonzalo Medina’s example, and also TeX.se which has some interesting (and abusing) examples).

Now, I don’t know nothing about this “arc”. If it is a true arc (part of a circle = constant radius) you can take three coordinates (preferable the most outer ones and the center one) and calculate the needed center and start and end angle), but if not, you can add more hidden circles as I did in the example below with {draw=none}/3cm.

If you want a correct smooth hyperbole, I’mma gonna need some math.

Code

\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={}
]

  \node[dot={right:\(P_1\)}] (P1) at ( 3,   0) {};
  \node[dot={\(P_2\)}]       (P2) at (-1.5, 2) {};
  \node[dot={below:\(F\)}]   (F)  at ( 0,   0) {};
  \path [blue] (F) edge (P1) edge (P2) (P1) edge (P2);

  \draw[dashed,gray] (F) -- ($(P1)!(F)!(P2)$) coordinate (F-center);
  \path ($(P1)!.5!(P2)$) coordinate (half-center);
  \foreach \sStyle/\xFocus in {{draw=gray}/F,{draw,thick}/half}
    \foreach \cPoint in {1,2}
     \foreach \sStyleR/\dDeltaRadius[count=\cRadius from 0] in {/0cm,/1cm/,/2cm,!/3cm}
       \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] {};

  \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);
%  
  \draw[green] (i-3-1) -- (i-2-1) -- (i-1-1) -- (F-center) -- (i-1-2) -- (i-2-2) -- (i-3-2); % These are straight line segments, but would you have known? ;)
\end{tikzpicture}
\end{document}

Output

Code (with plot/smooth)

\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}
\foreach \fRatio in {.05,.1,...,.96}{%
\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={}
]
  \node[dot={right:\(P_1\)}] (P1) at ( 3,   0) {};
  \node[dot={\(P_2\)}]       (P2) at (-1.5, 2) {};
  \node[dot={below:\(F\)}]   (F)  at ( 0,   0) {};
  \path [blue] (F) edge (P1) edge (P2) (P1) edge (P2);

  \path ($(P1)!\fRatio!(P2)$) coordinate (half-center);
  \foreach \sStyle/\xFocus in {{draw,thick}/half}
    \foreach \cPoint in {1,2}
     \foreach \sStyleR/\dDeltaRadius[count=\cRadius from 0] in {/0cm,!/.25cm,!/.5cm,!/.75cm,/1cm/,!/1.5cm,/2cm,!/2.5cm,/3cm,!/3.5cm} {
       \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
     }

  \let\maxCircles\cRadius
  \edef\maxCirclesMinus{\number\numexpr\maxCircles-1\relax}%
  \foreach \cSolution in {1,2}
    \foreach \cRadius in {1,...,\maxCircles}
      \coordinate (i-\cRadius-\cSolution) at (intersection cs: first node={half:1:\cRadius}, second node={half:2:\cRadius}, solution=\cSolution);

  \def\myList{}
  \foreach \cRadius in {\maxCircles,\maxCirclesMinus,...,1} {\xdef\myList{\myList(i-\cRadius-1)}}
  \edef\myList{\myList(half-center)}
  \foreach \cRadius in {1,...,\maxCircles} {\xdef\myList{\myList(i-\cRadius-2)}}
  \draw[ultra thick,green,smooth] plot coordinates {\myList};

  % for the bounding box:
    \path (P1) circle (8cm);
    \path (P2) circle (8cm);
\end{tikzpicture}}
\end{document}

Output, .gif

Answer 2

This might be a case where it makes sense to construct the drawing using external software like Geogebra. Geogebra actually exports surprisingly good TikZ code, which can be used as a starting point for manual adjustments:

\documentclass[10pt]{article}
\usepackage{tikz}
\usetikzlibrary{arrows}
\pagestyle{empty}
\begin{document}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1.0cm,y=1.0cm]
\clip(-4.3,-3.2) rectangle (7.4,6.3);
\draw [gray] (-0.36,3.16) circle (0.64cm);
\draw [gray] (1.48,1) circle (2.2cm);
\draw [gray] (-0.36,3.16) circle (1.27cm);
\draw [gray] (-0.36,3.16) circle (1.91cm);
\draw [gray] (1.48,1) circle (2.84cm);
\draw [gray] (1.48,1) circle (3.47cm);
\draw [samples=50,domain=-0.99:0.99,rotate around={130.43:(0.56,2.08)},xshift=0.56cm,yshift=2.08cm] plot ({0.78*(1+(\x)^2)/(1-(\x)^2)},{1.18*2*\x/(1-(\x)^2)});
\begin{scriptsize}
\fill [red] (1.48,1) circle (1.5pt);
\draw[red] (1.64,1.26) node {$A$};
\fill [red] (-0.36,3.16) circle (1.5pt);
\draw[red] (-0.2,3.42) node {$B$};
\end{scriptsize}
\end{tikzpicture}
\end{document}

Answer 3

The length of the segment joining P1 and P2 can be obtained with the help of the calc library,

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc}

\begin{document}
\begin{tikzpicture}[scale = .75]
  \node (P1) at (3, 0) {\(P_1\)};
  \node (P2) at (-1.5, 2) {\(P_2\)};
  \node (F) at (0, 0) {\(F\)};
  \draw (F) -- (P1);
  \draw (F) -- (P2);
  \draw (P1) -- (P2);
\foreach \radio in {0.5,0.8,1.1}
{
\draw[green] let \p1= ($ (P2) - (P1) $), \n2 = {veclen(\x1,\y1)}
 in (P1) circle [radius=\radio*\n2];
\draw[red] let \p1= ($ (P2) - (P1) $), \n2 = {veclen(\x1,\y1)}
 in (P2) circle [radius=\radio*\n2];
}
\end{tikzpicture}

\end{document}

The intersection points of the circles can be obtained with the help of the intersections library; then this points can be joined (I used a simple bend left, since I didn't know what kind of path should join those intersection points; now I know the path is part of an hyperbola, so perhaps tomorrow I'll do the calculations to find the equation of the hyperbola):

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc,intersections}

\begin{document}
\begin{tikzpicture}[scale = .75]
  \node (P1) at (3, 0) {\(P_1\)};
  \node (P2) at (-1.5, 2) {\(P_2\)};
  \node (F) at (0, 0) {\(F\)};
  \draw (F) -- (P1);
  \draw (F) -- (P2);
  \draw (P1) -- (P2);
\foreach \radioa/\radiob [count=\i] in {0.5/0.5,0.8/0.65,1.1/0.8}
{
\draw[green,name path global=cira\i] let \p1= ($ (P2) - (P1) $), \n2 = {veclen(\x1,\y1)}
 in (P1) circle [radius=\radioa*\n2];
\draw[red,name path global=cirb\i] let \p1= ($ (P2) - (P1) $), \n2 = {veclen(\x1,\y1)}
 in (P2) circle [radius=\radiob*\n2];
}
\foreach \i in {1,2,3}
\path[name intersections={of={cira\i} and {cirb\i},by={m\i,l\i}}]; 
\draw[blue] (m3) to[bend left=10] (m2) to[bend left=10] (m1) to[bend left=10] (l2) to[bend left=10] (l3);
\end{tikzpicture}

\end{document}

Answer 4

I have adapted Qrrbrbirlbel solutions and used Jake's comment on fitting a hyperbola to the concentric circles. Additionally, I changed the location of P2 to line on the x axis as Qrrbrbirlbel suggested to make the math a bit easier. Therefore, the location of F also changed. After that, I took the acrtan to get the angle of rotation to correctly orient the picture.

\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}