[Tex/LaTex] Tikz: drawing concentric circles with specified origins and the arc that connects the intersections

nodestikz-pgf

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:

enter image description here

Here is another extremely poor example:

enter image description here

Best Answer

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

enter image description here

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

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