Defining a vector of variables in Tikz

Question

I have the following animation in beamer:

with the following code (probably not optimal):

\documentclass{beamer}
\usepackage{tikz}
\usepackage{multimedia}
\begin{document}
\begin{frame}[label=persistence]
\animate<1-10>
\begin{columns}
    \begin{column}{5cm}
    \foreach \n in {1,...,10} {
    \begin{tikzpicture}[radius=2pt]
    \only<\n>{
        \node  at (-1,6){};
        \node  at (6.5,0){};
        \begin{scope}[fill opacity=0.2]
        \filldraw[fill=yellow,draw=black] (1,1) circle (2+3*\n pt);
        \filldraw[fill=yellow,draw=black] (2.3,1.1) circle (2+3*\n pt);
        \filldraw[fill=yellow,draw=black] (4.5,0.8) circle (2+3*\n pt);
        \filldraw[fill=yellow,draw=black] (5.1,1.8) circle (2+3*\n pt);
        \filldraw[fill=yellow,draw=black] (0.4,3.3) circle (2+3*\n pt);
        \filldraw[fill=yellow,draw=black] (2.1,2.8) circle (2+3*\n pt);
        \filldraw[fill=yellow,draw=black] (3.8,3.5) circle (2+3*\n pt);
        \filldraw[fill=yellow,draw=black] (4.8,4.2) circle (2+3*\n pt);
        \filldraw[fill=yellow,draw=black] (0.8,4.9) circle (2+3*\n pt);
        \filldraw[fill=yellow,draw=black] (2.1,4.1) circle (2+3*\n pt);
        \filldraw[fill=yellow,draw=black] (3.8,2.0) circle (2+3*\n pt);
        \filldraw[fill=yellow,draw=black] (3.5,0.6) circle (2+3*\n pt);
        \filldraw[fill=yellow,draw=black] (3.0,5.0) circle (2+3*\n pt);   
        \filldraw[fill=yellow,draw=black] (4.1,5.1) circle (2+3*\n pt);          
        \filldraw[fill=yellow,draw=black] (0.9,2.1) circle (2+3*\n pt);
        \end{scope}
        \filldraw[red] (1,1) circle ;
        \filldraw[red] (2.3,1.1) circle ;
        \filldraw[red] (4.5,0.8) circle ;
        \filldraw[red] (5.1,1.8) circle ;
        \filldraw[red] (0.4,3.3) circle ;
        \filldraw[red] (2.1,2.8) circle ;
        \filldraw[red] (3.8,3.5) circle ;
        \filldraw[red] (4.8,4.2) circle ;
        \filldraw[red] (0.8,4.9) circle ;
        \filldraw[red] (2.1,4.1) circle ;
        \filldraw[red] (3.8,2.0) circle ;
        \filldraw[red] (3.5,0.6) circle ;
        \filldraw[red] (3.0,5.0) circle ;   
        \filldraw[red] (4.1,5.1) circle ;          
        \filldraw[red] (0.9,2.1) circle ;
        }   
    \end{tikzpicture}
    }
    \end{column}
    \begin{column}{.4\textwidth}
    \begin{tikzpicture}     
    \filldraw[red] (1,1) circle ;
        \filldraw[red] (2.3,1.1) circle ;
        \filldraw[red] (4.5,0.8) circle ;
        \filldraw[red] (5.1,1.8) circle ;
        \filldraw[red] (0.4,3.3) circle ;
        \filldraw[red] (2.1,2.8) circle ;
        \filldraw[red] (3.8,3.5) circle ;
        \filldraw[red] (4.8,4.2) circle ;
        \filldraw[red] (0.8,4.9) circle ;
        \filldraw[red] (2.1,4.1) circle ;
        \filldraw[red] (3.8,2.0) circle ;
        \filldraw[red] (3.5,0.6) circle ;
        \filldraw[red] (3.0,5.0) circle ;   
        \filldraw[red] (4.1,5.1) circle ;          
        \filldraw[red] (0.9,2.1) circle ;
    \end{tikzpicture}
    \end{column}

\end{columns}
\end{frame}
\end{document}

Now, with the same set of points (the reds circles) I need to create an animate graph on the right side of the slide, where an edge between point $P$ and $Q$ will appear if the intersection between the neighborhoods is not empty. Is it possible with Tikz? The solution, that I have in mind is to define a vector for the points and then use two nested for. But I don't know how to do this.

Answer 1

For the first part of the requirements, the TikZ \foreach command can parse a list of coordinates which can be stored in a macro. The following illustrates how it can be done. It should be straightforward to adapt the code for the required use-case:

\documentclass[tikz,border=5]{standalone}
\begin{document}
\def\pointlist{
  (1.0,1.0), (2.3,1.1), (4.5,0.8), 
  (5.1,1.8), (0.4,3.3), (2.1,2.8),
  (3.8,3.5), (4.8,4.2), (0.8,4.9), 
  (2.1,4.1), (3.8,2.0), (3.5,0.6),
  (3.0,5.0), (4.1,5.1), (0.9,2.1) 
}
\begin{tikzpicture}[radius=2pt]
  \begin{scope}[fill opacity=0.2]
    \foreach \point in \pointlist
      \filldraw[fill=yellow,draw=black] \point circle [radius=5pt];
   \end{scope}
   \foreach \point in \pointlist
     \filldraw[red] \point circle;
\end{tikzpicture}
\end{document}

And assuming I understand the second part:

\documentclass[tikz,border=5]{standalone}
\begin{document}
\def\pointlist{
  (1.0,1.0), (2.3,1.1), (4.5,0.8), 
  (5.1,1.8), (0.4,3.3), (2.1,2.8),
  (3.8,3.5), (4.8,4.2), (0.8,4.9), 
  (2.1,4.1), (3.8,2.0), (3.5,0.6),
  (3.0,5.0), (4.1,5.1), (0.9,2.1) 
}
\foreach \N in {1,...,10}{
\begin{tikzpicture}[radius=2pt]
  \useasboundingbox (-1,-.5) rectangle (6.25,6.25);
  \begin{scope}[fill opacity=0.2]
    \foreach \point in \pointlist
      \filldraw[fill=yellow,draw=black] \point circle [radius=2pt+3*\N];
   \end{scope}
   \foreach \point in \pointlist
     \filldraw[red] \point circle;
   \foreach \P [count=\i] in \pointlist
     \foreach \Q [count=\j]in \pointlist {
       \ifnum\j>\i
       \else
         \path \P coordinate (P) \Q coordinate (Q);
         \pgfpointdiff{\pgfpointanchor{P}{center}}{\pgfpointanchor{Q}{center}}
         \pgfgetlastxy\x\y
         \pgfmathparse{int(veclen(\x,\y)/2 < 2+3*\N)}
         \ifnum\pgfmathresult=1
           \draw [thick] (P) -- (Q);
         \fi
       \fi
     }
\end{tikzpicture}
}
\end{document}

Following amorvincomni's answer, here is an alternative way of doing things using the math library:

\documentclass[tikz,border=5]{standalone}
\usetikzlibrary{math}
\begin{document}
\def\pointlist{
  (1.0,1.0), (2.3,1.1), (4.5,0.8), 
  (5.1,1.8), (0.4,3.3), (2.1,2.8),
  (3.8,3.5), (4.8,4.2), (0.8,4.9), 
  (2.1,4.1), (3.8,2.0), (3.5,0.6),
  (3.0,5.0), (4.1,5.1), (0.9,2.1) 
}
\foreach \N in {1,...,10}{
\begin{tikzpicture}[radius=2pt]
  \useasboundingbox (-1,-.5) rectangle (14.25,6.25);
  \begin{scope}[fill opacity=0.2]
    \foreach \point in \pointlist
      \filldraw[fill=yellow,draw=black] \point circle [radius=2pt+3*\N];
   \end{scope}
   \foreach \point in \pointlist
     \filldraw[red] \point circle;
   \begin{scope}[xshift=8cm]
     \foreach \point in \pointlist
       \filldraw[red] \point circle;
     \foreach \P [count=\i] in \pointlist
       \foreach \Q [count=\j]in \pointlist {
         \ifnum\j>\i
           \tikzmath{%
             coordinate \p, \q, \r;
             \p = \P; \q = \Q; 
             \pq = veclen(\px-\qx, \py-\qy)/2;
             \d = 2pt+3*\N;
             if (\pq < \d) then {
               { 
                 \draw \P -- \Q;
                 \foreach \R [count=\k] in \pointlist {
                   \ifnum\k>\j
                     \tikzmath{%
                       \r = \R;
                       \pr = veclen(\px-\rx, \py-\ry)/2;
                       \qr = veclen(\qx-\rx, \qy-\ry)/2;             
                       if (\pr < \d) && (\qr < \d) then {
                         { 
                           \fill [fill=yellow, fill opacity=.2] \P -- \Q -- \R;                        
                         };
                       };
                     }
                   \fi
                 }
               };
             };
           }
        \fi
      }
  \end{scope}
\end{tikzpicture}
}
\end{document}

Answer 2

Just for fun: from the good answer of Mark Wibrow I have made the following code. It is not optimal, but it creates filled triangles when the intersection of the three neighbourhood corresponding to the vertices of the triangle is not empty:

\begin{frame}[label=persistence]
\animate<1-27>
\scalebox{.8}{%
\begin{columns}
    \begin{column}{5cm}
    \begin{tikzpicture}[radius=2pt]%        
    \foreach \n in {1,...,27}% 
    {%      
    \only<\n>{%
        \useasboundingbox (-1,-1.5) rectangle (6.25,7.25);      
        \begin{scope}[fill opacity=0.2]
            \foreach \point in \pointlist {
        \filldraw[fill=yellow,draw=black] \point circle (5+\n pt);
        }
        \end{scope}
        \foreach \point in \pointlist {
        \filldraw[red] \point circle;}
        }
    }
    \end{tikzpicture}

    \end{column}
    \begin{column}{5cm}
        \begin{tikzpicture}[radius=2pt]
            \useasboundingbox (-1,-1.5) rectangle (6.25,7.25);              
            \foreach \n in {1,...,27}{%
                \only<\n>{%
                    \foreach \point in \pointlist
                        \filldraw[red] \point circle;
                    \foreach \P [count=\i] in \pointlist
                        \foreach \Q [count=\j]in \pointlist {
                        \ifnum\j>\i
                            \path \P coordinate (P) \Q coordinate (Q);
                            \pgfpointdiff{\pgfpointanchor{P}{center}}{\pgfpointanchor{Q}{center}}
                            \pgfgetlastxy\x\y
                            \pgfmathparse{int(veclen(\x,\y)/2 < 5+\n)}
                            \ifnum\pgfmathresult=1
                                \draw [thick] (P) -- (Q);
                                \foreach \T [count=\k] in \pointlist {  
                                \ifnum\k>\j
                                \path \P coordinate (P) \T coordinate (T);
                                \pgfpointdiff{\pgfpointanchor{P}{center}}{\pgfpointanchor{T}{center}}
                                \pgfgetlastxy\x\y
                                \pgfmathparse{int(veclen(\x,\y)/2 < 5+\n)}
                                \ifnum\pgfmathresult=1
                                \coordinate  (A) at ($(P)!0.5!(Q)$);
                                \coordinate  (C) at ($(P)!0.5!(T)$);
                                \coordinate  (A') at ($(A)!2cm!90:(P)$);
                                \coordinate  (C') at ($(C)!2cm!90:(P)$);
                                \coordinate  (O) at (intersection of A--A' and C--C');
                                \pgfpointdiff{\pgfpointanchor{O}{center}}{\pgfpointanchor{T}{center}}
                                \pgfgetlastxy\x\y
                                \pgfmathparse{int(veclen(\x,\y) < 5+\n)}
                                \ifnum\pgfmathresult=1
                                \pgfpointdiff{\pgfpointanchor{O}{center}}{\pgfpointanchor{Q}{center}}
                                \pgfgetlastxy\x\y
                                \pgfmathparse{int(veclen(\x,\y) < 5+\n)}
                                \ifnum\pgfmathresult=1
                                \pgfpointdiff{\pgfpointanchor{O}{center}}{\pgfpointanchor{P}{center}}
                                \pgfgetlastxy\x\y
                                \pgfmathparse{int(veclen(\x,\y) < 5+\n)}
                                \ifnum\pgfmathresult=1
                                \begin{scope}[fill opacity=0.3]
                                    \draw[fill=yellow] (P) --  (Q) --  (T) -- cycle;
                                    \end{scope}
                                \fi
                                \fi
                                \fi
                                \fi
                                \fi
                                }%
                            \fi
                        \fi
                        }%
                }
            }
        \end{tikzpicture}

    \end{column}
\end{columns}
}
\end{frame}

\end{document}

This is the results: