Draw uniform lines/grid on a distorted square/axes?

Question

I want to draw a square discretised by equally spaced lines. However, if I move one of the nodes, say node 4, the grid lines will keep it's relative spacing (see the image). I have no idea what to search for. One idea would be to calculate coordinates on the path (1)--(4) and (4)--(3), then re-calculate the coordinates when ever I move node 4 before I draw lines between the two opposite sides. I'm wondering if there is a better way to do this.

Many thanks.

Answer 1

Here is my attempt. Initially I started looking for a solution similar to the one proposed by John Kormylo in his comment, and ended up using the decorations.markings library. I hope it is not such an overkill :-) ! In every case the grid is self-adapting, as you can see from the following images:

My code is:

\documentclass{standalone}

\usepackage{tikz}
\usetikzlibrary{decorations.markings}

\def\NrLines{8}
\pgfmathsetmacro{\UnitSegment}{1/(\NrLines+1)}

\begin{document}
\begin{tikzpicture}[draw=red]

% Nodes definition
\node[inner sep=0pt,minimum size=0pt,label=below:1] (a) at (2,5) {};
\node[inner sep=0pt,minimum size=0pt,label=below:2] (b) at (5,1) {};
\node[inner sep=0pt,minimum size=0pt,label=above:3] (c) at (6,2) {};
\node[inner sep=0pt,minimum size=0pt,label=above:4] (d) at (5,3) {};

% Paths between nodes
\draw [postaction={decorate},decoration={markings,
mark=between positions 0 and 1 step \UnitSegment with {
\node [inner sep=0pt,minimum size=0pt,
name=mark-1-\pgfkeysvalueof{/pgf/decoration/mark info/sequence number}]
{};}}] (a) -- (b);

\draw [postaction={decorate},decoration={markings,
mark=between positions 0 and 1 step \UnitSegment with {
\node [inner sep=0pt,minimum size=0pt,
name=mark-2-\pgfkeysvalueof{/pgf/decoration/mark info/sequence number}]
{};}}] (b) -- (c);

\draw [postaction={decorate},decoration={markings,
mark=between positions 0 and 1 step \UnitSegment with {
\node [inner sep=0pt,minimum size=0pt,
name=mark-3-\pgfkeysvalueof{/pgf/decoration/mark info/sequence number}]
{};}}] (c) -- (d);

\draw [postaction={decorate},decoration={markings,
mark=between positions 0 and 1 step \UnitSegment with {
\node [inner sep=0pt,minimum size=0pt,
name=mark-4-\pgfkeysvalueof{/pgf/decoration/mark info/sequence number}]
{};}}] (d) -- (a);

% Grid
\draw (mark-1-2) -- (mark-3-9);
\draw (mark-1-3) -- (mark-3-8);
\draw (mark-1-4) -- (mark-3-7);
\draw (mark-1-5) -- (mark-3-6);
\draw (mark-1-6) -- (mark-3-5);
\draw (mark-1-7) -- (mark-3-4);
\draw (mark-1-8) -- (mark-3-3);
\draw (mark-1-9) -- (mark-3-2);

\draw (mark-2-2) -- (mark-4-9);
\draw (mark-2-3) -- (mark-4-8);
\draw (mark-2-4) -- (mark-4-7);
\draw (mark-2-5) -- (mark-4-6);
\draw (mark-2-6) -- (mark-4-5);
\draw (mark-2-7) -- (mark-4-4);
\draw (mark-2-8) -- (mark-4-3);
\draw (mark-2-9) -- (mark-4-2);

% Nodes circles
\foreach \a in {a,...,d}{%
    \fill[black] (\a) circle (2pt);
}

\end{tikzpicture}
\end{document}

In this code \NrLines is defined as the number of internal lines, which are separated by a distance equal to \UnitSegment.

Unfortunately I could not find a way to draw the grid automatically; I tried with:

\pgfmathsetmacro{\End}{\NrLines+1}

\foreach \x in {1,2}{%
    \pgfmathsetmacro{\y}{\x+2}
    \foreach \i/\j in {2/\End,...,\End/2}{%
        \draw (mark-\x-\i) -- (mark-\y-\j);
    }
}

but there was a problem with using the result of \pgfsetmacro{} inside a node name. Maybe someone more expert than me may suggest a solution to use this \foreach loop.

EDIT: Finally, I improved the code to draw the grid automatically! There were two issues:

• the use of \pgfmathsetmacro inside a node name, solved thanks to Stefan Kottwitz's answer "Pointing to a node with calculated name, why do I get to the east, not to the center?": instead of pgfmathsetmacro, one should use \pgfmathtruncatemacro to remove decimal points in the result;
• the internal \foreach loop can not work using \foreach \i/\j in {2/\End,...,\End/2} since it is not clear (for the solver :-) ) how \i and \j are changing. Therefore, \j is evaluated in each step as a function of \i, which is the only counter for the loop.

The new improved code is:

\documentclass{standalone}

\usepackage{tikz}
\usetikzlibrary{decorations.markings}

\def\NrLines{8}
\pgfmathsetmacro{\UnitSegment}{1/(\NrLines+1)}
\pgfmathsetmacro{\End}{\NrLines+1}

\begin{document}
\begin{tikzpicture}[draw=red]

% Nodes definition
\node[inner sep=0pt,minimum size=0pt,label=below:1] (a) at (2,0) {};
\node[inner sep=0pt,minimum size=0pt,label=below:2] (b) at (5,0) {};
\node[inner sep=0pt,minimum size=0pt,label=above:3] (c) at (4,3) {};
\node[inner sep=0pt,minimum size=0pt,label=above:4] (d) at (3,3) {};

% Paths between nodes
\draw [postaction={decorate},decoration={markings,
mark=between positions 0 and 1 step \UnitSegment with {
\node [inner sep=0pt,minimum size=0pt,
name=mark-1-\pgfkeysvalueof{/pgf/decoration/mark info/sequence number}]
{};}}] (a) -- (b);

\draw [postaction={decorate},decoration={markings,
mark=between positions 0 and 1 step \UnitSegment with {
\node [inner sep=0pt,minimum size=0pt,
name=mark-2-\pgfkeysvalueof{/pgf/decoration/mark info/sequence number}]
{};}}] (b) -- (c);

\draw [postaction={decorate},decoration={markings,
mark=between positions 0 and 1 step \UnitSegment with {
\node [inner sep=0pt,minimum size=0pt,
name=mark-3-\pgfkeysvalueof{/pgf/decoration/mark info/sequence number}]
{};}}] (c) -- (d);

\draw [postaction={decorate},decoration={markings,
mark=between positions 0 and 1 step \UnitSegment with {
\node [inner sep=0pt,minimum size=0pt,
name=mark-4-\pgfkeysvalueof{/pgf/decoration/mark info/sequence number}]
{};}}] (d) -- (a);

% Grid
\foreach \x in {1,2}{%
    \pgfmathtruncatemacro{\y}{\x+2}
    \foreach \i in {2,...,\End}{%
        \pgfmathtruncatemacro{\j}{\End-\i+2}
        \draw (mark-\x-\i) -- (mark-\y-\j);
    }
}

% Nodes circles
\foreach \a in {a,...,d}{%
    \fill[black] (\a) circle (2pt);
}

\end{tikzpicture}
\end{document}

Now you can easily modify the grid setting the number of internal lines \NrLines, and the code will work without problems!

Answer 2

Here's one better way, using Metapost, using the point x of y notation. TikZ has the same idea, but using a rather more compact notation. With this set up you can move your four points as much as you like, and the grid will adapt itself accordingly.

prologues := 3;
outputtemplate := "%j%c.eps";

beginfig(1);

z1 = origin;
z2 = (80,0);
z3 = (96,80);
z4 = (-10,40);

n = 8;

for i=0 upto n:
  draw point i/n of (z1--z2) -- point i/n of (z4--z3) withcolor .67 red;
  draw point i/n of (z1--z4) -- point i/n of (z2--z3) withcolor .67 red;
endfor

dotlabel.llft("1", z1);
dotlabel.lrt ("2", z2);
dotlabel.urt ("3", z3);
dotlabel.ulft("4", z4);

endfig;
end.

Now I've moved z3 and z4 a bit, but changed nothing else.

Answer 3

A generalisation of John Kormylo's comment above, all tied up in pic.

\documentclass[tikz, border=5]{standalone}
\usetikzlibrary{calc}
\tikzset{pics/.cd,
grid/.style args={(#1)#2(#3)#4(#5)#6(#7)#8}{code={%
\tikzset{pics/grid/dimensions=#8}%
\foreach \i in {0,...,\y}
  \draw [pic actions/.try] ($(#1)!\i/\y!(#7)$) -- ($(#3)!\i/\y!(#5)$);
\foreach \i in {0,...,\x}
  \draw [pic actions/.try] ($(#1)!\i/\x!(#3)$) -- ($(#7)!\i/\x!(#5)$);
\path (#1) coordinate (-1) (#3) coordinate (-2)
      (#5) coordinate (-3) (#7) coordinate (-4);
}},
grid/dimensions/.code args={#1x#2}{\def\x{#1}\def\y{#2}}}
\begin{document}
\begin{tikzpicture}
\pic (A) at (0,0) [black] {grid={(0,0) (4,0)  (4,4) (0,4)  8x8}};
\pic (B) at (7,0) [red]   {grid={(0,0) (4,0)  (4,3) (0,5)  8x8}};
\pic (C) at (0,7) [green] {grid={(0,0) (4,0)  (5,5) (0,3)  8x8}};
\pic (D) at (7,7) [blue]  {grid={(0,0) (3,-1) (4,5) (-1,3) 8x8}};
\foreach \i in {A,...,D}
  \foreach \j in {1,...,4}
    \fill (\i-\j) circle [radius=.1] node [anchor=\j*90-45] {\j};
\end{tikzpicture}
\end{document}