paths and polygons on a lattice (or in general, I guess)

lattice_paths_demo.tex

\documentclass{article}

\usepackage{wcsetup}  % github:wchargin/wclatex
\usepackage{tikz}

\begin{document}

\tikzset{
  latticept/.style={
    red,
  },
  latticepath/.style={
    blue,
  },
}

% usage: \latticepath{(x0, y0){(x1, y1), (x2, y2), ...}
% (the points can actually be any TikZ coordinates)
\newcommand\latticepath[2]{%
  \draw[latticepath] #1 \foreach \point in {#2} {-- \point };
  \foreach \point in {#1, #2} \filldraw[latticept] \point circle[radius=2pt];
}

% like \latticepath, but will automatically close the path
\newcommand\latticegon[2]{\latticepath{#1}{#2, #1}}

\begin{tikzpicture}
  \latticepath{(-1, 0)}{(0, 3), (1, 0), (-1.5, 2), (1.5, 2), (-1, 0)}
  %
  % works with all your usual commands...
  \begin{scope}[
      opacity=0.8, transparency group,
      xshift=5cm,
      latticept/.style={orange,fill=white},
      latticepath/.append style={thick,dotted},
    ]
    \latticegon{(-1, 0)}{(0, 3), (1, 0), (-1.5, 2), (1.5, 2)}
  \end{scope}
\end{tikzpicture}

\end{document}