% \iffalse
%% File: randomwalk.dtx Copyright (C) 2011-2017 Bruno Le Floch
%%
%% This work may be distributed and/or modified under the
%% conditions of the LaTeX Project Public License, either version 1.3
%% of this license or (at your option) any later version.
%% The latest version of this license is in
%%   http://www.latex-project.org/lppl.txt
%%
%% This work has the LPPL maintenance status 'maintained'
%% and the current maintainer is Bruno Le Floch.
%%
%% This work consists of the files randomwalk.dtx and randomwalk.ins and
%% derived files randomwalk.sty and randomwalk.pdf.
%% -----------------------------------------------------------------------
%
%<*driver|package>
%</driver|package>
%<*driver>
\RequirePackage{expl3}
\documentclass[full]{l3doc}
\usepackage{randomwalk}
\AtBeginDocument{\chgrand[seed = 411500]\rand}
\usepackage{amsmath}
\begin{document}
  \DocInput{randomwalk.dtx}
\end{document}
%</driver>
% \fi
%
%^^A The date here also is in \ProvidesExplPackage, and in the copyright.
% \def\fileversion{v0.5}
% \def\filedate{2017/11/29}
%
% \title{The \pkg{randomwalk} package: \\
%   customizable random walks\thanks{This file describes version
%     \fileversion, last revised \filedate.}}
% \author{Bruno Le Floch\thanks{E-mail blflatex@gmail.com}}
% \date{Released on \filedate}
%
% \maketitle
% \tableofcontents
%
% \begin{documentation}
%
% \begin{abstract}
%
% The \pkg{randomwalk} package draws random walks. The
% following parameters can be customized:
% \begin{itemize}
%   \item The number of steps, of course.
%   \item The length of the steps, either a fixed length, or a length
%     taken uniformly at random from a given list.
%   \item The angle of each step, either taken uniformly at random
%     from a given list, or uniformly distributed between $0$~and
%     $360$ degrees.
% \end{itemize}
%
% \end{abstract}
%
%
% \section{How to use \pkg{randomwalk}}
%
% \newcommand{\examplei}
%   {\RandomWalk {number = 200, length = {4pt, 10pt}}}
% \newcommand{\exampleii}
%   {\RandomWalk {number = 100, angles = {0,60,120,180,240,300}, degree}}
% \newcommand{\exampleiii}
%   {\RandomWalk {number = 50, length = 1ex, angles = {0,24,48,-24,-48}, degree, angles-relative}}
% \begin{function}{\RandomWalk}
% The \pkg{randomwalk} package has a single user command:
% \cs{RandomWalk}, which takes a list of key-value pairs as its
% argument. A few examples are given in Figures~\ref{examplei}, \ref{exampleii}, and~\ref{exampleiii}:
% \begin{quote}\ttfamily
% \detokenize\expandafter{\examplei}\\
% \detokenize\expandafter{\exampleii}\\
% \detokenize\expandafter{\exampleiii}
% \end{quote}
% Here is a list of all the keys, and their meaning:
% \begin{itemize}
%   \item \texttt{number}: the number of steps (default \(10\))
%   \item \texttt{length}: the length of each step: either one dimension
%     (\emph{e.g.}, |1ex|), or a comma-separated list of dimensions
%     (\emph{e.g.}, |{2pt, 5pt}|), by default |10pt|. The length of each
%     step is a (uniformly distributed) random element in this set of
%     possible dimensions.
%   \item \texttt{angles}: the polar angle for each step: a
%     comma-separated list of angles, and each step takes a random angle
%     in the list.  If this is not specified, then the angle is
%     uniformly distributed along the circle.
%   \item \texttt{degree} or \texttt{degrees}: specify that the angles
%     are given in degrees (by default, they are in radians).
%   \item \texttt{angles-relative}: instead of being absolute, the
%     angles are relative to the direction of the previous step.
%   \item \texttt{revert-random} (boolean, false by default): revert the
%     seed of the random number generator to its original value after
%     the random walk.
% \end{itemize}
% \end{function}
%
% \begin{figure}
%   \begin{center}
%     \framebox{\examplei}
%     \caption{\label{examplei}A \(200\) steps long
%       walk, where each step has one of two lengths:
%       \texttt{\detokenize\expandafter{\examplei}}}
%   \end{center}
% \end{figure}
%
% \begin{figure}
%   \begin{center}
%     \framebox{\exampleii}
%     \caption{\label{exampleii}A walk with constrained angles:
%       \texttt{\detokenize\expandafter{\exampleii}}}
%   \end{center}
% \end{figure}
%
% \begin{figure}
%   \begin{center}
%     \framebox{\exampleiii}
%     \caption{\label{exampleiii}A last example, with small relative
%       angles: \texttt{\detokenize\expandafter{\exampleiii}}}
%   \end{center}
% \end{figure}
%
% \end{documentation}
%
% \begin{implementation}
%
% \section{\pkg{randomwalk} implementation}
%
% \subsection{Packages}
%
% The \pkg{expl3} bundle is loaded first.
%
%<*package>
%    \begin{macrocode}
%<@@=randomwalk>
%    \end{macrocode}
%
%    \begin{macrocode}
\RequirePackage{expl3}[2017/11/14]
\ProvidesExplPackage
  {randomwalk} {2017/11/29} {0.5} {Customizable random walks}
\RequirePackage{xparse}[2017/11/14]
%    \end{macrocode}
%
% Load \pkg{pgfcore} for figures.
%    \begin{macrocode}
\RequirePackage{pgfcore}
%    \end{macrocode}
%
% Load \pkg{lcg} for random numbers.
% It needs to know the smallest and biggest random numbers that
% should be produced, which we take to be $0$ and $\cs{c_@@_lcg_last_int}
% = 2^{31}-2$.  It will then store them in \cs{c@lcg@rand}: the |\c@| is
% there because of how \LaTeXe{} defines counters. To make it clear that
% |\c| has a very special meaning here, I do not follow \LaTeX3 naming
% conventions.  Also of note is that I use \cs{cr@nd} in \cs{@@_walk:}.
%
% It seems that the \pkg{lcg} package has to be loaded after the
% document class, hence we do it \cs{AtBeginDocument}.  Also worth noting
% is the call to \cs{rand}, which avoids some very odd bug.
%    \begin{macrocode}
\int_const:Nn \c_@@_lcg_last_int { \c_max_int - \c_one }
\AtBeginDocument
  {
    \RequirePackage
      [
        first= \c_zero ,
        last = \c_@@_lcg_last_int ,
        counter = lcg@rand
      ]
      { lcg }
    \rand
  }
%    \end{macrocode}
%
% \subsection{Variables}
%
% \begin{variable}[aux]{\l_@@_internal_tl, \l_@@_internal_int}
%   Used for scratch assignments.
%    \begin{macrocode}
\tl_new:N \l_@@_internal_tl
\int_new:N \l_@@_internal_int
%    \end{macrocode}
% \end{variable}
%
% \begin{variable}[aux]{\l_@@_step_number_int}
%   The number of steps requested by the caller.
%    \begin{macrocode}
\int_new:N \l_@@_step_number_int
%    \end{macrocode}
% \end{variable}
%
% \begin{variable}[aux]{\l_@@_relative_angles_bool, \l_@@_degrees_bool}
%   Booleans for whether angles are relative (keyval option),
%   and whether they are in degrees.
%    \begin{macrocode}
\bool_new:N \l_@@_relative_angles_bool
\bool_new:N \l_@@_degrees_bool
%    \end{macrocode}
% \end{variable}
%
% \begin{variable}[aux]{\l_@@_revert_random_bool}
%   Booleans for whether to revert the random seed to its original value
%   or keep the last value reached at the end of a random path.
%    \begin{macrocode}
\bool_new:N \l_@@_revert_random_bool
%    \end{macrocode}
% \end{variable}
%
% \begin{macro}[aux]{\@@_next_angle:, \@@_next_length:}
%   Set the \cs{l_@@_angle_fp} and \cs{l_@@_length_fp} of the next step,
%   most often randomly.
%    \begin{macrocode}
\cs_new_protected:Npn \@@_next_angle: { }
\cs_new_protected:Npn \@@_next_length: { }
%    \end{macrocode}
% \end{macro}
%
% \begin{variable}[aux]{\l_@@_angle_fp, \l_@@_length_fp}
%   Angle and length of the next step.
%    \begin{macrocode}
\fp_new:N \l_@@_angle_fp
\fp_new:N \l_@@_length_fp
%    \end{macrocode}
% \end{variable}
%
% \begin{variable}[aux]{\l_@@_x_dim, \l_@@_y_dim}
%   Current coordinates: each \cs{pgfpathlineto} statement
%   goes from the previous value of these to the next.  See
%   \cs{@@_walk_step:}.
%    \begin{macrocode}
\dim_new:N \l_@@_x_dim
\dim_new:N \l_@@_y_dim
%    \end{macrocode}
% \end{variable}
%
% \begin{variable}[aux]{\l_@@_angles_seq, \l_@@_lengths_seq}
%   Sequences containing all allowed angles and lengths, as
%   floating point numbers.
%    \begin{macrocode}
\seq_new:N \l_@@_angles_seq
\seq_new:N \l_@@_lengths_seq
%    \end{macrocode}
% \end{variable}
%
% \subsection{User command and key-value list}
%
% \begin{macro}{\RandomWalk, \randomwalk:n}
%   The user command \cs{RandomWalk} is based on the code-level
%   command \cs{randomwalk:n}, which simply does the setup and calls
%   the internal macro \cs{@@_walk:}.
%    \begin{macrocode}
\DeclareDocumentCommand \RandomWalk { m }
  { \randomwalk:n {#1} }
\cs_new_protected:Npn \randomwalk:n #1
  {
    \@@_setup_defaults:
    \keys_set:nn { randomwalk } {#1}
    \@@_walk:
  }
%    \end{macrocode}
% \end{macro}
%
% We introduce the keys for the package.
%    \begin{macrocode}
\keys_define:nn { randomwalk }
  {
    number .value_required:n = true ,
    length .value_required:n = true ,
    angles .value_required:n = true ,
    number .int_set:N = \l_@@_step_number_int ,
    length .code:n = { \@@_setup_length:n {#1} } ,
    angles .code:n = { \@@_setup_angles:n {#1} } ,
    degree .bool_set:N = \l_@@_degrees_bool ,
    degrees .bool_set:N = \l_@@_degrees_bool ,
    angles-relative .bool_set:N = \l_@@_relative_angles_bool ,
    revert-random .bool_set:N = \l_@@_revert_random_bool ,
  }
%    \end{macrocode}
%
% \subsection{Setup}
%
% \begin{macro}[aux]{\@@_setup_defaults:}
%   The package treats the length of steps, and the angle,
%   completely independently.  The function \cs{@@_next_length:}
%   contains the action that decides the length of the next step, while
%   the function \cs{@@_next_angle:} pertains to the angle.
%
%   \cs{@@_setup_defaults:} sets the default values before processing the
%   user's key-value input.  This also sets initial values of variables
%   that currently cannot be altered through keys, because it might be
%   good to provide keys for their initial values too later on.
%    \begin{macrocode}
\cs_new_protected:Npn \@@_setup_defaults:
  {
    \int_set:Nn \l_@@_step_number_int {10}
    \cs_gset_protected:Npn \@@_next_angle:
      { \@@_fp_set_rand:Nnn \l_@@_angle_fp { 0 } { 360 } }
    \cs_gset_protected:Npn \@@_next_length:
      { \fp_set:Nn \l_@@_length_fp {10} }
    \bool_set_false:N \l_@@_revert_random_bool
    \bool_set_false:N \l_@@_relative_angles_bool
    \fp_zero:N \l_@@_angle_fp
    \fp_zero:N \l_@@_length_fp
    \dim_zero:N \l_@@_x_dim
    \dim_zero:N \l_@@_y_dim
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[aux]{\@@_setup_length:n}
%   Convert each item in the comma list into a floating point, then
%   define \cs{@@_next_length:} to set \cs{l_@@_length_fp} to a random
%   floating point in the list.
%    \begin{macrocode}
\cs_new_protected:Npn \@@_setup_length:n #1
  {
    \seq_set_split:Nnn \l_@@_lengths_seq { , } {#1}
    \seq_set_map:NNn \l_@@_lengths_seq
      \l_@@_lengths_seq { \dim_to_fp:n {##1} }
    \cs_gset_protected:Npn \@@_next_length:
      {
        \@@_get_rand_seq_item:NN
          \l_@@_lengths_seq \l_@@_internal_tl
        \fp_set:Nn \l_@@_length_fp { \l_@@_internal_tl }
      }
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[aux]{\@@_setup_angles:n}
%   Two complications compared to \cs{@@_setup_length:n}.  First, the
%   angle can be given in radians rather than degrees: then add |rad|
%   after the randomly chosen value (in principle it would be better to
%   convert angles once and for all at the beginning, but that
%   interacts in a complicated way with the fact that keys can be given
%   in any order).  Second, angles can be relative, in which case we
%   use \cs{fp_add:Nn} to take the last angle into account.
%    \begin{macrocode}
\cs_new_protected:Npn \@@_setup_angles:n #1
  {
    \seq_set_split:Nnn \l_@@_angles_seq { , } {#1}
    \seq_set_map:NNn \l_@@_angles_seq
      \l_@@_angles_seq { \fp_to_tl:n {##1} }
    \cs_gset_protected:Npn \@@_next_angle:
      {
        \@@_get_rand_seq_item:NN
          \l_@@_angles_seq \l_@@_internal_tl
        \bool_if:NF \l_@@_degrees_bool
          { \tl_put_right:Nn \l_@@_internal_tl { rad } }
        \bool_if:NTF \l_@@_relative_angles_bool
          { \fp_add:Nn } { \fp_set:Nn }
          \l_@@_angle_fp { \l_@@_internal_tl }
      }
  }
%    \end{macrocode}
% \end{macro}
%
% \subsection{Drawing}
%
% \begin{macro}[aux]{\@@_walk:}
%   We are ready to define \cs{@@_walk:}, which draws a \pkg{pgf}
%   picture of a random walk with the parameters set up by the
%   \texttt{keys}.  We reset coordinates to zero originally.
%   Then draw the relevant \pkg{pgf} picture by repeatedly calling
%   \cs{@@_walk_step:}.
%    \begin{macrocode}
\cs_new_protected:Npn \@@_walk:
  {
    \@@_walk_start:
    \prg_replicate:nn { \l_@@_step_number_int }
      { \@@_walk_step: }
    \bool_if:NF \l_@@_revert_random_bool
      { \int_gset_eq:NN \cr@nd \cr@nd }
    \@@_walk_stop:
  }
%    \end{macrocode}
%   \cs{cr@nd} is internal to the lcg package.
% \end{macro}
%
% \begin{macro}[aux]{\@@_walk_start:, \@@_walk_line:, \@@_walk_stop:}
%   These functions encapsulate all of the \pkg{pgf}-related code.  The
%   \texttt{start} function begins the pgfpicture environment and
%   starts a path at position (x,y).  The \texttt{line} function adds
%   to the path a line from the previous position to the new (x,y).
%   The \texttt{stop} function draws the path constructed by
%   \cs{@@_walk_step:} and ends the pgfpicture environment.
%    \begin{macrocode}
\cs_new_protected:Npn \@@_walk_start:
  {
    \begin{pgfpicture}
      \pgfpathmoveto
        { \pgfpoint { \l_@@_x_dim } { \l_@@_y_dim } }
  }
\cs_new_protected:Npn \@@_walk_line:
  {
    \pgfpathlineto
      { \pgfpoint { \l_@@_x_dim } { \l_@@_y_dim } }
  }
\cs_new_protected:Npn \@@_walk_stop:
  {
      \pgfusepath { stroke }
    \end{pgfpicture}
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[aux]{\@@_walk_step:}
%   \cs{@@_walk_step:} calls \cs{@@_next_length:} and
%   \cs{@@_next_angle:} to determine the length and angle of the new
%   step.  This is then converted to cartesian coordinates and added to
%   the previous end-point.  Finally, we call \pkg{pgf}'s \cs{pgfpathlineto} to
%   produce a line to the new point.
%    \begin{macrocode}
\cs_new_protected:Npn \@@_walk_step:
  {
    \@@_next_length:
    \@@_next_angle:
    \dim_add:Nn \l_@@_x_dim
      {
        \fp_to_dim:n
          { \l_@@_length_fp * cosd ( \l_@@_angle_fp ) }
      }
    \dim_add:Nn \l_@@_y_dim
      {
        \fp_to_dim:n
          { \l_@@_length_fp * sind ( \l_@@_angle_fp ) }
      }
    \@@_walk_line:
  }
%    \end{macrocode}
% \end{macro}
%
% \subsection{On random numbers and items}
%
% For random numbers, the interface of \pkg{lcg} is not quite enough, so
% we provide our own \LaTeX3-y functions.  Also, this will allow us to
% change quite easily our source of random numbers.
%
% \begin{macro}[aux]{\@@_fp_set_rand:Nnn}
%   We also need floating point random numbers, assigned
%   to the variable |#1|.
%    \begin{macrocode}
\cs_new_protected:Npn \@@_fp_set_rand:Nnn #1#2#3
  {
    \rand
    \fp_set:Nn #1
      { #2 + (#3 - (#2)) * \c@lcg@rand / \c_@@_lcg_last_int }
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[aux]{\@@_get_rand_seq_item:NN}
%   We can now pick an element at random from a sequence.  If the
%   sequence has a single element, no need for randomness.
%    \begin{macrocode}
\cs_new_protected:Npn \@@_get_rand_seq_item:NN #1#2
  {
    \int_set:Nn \l_@@_internal_int { \seq_count:N #1 }
    \int_compare:nTF { \l_@@_internal_int = 1 }
      { \tl_set:Nx #2 { \seq_item:Nn #1 { 1 } } }
      {
        \rand
        \tl_set:Nx #2
          {
            \seq_item:Nn #1
              {
                1 +
                \int_mod:nn { \c@lcg@rand } { \l_@@_internal_int }
              }
          }
      }
  }
%    \end{macrocode}
% \end{macro}
%
%    \begin{macrocode}
%</package>
%    \end{macrocode}
%
% \end{implementation}
%
% \endinput
