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% Copyright 2018 by Mark Wibrow
%
% This file may be distributed and/or modified
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% 1. under the LaTeX Project Public License and/or
% 2. under the GNU Free Documentation License.
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% See the file doc/generic/pgf/licenses/LICENSE for more details.
\section{Additional Mathematical Commands}
\label{pgfmath-commands}
Instead of parsing and evaluating complex expressions, you can also use the
mathematical engine to evaluate a single mathematical operation. The macros
used for many of these computations are listed above in
Section~\ref{pgfmath-functions}. \pgfname{} also provides some additional
commands which are shown below:
\subsection{Basic arithmetic functions}
\label{pgfmath-commands-basic}
In addition to the commands described in Section~\ref{pgfmath-functions-basic},
the following command is provided:
\begin{command}{\pgfmathreciprocal\marg{x}}
Defines |\pgfmathresult| as $1\div\meta{x}$. This provides greatest
accuracy when \mvar{x} is small.
\end{command}
\subsection{Comparison and logical functions}
In addition to the commands described in
Section~\ref{pgfmath-functions-comparison}, the following command was provided
by Christian Feuers\"anger:
\begin{command}{\pgfmathapproxequalto\marg{x}\marg{y}}
Defines |\pgfmathresult| 1.0 if $ \rvert \meta{x} - \meta{y} \lvert <
0.0001$, but 0.0 otherwise. As a side-effect, the global boolean
|\ifpgfmathcomparison| will be set accordingly.
\end{command}
\subsection{Pseudo-Random Numbers}
\label{pgfmath-random}
In addition to the commands described in
Section~\ref{pgfmath-functions-random}, the following commands are provided:
\begin{command}{\pgfmathgeneratepseudorandomnumber}
Defines |\pgfmathresult| as a pseudo-random integer between 1 and
$2^{31}-1$. This uses a linear congruency generator, based on ideas of
Erich Janka.
\end{command}
\begin{command}{\pgfmathrandominteger\marg{macro}\marg{minimum}\marg{maximum}}
This defines \meta{macro} as a pseudo-randomly generated integer from the
range \meta{minimum} to \meta{maximum} (inclusive).
%
\begin{codeexample}[]
\begin{pgfpicture}
\foreach \x in {1,...,50}{
\pgfmathrandominteger{\a}{1}{50}
\pgfmathrandominteger{\b}{1}{50}
\pgfpathcircle{\pgfpoint{+\a pt}{+\b pt}}{+2pt}
\color{blue!40!white}
\pgfsetstrokecolor{blue!80!black}
\pgfusepath{stroke, fill}
}
\end{pgfpicture}
\end{codeexample}
%
\end{command}
\begin{command}{\pgfmathdeclarerandomlist\marg{list name}{\ttfamily\{}\marg{item-1}\marg{item-2}...{\ttfamily\}}}
This creates a list of items with the name \meta{list name}.
\end{command}
\begin{command}{\pgfmathrandomitem\marg{macro}\marg{list name}}
Select an item from a random list \meta{list name}. The
selected item is placed in \meta{macro}.
\end{command}
\begin{codeexample}[]
\begin{pgfpicture}
\pgfmathdeclarerandomlist{color}{{red}{blue}{green}{yellow}{white}}
\foreach \a in {1,...,50}{
\pgfmathrandominteger{\x}{1}{85}
\pgfmathrandominteger{\y}{1}{85}
\pgfmathrandominteger{\r}{5}{10}
\pgfmathrandomitem{\c}{color}
\pgfpathcircle{\pgfpoint{+\x pt}{+\y pt}}{+\r pt}
\color{\c!40!white}
\pgfsetstrokecolor{\c!80!black}
\pgfusepath{stroke, fill}
}
\end{pgfpicture}
\end{codeexample}
\begin{command}{\pgfmathsetseed\marg{integer}}
Explicitly sets the seed for the pseudo-random number generator. By default
it is set to the value of |\time|$\times$|\year|.
\end{command}
\subsection{Base Conversion}
\label{pgfmath-bases}
\pgfname{} provides limited support for conversion between
\emph{representations} of numbers. Currently the numbers must be positive
integers in the range $0$ to $2^{31}-1$, and the bases in the range $2$ to
$36$. All digits representing numbers greater than 9 (in base ten), are
alphabetic, but may be upper or lower case.
In addition to the commands described in Section~\ref{pgfmath-functions-base},
the following commands are provided:
\begin{command}{\pgfmathbasetodec\marg{macro}\marg{number}\marg{base}}
Defines \meta{macro} as the result of converting \meta{number} from base
\meta{base} to base 10. Alphabetic digits can be upper or lower case.
\medskip{\def\medskip{}
\begin{codeexample}[]
\pgfmathbasetodec\mynumber{107f}{16} \mynumber
\end{codeexample}
\noindent Note that, as usual in \TeX, the braces around an argument can be
omitted if the argument is just a single token (a macro name is a single
token).
%
\begin{codeexample}[]
\pgfmathbasetodec\mynumber{33FC}{20} \mynumber
\end{codeexample}
}\medskip
%
\end{command}
\begin{command}{\pgfmathdectobase\marg{macro}\marg{number}\marg{base}}
Defines \meta{macro} as the result of converting \meta{number} from base 10
to base \meta{base}. Any resulting alphabetic digits are in \emph{lower
case}.
%
\begin{codeexample}[]
\pgfmathdectobase\mynumber{65535}{16} \mynumber
\end{codeexample}
%
\end{command}
\begin{command}{\pgfmathdectoBase\marg{macro}\marg{number}\marg{base}}
Defines \meta{macro} as the result of converting \meta{number} from base 10
to base \meta{base}. Any resulting alphabetic digits are in \emph{upper
case}.
%
\begin{codeexample}[]
\pgfmathdectoBase\mynumber{65535}{16} \mynumber
\end{codeexample}
%
\end{command}
\begin{command}{\pgfmathbasetobase\marg{macro}\marg{number}\marg{base-1}\marg{base-2}}
Defines \meta{macro} as the result of converting \meta{number} from base
\meta{base-1} to base \meta{base-2}. Alphabetic digits in \meta{number} can
be upper or lower case, but any resulting alphabetic digits are in
\emph{lower case}.
%
\begin{codeexample}[]
\pgfmathbasetobase\mynumber{11011011}{2}{16} \mynumber
\end{codeexample}
%
\end{command}
\begin{command}{\pgfmathbasetoBase\marg{macro}\marg{number}\marg{base-1}\marg{base-2}}
Defines \meta{macro} as the result of converting \meta{number} from base
\meta{base-1} to base \meta{base-2}. Alphabetic digits in \meta{number} can
be upper or lower case, but any resulting alphabetic digits are in
\emph{upper case}.
%
\begin{codeexample}[]
\pgfmathbasetoBase\mynumber{121212}{3}{12} \mynumber
\end{codeexample}
%
\end{command}
\begin{command}{\pgfmathsetbasenumberlength\marg{integer}}
Sets the number of digits in the result of a base conversion to
\meta{integer}. If the result of a conversion has less digits than this
number, it is prefixed with zeros.
%
\begin{codeexample}[]
\pgfmathsetbasenumberlength{8}
\pgfmathdectobase\mynumber{15}{2} \mynumber
\end{codeexample}
%
\end{command}
\begin{command}{\pgfmathtodigitlist\marg{macro}\marg{number}}
This command converts \meta{number} into a comma-separated list of digits
and stores the result in \meta{macro}. The \marg{number} is \emph{not}
parsed before processing.
%
\begin{codeexample}[]
\pgfmathsetbasenumberlength{8}
\begin{tikzpicture}[x=0.25cm, y=0.25cm]
\foreach \n [count=\y] in {0, 60, 102, 102, 126, 102, 102, 102, 0}{
\pgfmathdectobase{\binary}{\n}{2}
\pgfmathtodigitlist{\digitlist}{\binary}
\foreach \digit [count=\x, evaluate={\c=\digit*50+15;}] in \digitlist
\fill [fill=black!\c] (\x, -\y) rectangle ++(1,1);
}
\end{tikzpicture}
\end{codeexample}
%
\end{command}
\subsection{Angle Computations}
Unlike the rest of the math engine, which is a ``standalone'' package, the
following commands only work in conjunction with the core of \pgfname.
\begin{command}{\pgfmathanglebetweenpoints\marg{p}\marg{q}}
Returns the angle of a line from \meta{p} to \meta{q} relative to a line
going straight right from \meta{p}.
%
\begin{codeexample}[]
\pgfmathanglebetweenpoints{\pgfpoint{1cm}{3cm}}{\pgfpoint{2cm}{4cm}}
\pgfmathresult
\end{codeexample}
%
\end{command}
\begin{command}{\pgfmathanglebetweenlines\marg{$p_1$}\marg{$q_1$}\marg{$p_2$}\marg{$q_2$}}
Returns the clockwise angle between a line going through $p_1$ and $q_1$
and a line going through $p_2$ and $q_2$.
%
\begin{codeexample}[]
\pgfmathanglebetweenlines{\pgfpoint{1cm}{3cm}}{\pgfpoint{2cm}{4cm}}
{\pgfpoint{0cm}{1cm}}{\pgfpoint{1cm}{0cm}}
\pgfmathresult
\end{codeexample}
%
\end{command}