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+% Copyright 2019 by Mark Wibrow
+%
+% This file may be distributed and/or modified
+%
+% 1. under the LaTeX Project Public License and/or
+% 2. under the GNU Free Documentation License.
+%
+% See the file doc/generic/pgf/licenses/LICENSE for more details.
+
+
+\section{Math Library}
+\label{section-library-math}
+
+\begin{tikzlibrary}{math}
+ This library defines a simple mathematical language to define simple
+ functions and perform sequences of basic mathematical operations.
+\end{tikzlibrary}
+
+
+\subsection{Overview}
+
+\pgfname\ and \tikzname\ both use the \pgfname\ mathematical engine which
+provides many commands for parsing expressions. Unfortunately the \pgfname\
+math engine is somewhat cumbersome for long sequences of mathematical
+operations, particularly when assigning values to multiple variables. The
+\tikzname\ |calc| library provides some additional ``convenience'' operations
+for doing calculations (particularly with coordinates), but this can only be
+used inside \tikzname\ path commands.
+
+This |math| library provides a means to perform sequences of mathematical
+operations in a more `user friendly' manner than the \pgfname\ math engine. In
+addition, the coordinate calculations of the |calc| library can be accessed
+(provided it is loaded).
+%
+However as the |math| library uses the \pgfname\ math engine -- which uses pure
+\TeX\ to perform all its calculations -- it is subject to the same speed and
+accuracy limitations. It is worth bearing this in mind, before trying to
+implement algorithms requiring intensive and highly accurate computation. You
+can, of course use the |fp| or the |fpu| libraries to increase the accuracy
+(but not necessarily the speed) of computations.
+
+For most purposes, the features provided by this library are accessed using the
+following command:
+
+\begin{command}{\tikzmath\texttt{\{}\meta{statements}\texttt{\}}}
+ This command process a series of \meta{statements} which can represent
+ assignments, function definitions, conditional evaluation, and iterations.
+ It provides, in effect, a miniature mathematical language to perform basic
+ mathematical operations. Perhaps the most important thing to remember is
+ that \emph{every statement should end with a semi-colon}. This is likely to
+ be the most common reason why the |\tikzmath| command fails.
+ %
+\begin{codeexample}[preamble={\usetikzlibrary{math}}]
+\tikzmath{
+ % Adapted from http://www.cs.northwestern.edu/academics/courses/110/html/fib_rec.html
+ function fibonacci(\n) {
+ if \n == 0 then {
+ return 0;
+ } else {
+ return fibonacci2(\n, 0, 1);
+ };
+ };
+ function fibonacci2(\n, \p, \q) {
+ if \n == 1 then {
+ return \q;
+ } else {
+ return fibonacci2(\n-1, \q, \p+\q);
+ };
+ };
+ int \f, \i;
+ for \i in {0,1,...,20} {
+ \f = fibonacci(\i);
+ print {\f, };
+ };
+}
+\end{codeexample}
+ %
+\end{command}
+
+In addition to this command the following key is provided:
+
+\begin{key}{/tikz/evaluate={\meta{statements}}}
+ This key simply executes |\tikzmath{|\meta{statements}|}|.
+ %
+\begin{codeexample}[preamble={\usetikzlibrary{math}}]
+\tikz[x=0.25cm,y=0.25cm,
+ evaluate={
+ int \i, \j;
+ for \i in {0,...,10} {
+ for \j in {0,...,10} {
+ \a{\i,\j} = (\i+\j)*5;
+ };
+ };
+ }
+]
+\foreach \i in {0,...,10}
+ \foreach \j in {0,...,10}
+ \fill [red!\a{\i,\j}!yellow] (\i,\j) rectangle ++(1, 1);
+\end{codeexample}
+ %
+\end{key}
+
+The following sections describe the miniature language that this library
+provides and can be used in the |\tikzmath| command and the |evaluate| key.
+The language consists only of simple keywords and expressions but the
+mini-parser allows you to format code in a reasonably versatile way (much like
+the |tikz| parser) except that \emph{all the keywords must be followed by at
+least one space}. This is the second most important thing to remember (after
+remembering to insert semi-colons at the end of every statement).
+
+
+\subsection{Assignment}
+
+In the simplest case, you will want to evaluate an expression and assign it to
+a macro, or a \TeX\ count or dimension register. In this case, use of the
+|math| library is straightforward:
+%
+\begin{codeexample}[preamble={\usetikzlibrary{math}}]
+\newcount\mycount
+\newdimen\mydimen
+\tikzmath{
+ \a = 4*5+6;
+ \b = sin(30)*4;
+ \mycount = log10(2048) / log10(2);
+ \mydimen = 15^2;
+}
+\a, \b, \the\mycount, \the\mydimen
+\end{codeexample}
+
+In addition, \TeX-macros (\emph{not} \TeX\ registers) can be suffixed with an
+index, similar to indices in mathematical notation, for example, $x_1$, $x_2$,
+$x_3$:
+%
+\begin{codeexample}[preamble={\usetikzlibrary{math}}]
+\tikzmath{
+ \x1 = 3+4; \x2 = 30+40; \x3 = 300+400;
+}
+\x1, \x2, \x3
+\end{codeexample}
+
+The index does not have to be a number. By using braces |{}|, more
+sophisticated indices can be created:
+%
+\begin{codeexample}[preamble={\usetikzlibrary{math}}]
+\tikzmath{
+ \c{air} = 340; \c{water} = 1435; \c{steel} = 6100;
+}
+\foreach \medium in {air,steel}{The speed of sound in \medium\ is \c{\medium} m/s. }
+\end{codeexample}
+
+You should not, however, try to mix indexed and non-indexed variables. Once an
+assignment is made using an index, the |math| library expects all instances of
+the variable on the right hand side of an assignment to be followed by an
+index. This effect is reversed if you subsequently make an assignment to the
+variable without an index: the |math| library (or to be precise the \pgfname\
+math-engine) will then ignore any index following the variable on the right
+hand side of an assignment.
+
+In some cases, you may wish to assign a value or expression to a variable
+without evaluating it with the \pgfname\ math-engine. In this case, you can use
+the following keyword:
+
+\begin{math-keyword}{{let} \meta{variable} \texttt{=} \meta{expression}\texttt{;}}
+ This keyword assigns \meta{expression} to \meta{variable} without
+ evaluation. The \meta{expression} is however fully expanded using |\edef|.
+ Any spaces preceding \meta{expression} are removed, but any trailing spaces
+ (before the semi-colon) are included.
+ %
+\begin{codeexample}[preamble={\usetikzlibrary{math}}]
+\tikzmath{
+ let \x = (5*4)+1;
+ let \c1 = blue;
+}
+\x, ``\c1''
+\end{codeexample}
+ %
+\end{math-keyword}
+
+
+\subsection{Integers, ``Real'' Numbers, and Coordinates}
+
+By default, assignments are made by evaluating expressions using the \pgfname\
+math-engine and results are usually returned as number with a decimal point
+(unless you are assigning to a count register or use the |int| function).
+%
+As this is not always desirable, the |math| library allows variables -- which
+\emph{must} be \TeX\ macros -- to be `declared' as being a particular `type'.
+The library recognizes three types: integers (numbers without a decimal point),
+real numbers (numbers with a decimal point\footnote{Strictly speaking, due to
+the finite range and precision of \TeX\ numerical capabilities, the term
+``real'' is not correct.}), and coordinates.
+
+To declare a variable as being one of the three types, you can use the
+keywords shown below. It is important to remember that by telling the |math|
+library you want it to do a particular assignment for a variable, it will also
+do the same assignment when the variable is indexed.
+%
+\begin{codeexample}[preamble={\usetikzlibrary{math}}]
+\tikzmath{
+ integer \x;
+ \x1 = 3+4; \x2 = 30+40; \x3 = 300+400;
+}
+\x1, \x2, \x3
+\end{codeexample}
+
+%But, if you want integer results without using a count register or the
+%|int| function, you can use a keyword to indicate this:
+
+\begin{math-keyword}{{integer} \meta{variable}\opt{\texttt{,} \meta{additional variables}}\texttt{;}}
+ The |integer| keyword indicates that assignments to the \meta{variable} or
+ the comma separated list of \meta{additional variables} should be truncated
+ (not rounded) to integers. The variables should be ordinary macros --
+ \emph{not} \TeX\ registers. In addition the variables should \emph{not} be
+ indexed.
+ %
+\begin{codeexample}[preamble={\usetikzlibrary{math}}]
+\tikzmath{
+ integer \x, \y, \z;
+ \x = 4*5+6;
+ \y = sin(30)*4;
+ \z = log10(512) / log10(2);
+ print {$x=\x$, $y=\y$, $z=\z$};
+}
+\end{codeexample}
+ %
+\end{math-keyword}
+
+\begin{math-keyword}{{int} \meta{variable}\opt{\texttt{,} \meta{additional variables}}\texttt{;}}
+ Short version of the |integer| keyword.
+\end{math-keyword}
+
+Having declared a variable as an integer, the |math| library will continue to
+assign only integers to that variable within the current \TeX\ scope. If you
+wish to assign non-integer (i.e., \emph{real}) numbers to the same variable,
+the following keyword can be used.
+
+\begin{math-keyword}{{real} \meta{variable}\opt{\texttt{,} \meta{additional variables}}\texttt{;}}
+ The |real| keyword ensures that assignments \meta{variable} (and
+ \meta{additional variables}) will not be truncated to integers.
+\end{math-keyword}
+
+In order to take advantage of |math| library interface to the |calc| library
+you must indicate that a variable is to be assigned coordinates, using the
+following keyword.
+
+\begin{math-keyword}{{coordinate} \meta{variable}\opt{\texttt{,} \meta{additional variables}}\texttt{;}}
+ This keyword enables \tikzname-style coordinates such as |(2cm,3pt)| or
+ |(my node.east)| to be parsed and assigned to \meta{variable} in the form
+ $x,y$, which can then be used in a |tikzpicture|:
+ %
+\begin{codeexample}[preamble={\usetikzlibrary{math}}]
+\tikzmath{
+ coordinate \c;
+ \c = (45:10pt);
+}
+\tikz\draw (0,0) -- (\c);
+\end{codeexample}
+
+ If the \tikzname\ |calc| library is loaded, coordinate calculations can be
+ performed; the coordinate expression does not have to be surrounded by
+ |($|\ldots|$)|.
+ %
+\begin{codeexample}[preamble={\usetikzlibrary{math}}]
+\tikzmath{
+ coordinate \c, \d;
+ \c = (-1,2)+(1,-1);
+ \d = (4,1)-(2,-1);
+}
+\tikz\draw (\c) -- (\d);
+\end{codeexample}
+
+ In addition to assigning the $x$ and $y$ coordinates to \meta{variable}
+ (possibly with an optional index), two further variables are defined. The
+ first takes the name of \meta{variable} (e.g., |\c|) suffixed with |x|
+ (i.e., |\cx|) and is assigned the $x$ coordinate of |\c|. The second takes
+ the name of \meta{variable} suffixed with |y| (i.e., |\cy|) and is assigned
+ the $y$ coordinate of |\c|.
+ %
+\begin{codeexample}[preamble={\usetikzlibrary{math}}]
+\tikzmath{
+ coordinate \c;
+ \c1 = (30:20pt);
+ \c2 = (210:20pt);
+}
+\tikz\draw (\cx1,\cy1) -- (\cx2,\cy1) -- (\cx2,\cy2) -- (\cx1,\cy2);
+\end{codeexample}
+ %
+\end{math-keyword}
+
+%\begin{math-keyword}{{point} \meta{variable}\opt{\texttt{,} \meta{additional variables}}\texttt{;}}
+% The |point| keyword is a synonym for the |coordinate| keyword and performs
+% the same function.
+%\end{math-keyword}
+
+
+\subsection{Repeating Things}
+
+\begin{math-keyword}{{for} \meta{variable} \texttt{in \{}\meta{list} \texttt{\}\{}\meta{expressions}\texttt{\};}}
+ This is a ``trimmed down'' version of the |\foreach| command available as
+ part of \pgfname\ and \tikzname, but cannot currently be used outside of
+ the |\tikzmath| command. It is important to note the following:
+ %
+ \begin{itemize}
+ \item Every value in \meta{list} is evaluated using the \pgfname\
+ mathematical engine. However, if an item in \meta{list} contains a
+ comma, it \emph{must} be surrounded by braces, for example,
+ |{mod(5, 2)}|.
+ %
+\begin{codeexample}[pre={\pgfmathsetseed{1}},preamble={\usetikzlibrary{math}}]
+\tikzmath{
+ int \x, \v;
+ \v=1;
+ for \x in {1,...,{random(3,10)}} {
+ \v=\v*2;
+ };
+ print {$x=\x, v=\v$};
+}
+\end{codeexample}
+ %
+ \item Because each item is evaluated, you cannot use \tikzname\
+ coordinates in \meta{list}.
+ \item Only single variable assignment is supported.
+ \item The ``dots notation'' (e.g., |1,2,...,9|) can be used in
+ \meta{list}, but is not as sophisticated as the \pgfname\
+ |\foreach| command. In particular, contextual replacement is not
+ possible.
+ \item Assignments that occur in the loop body \emph{are not scoped}.
+ They last beyond the body of each iteration and the end of the
+ |for| statement. This includes the values assigned to the
+ \meta{variable}.
+ %
+\begin{codeexample}[preamble={\usetikzlibrary{math}}]
+\tikzmath{
+ int \x, \y;
+ \y = 0;
+ for \x1 in {1,...,5} {
+ for \x2 in {10,20,...,50} {
+ \y = \y+\x1*\x2;
+ };
+ };
+}
+$x_1=\x1, x_2=\x2, y=\y$
+\end{codeexample}
+ \end{itemize}
+\end{math-keyword}
+
+
+\subsection{Branching Statements}
+
+Sometimes you may wish to execute different statements depending on the value
+of an expression. In this case the following keyword can be used:
+
+\begin{math-keyword}{{if} \meta{condition} \texttt{then \{}\meta{if-non-zero-statements}\texttt{\};}}
+ This keyword executes \meta{if-non-zero-statements} if the expression in
+ \meta{condition} evaluates to any value other than zero.
+\end{math-keyword}
+
+\begin{math-keyword}{{if} \meta{condition} \texttt{then \{}\meta{if-non-zero-statements}\texttt{\}} \texttt{else} \texttt{\{}\meta{if-zero-statements}\texttt{\}}\texttt{;}}
+ This keyword executes \meta{if-non-zero-statements} if the expression in
+ \meta{condition} evaluates to any value other than zero and the
+ \meta{if-zero-statements} are executed if the expression in
+ \meta{condition} evaluates to zero.
+ %
+\begin{codeexample}[preamble={\usetikzlibrary{math}}]
+ \begin{tikzpicture}
+ \tikzmath{
+ int \x;
+ for \k in {0,10,...,350} {
+ if \k>260 then { let \c = orange; } else {
+ if \k>170 then { let \c = blue; } else {
+ if \k>80 then { let \c = red; } else {
+ let \c = green; }; }; };
+ {
+ \path [fill=\c!50, draw=\c] (\k:0.5cm) -- (\k:1cm) --
+ (\k+5:1cm) -- (\k+5:0.5cm) -- cycle;
+ };
+ };
+ }
+ \end{tikzpicture}
+\end{codeexample}
+ %
+\end{math-keyword}
+
+
+\subsection{Declaring Functions}
+
+You can add functions by using the following keywords:
+
+\begin{math-keyword}{{function} \meta{name}\texttt{(}\meta{arguments}\texttt{) \{} \meta{definition} \texttt{\};}}
+ This keyword works much like the |declare function| provided by the
+ \pgfname\ math-engine. The function \meta{name} can be any name that is not
+ already a function name in the current scope. The list of \meta{arguments}
+ are comma separated \TeX\ macros such as |\x|, or |\y| (it is not possible
+ to declare functions that take variable numbers of arguments). If the
+ function takes no arguments then the parentheses need not be used. It is
+ very important to note that the arrays that the \pgfname\ math engine
+ supports \emph{cannot currently be passed as arguments to functions}.
+
+ The function \meta{definition} should be a sequence of statements that can
+ be parsed by the |\tikzmath| command and should use the commands specified
+ in the \meta{arguments}. The |return| keyword (described below) should be
+ used to indicate the value returned by the function.
+ %
+ Although \meta{definition} can take any statements accepted by |\tikzmath|,
+ it is not advisable try to define functions inside other functions.
+ %
+\begin{codeexample}[pre={\pgfmathsetseed{1}},preamble={\usetikzlibrary{math}}]
+\tikzmath{
+ function product(\x,\y) {
+ return \x*\y;
+ };
+ int \i, \j, \k;
+ \i = random(1,10);
+ \j = random(20, 40);
+ \k = product(\i, \j);
+ print { $\i\times \j = \k$ };
+}
+\end{codeexample}
+ %
+\end{math-keyword}
+
+\begin{math-keyword}{{return} \meta{expression}\texttt{;}}
+ This keyword should be used as the last executed statement in a function
+ definition to indicate the value that should be returned.
+\end{math-keyword}
+
+
+\subsection{Executing Code Outside the Parser}
+
+Sometimes you may wish to do ``something'' outside the parser, perhaps display
+some intermediate result or execute some code. In this case you have two
+options. Firstly, the following keyword can be used:
+
+\begin{math-keyword}{{print} \texttt{\{}\meta{code}\texttt{\};}}
+ Execute \meta{code} immediately. This is intended as convenience keyword
+ for displaying information in a document (analogous to the |print| command
+ in real programming languages). The \meta{code} is executed inside a \TeX\
+ group.
+ %
+\begin{codeexample}[pre={\pgfmathsetseed{1}},preamble={\usetikzlibrary{math}}]
+\tikzmath{
+ int \x, \y, \z;
+ \x = random(2, 5);
+ for \y in {0,...,6} {
+ \z = \x^\y;
+ print {$\x^\y=\z$, };
+ };
+}
+\end{codeexample}
+ %
+\end{math-keyword}
+
+Secondly, if a statement begins with a brace |{|, then everything up to the
+closing brace |}| is collected and executed (the closing brace \emph{must} be
+followed by a semi-colon). Like the |print| keyword, the contents of the braces
+is executed inside a \TeX\ group. Unlike the |print| keyword, the brace
+notation can be used in functions so that |tikz| path commands can be safely
+executed inside a |tikzpicture|.
+%
+\begin{codeexample}[preamble={\usetikzlibrary{math}}]
+\begin{tikzpicture}
+\draw [help lines] grid (3,2);
+\tikzmath{
+ coordinate \c;
+ for \x in {0,10,...,360} {
+ \c = (1.5cm, 1cm) + (\x:1cm and 0.5cm);
+ { \fill (\c) circle [radius=1pt]; };
+ };
+}
+\end{tikzpicture}
+\end{codeexample}