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diff --git a/Master/texmf-dist/doc/generic/pgf/pgfmanual-en-library-math.tex b/Master/texmf-dist/doc/generic/pgf/pgfmanual-en-library-math.tex new file mode 100644 index 00000000000..8fabb4998e5 --- /dev/null +++ b/Master/texmf-dist/doc/generic/pgf/pgfmanual-en-library-math.tex @@ -0,0 +1,471 @@ +% 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} |