From e0c6872cf40896c7be36b11dcc744620f10adf1d Mon Sep 17 00:00:00 2001 From: Norbert Preining Date: Mon, 2 Sep 2019 13:46:59 +0900 Subject: Initial commit --- macros/latex-dev/required/graphics/trig.dtx | 329 ++++++++++++++++++++++++++++ 1 file changed, 329 insertions(+) create mode 100644 macros/latex-dev/required/graphics/trig.dtx (limited to 'macros/latex-dev/required/graphics/trig.dtx') diff --git a/macros/latex-dev/required/graphics/trig.dtx b/macros/latex-dev/required/graphics/trig.dtx new file mode 100644 index 0000000000..712db13ba2 --- /dev/null +++ b/macros/latex-dev/required/graphics/trig.dtx @@ -0,0 +1,329 @@ +% \iffalse +%% File: trig.dtx Copyright (C) 1993 1994 1995 1996 1997 1999 David Carlisle +%% Copyright (C) 2000--2015 David Carlisle, LaTeX3 Project +%% +%% This file is part of the Standard LaTeX `Graphics Bundle'. +%% It may be distributed under the terms of the LaTeX Project Public +%% License, as described in lppl.txt in the base LaTeX distribution. +%% Either version 1.3c or, at your option, any later version. +%% +% +%<*dtx> + \ProvidesFile{trig.dtx} +% +%<*!plain> +%\NeedsTeXFormat{LaTeX2e} +%\ProvidesPackage{trig} +% \ProvidesFile{trig.drv} +% \fi +% \ProvidesFile{trig.dtx} + [2016/01/03 v1.10 sin cos tan (DPC)] +% +% \iffalse +% +%<*driver> +\documentclass{ltxdoc} +\usepackage{trig} +\begin{document} + \DocInput{trig.dtx} +\end{document} +% +% \fi +% +% \GetFileInfo{trig.dtx} +% \title{The \textsf{trig} package\thanks{This file +% has version number \fileversion, last +% revised \filedate.}} +% \author{David Carlisle} +% \date{\filedate} +% \MaintainedByLaTeXTeam{graphics} +% \maketitle +% +% +% \changes{v1.00}{1993/00/00}{Undocumented versions} +% \changes{v1.05}{1993/10/07}{Documented, added tan} +% \changes{v1.06}{1994/02/01}{Update for LaTeX2e} +% \changes{v1.07}{1994/03/15}{Use ltxdoc} +% \changes{v1.08}{1994/10/16}{Change \cs{@xc} to \cs{nin@ty}} +% +% \section{Introduction} +% +% These macros implement the trigonometric functions, sin, cos and tan. +% In each case two commands are defined. For instance the command +% |\CalculateSin{33}| may be issued at some point, and then anywhere +% later in the document, the command |\UseSin{33}| will return the +% decimal expansion of $\sin(33^\circ)$. +% +% The arguments to these macros do not have to be whole numbers, +% although in the case of whole numbers, \LaTeX\ or plain \TeX\ counters +% may be used. In \TeX{}Book syntax, arguments must be of type: +% \meta{optional signs}\meta{factor} +% +% Some other examples are:\\ +% |\CalculateSin{22.5}|, |\UseTan{\value{mycounter}}|, +% |\UseCos{\count@}|. +% +% Note that unlike the psfig macros, these save all previously +% computed values. This could easily be changed, but I thought that in +% many applications one would want many instances of the +% same value. (eg rotating all the headings of a table by the +% \emph{same} amount). +% +% I don't really like this need to pre-calculate the values, I +% originally implemented |\UseSin| so that it automatically calculated +% the value if it was not pre-stored. This worked fine in testing, until +% I remembered why one needs these values. You want to be able to say +% |\dimen2=\UseSin{30}\dimen0|. Which means that |\UseSin| must +% \emph{expand} to a \meta{factor}. +% +% \StopEventually{} +% +% \section{The Macros} +% +% \begin{macrocode} +%<*package> +% \end{macrocode} +% +% \begin{macro}{\nin@ty}\begin{macro}{\@clxx} +% \begin{macro}{\@lxxi}\begin{macro}{\@mmmmlxviii} +% Some useful constants for converting between degrees and radians. +% $$\frac{\pi}{180}\simeq\frac{355}{113\times180}=\frac{71}{4068}$$ +% \begin{macrocode} +\chardef\nin@ty=90 +\chardef\@clxx=180 +\chardef\@lxxi=71 +\mathchardef\@mmmmlxviii=4068 +% \end{macrocode} +% \end{macro}\end{macro}\end{macro}\end{macro} +% +% The approximation to $\sin$. I experimented with various +% approximations based on Tchebicheff polynomials, and also some +% approximations from a SIAM handbook `Computer Approximations' However +% the standard Taylor series seems sufficiently accurate, and used by +% far the fewest \TeX\ tokens, as the coefficients are all rational. +% \begin{eqnarray*} +% \sin(x)& \simeq& x - (1/3!)x^3 + (1/5!)x^5 - (1/7!)x^7 + (1/9!)x^9\\ +% &\simeq&\frac{((((7!/9!x^2-7!/7!)x^2+7!/5!)x^2 +7!/3!)x^2+7!/1!)x} +% {7!}\\ +% &=&\frac{((((1/72x^2-1)x^2+42)x^2 +840)x^2+5040)x} +% {5040} +% \end{eqnarray*} +% The nested form used above reduces the number of operations required. +% In order to further reduce the number of operations, and more +% importantly reduce the number of tokens used, we can precompute the +% coefficients. Note that we can not use $9!$ as the denominator as +% this would cause overflow of \TeX's arithmetic. +% \begin{macro}{\@coeffz}\begin{macro}{\@coeffa}\begin{macro}{\@coeffb} +% \begin{macro}{\@coeffc}\begin{macro}{\@coeffd} +% Save the coefficients as |\|(|math|)|char|s. +% \begin{macrocode} +\chardef\@coeffz=72 +%\chardef\@coefa=1 +\chardef\@coefb=42 +\mathchardef\@coefc=840 +\mathchardef\@coefd=5040 +% \end{macrocode} +% \end{macro}\end{macro}\end{macro}\end{macro}\end{macro} +% +% \begin{macro}{\TG@rem@pt} +% The standard trick of getting a real number out of a \meta{dimen}. +% This gives a maximum accuracy of approx.\ 5 decimal places, which +% should be sufficient. It puts a space after the number, perhaps it +% shouldn't. +% \begin{macrocode} +{\catcode`t=12\catcode`p=12\gdef\noPT#1pt{#1}} +\def\TG@rem@pt#1{\expandafter\noPT\the#1\space} +% \end{macrocode} +% \end{macro} +% +% \begin{macro}{\TG@term} +% Compute one term of the above nested series. Multiply the previous sum +% by $x^2$ (stored in |\@tempb|, then add the next coefficient, |#1|. +% \begin{macrocode} +\def\TG@term#1{% + \dimen@\@tempb\dimen@ + \advance\dimen@ #1\p@} +% \end{macrocode} +% \end{macro} +% +% \begin{macro}{\TG@series} +% Compute the above series. the value in degrees will be in |\dimen@| +% before this is called. +% \begin{macrocode} +\def\TG@series{% + \dimen@\@lxxi\dimen@ + \divide \dimen@ \@mmmmlxviii +% \end{macrocode} +% |\dimen@| now contains the angle in radians, as a \meta{dimen}. We +% need to remove the units, so store the same value as a \meta{factor} +% in |\@tempa|. +% \begin{macrocode} + \edef\@tempa{\TG@rem@pt\dimen@}% +% \end{macrocode} +% Now put $x^2$ in |\dimen@| and |\@tempb|. +% \begin{macrocode} + \dimen@\@tempa\dimen@ + \edef\@tempb{\TG@rem@pt\dimen@}% +% \end{macrocode} +% The first coefficient is $1/72$. +% \begin{macrocode} + \divide\dimen@\@coeffz + \advance\dimen@\m@ne\p@ + \TG@term\@coefb + \TG@term{-\@coefc}% + \TG@term\@coefd +% \end{macrocode} +% Now the cubic in $x^2$ is completed, so we need to multiply by $x$ and +% divide by $7!$. +% \begin{macrocode} + \dimen@\@tempa\dimen@ + \divide\dimen@ \@coefd} +% \end{macrocode} +% \end{macro} +% +% \begin{macro}{\CalculateSin} +% If this angle has already been computed, do nothing, else store the +% angle, and call |\TG@@sin|. +% \begin{macrocode} +\def\CalculateSin#1{{% + \expandafter\ifx\csname sin(\number#1)\endcsname\relax + \dimen@=#1\p@\TG@@sin + \expandafter\xdef\csname sin(\number#1)\endcsname + {\TG@rem@pt\dimen@}% + \fi}} +% \end{macrocode} +% \end{macro} +% +% \begin{macro}{\CalculateCos} +% As above, but use the relation $\cos(x) = \sin(90-x)$. +% \begin{macrocode} +\def\CalculateCos#1{{% + \expandafter\ifx\csname cos(\number#1)\endcsname\relax + \dimen@=\nin@ty\p@ + \advance\dimen@-#1\p@ + \TG@@sin + \expandafter\xdef\csname cos(\number#1)\endcsname + {\TG@rem@pt\dimen@}% + \fi}} +% \end{macrocode} +% \end{macro} +% +% \begin{macro}{\TG@reduce} +% Repeatedly use one of the the relations +% $\sin(x)=\sin(180-x)=\sin(-180-x)$ to get $x$ in the range $-90 \leq +% x\leq 90$. Then call |\TG@series|. +% \begin{macrocode} +\def\TG@reduce#1#2{% +\dimen@#1#2\nin@ty\p@ + \advance\dimen@#2-\@clxx\p@ + \dimen@-\dimen@ + \TG@@sin} +% \end{macrocode} +% \end{macro} +% +% \begin{macro}{\TG@@sin} +% Slightly cryptic, but it seems to work\ldots +% \begin{macrocode} +\def\TG@@sin{% + \ifdim\TG@reduce>+% + \else\ifdim\TG@reduce<-% + \else\TG@series\fi\fi}% +% \end{macrocode} +% \end{macro} +% +% \begin{macro}{\UseSin} +% \begin{macro}{\UseCos} +% Use a pre-computed value. +% \begin{macrocode} +\def\UseSin#1{\csname sin(\number#1)\endcsname} +\def\UseCos#1{\csname cos(\number#1)\endcsname} +% \end{macrocode} +% \end{macro} +% \end{macro} +% +% A few shortcuts to save space. +% \changes{v1.10}{2016/01/03}{make these expand to digits so safe to use in ctm} +% \begin{macrocode} +\def\z@num{0 } +\def\@tempa{1 } +\def\@tempb{-1 } +% \end{macrocode} +% +% \begin{macrocode} +\expandafter\let\csname sin(0)\endcsname\z@num +\expandafter\let\csname cos(0)\endcsname\@tempa +\expandafter\let\csname sin(90)\endcsname\@tempa +\expandafter\let\csname cos(90)\endcsname\z@num +\expandafter\let\csname sin(-90)\endcsname\@tempb +\expandafter\let\csname cos(-90)\endcsname\z@num +\expandafter\let\csname sin(180)\endcsname\z@num +\expandafter\let\csname cos(180)\endcsname\@tempb +% \end{macrocode} +% +% A few more added in 1.10 (previously in pdftex.def) +% \begin{macrocode} +\expandafter\let\csname sin(270)\endcsname\@tempb +\expandafter\let\csname cos(270)\endcsname\z@num +\expandafter\let\csname sin(360)\endcsname\z@num +\expandafter\let\csname cos(360)\endcsname\@tempa +\expandafter\let\csname sin(-180)\endcsname\z@num +\expandafter\let\csname cos(-180)\endcsname\@tempb +\expandafter\let\csname sin(-270)\endcsname\@tempa +\expandafter\let\csname cos(-270)\endcsname\z@num +\expandafter\let\csname sin(-360)\endcsname\z@num +\expandafter\let\csname cos(-360)\endcsname\@tempa +% \end{macrocode} +% +% \begin{macro}{\CalculateTan} +% Originally I coded the Taylor series for tan, but it seems to be +% more accurate to just take the ratio of the sine and cosine. +% This is accurate to 4 decimal places for angles up to +% $50^\circ$, after that the accuracy tails off, giving +% 57.47894 instead of 57.2900 for $89^\circ$. +% \begin{macrocode} +\def\CalculateTan#1{{% + \expandafter\ifx\csname tan(\number#1)\endcsname\relax + \CalculateSin{#1}% + \CalculateCos{#1}% + \@tempdima\UseCos{#1}\p@ + \divide\@tempdima\@iv + \@tempdimb\UseSin{#1}\p@ + \@tempdimb\two@fourteen\@tempdimb + \divide\@tempdimb\@tempdima + \expandafter\xdef\csname tan(\number#1)\endcsname + {\TG@rem@pt\@tempdimb}% + \fi}} +% \end{macrocode} +% \end{macro} +% +% \begin{macro}{\UseTan} +% Just like |\UseSin|. +% \begin{macrocode} +\def\UseTan#1{\csname tan(\number#1)\endcsname} +% \end{macrocode} +% \end{macro} +% +% \begin{macro}{\two@fourteen} +% \begin{macro}{\@iv} +% two constants needed to keep the division within \TeX's range. +% \begin{macrocode} +\mathchardef\two@fourteen=16384 +\chardef\@iv=4 +% \end{macrocode} +% \end{macro} +% \end{macro} +% +% Predefine $\tan(\pm90)$ to be an error. +% \begin{macrocode} +\expandafter\def\csname tan(90)\endcsname{\errmessage{Infinite tan !}} +\expandafter\let\csname tan(-90)\expandafter\endcsname + \csname tan(90)\endcsname +% \end{macrocode} +% +% \begin{macrocode} +% +% \end{macrocode} +% +% \Finale +% +\endinput -- cgit v1.2.3