summaryrefslogtreecommitdiff
path: root/macros/latex-dev/required/graphics/trig.dtx
blob: 83e774ff20b0d9d3adeaedcca017c6358062c368 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
% \iffalse
%% File: trig.dtx Copyright (C) 1993-1999 David Carlisle
%%                Copyright (C) 2000-2021
%% The LaTeX Project and any individual authors listed elsewhere
%% in this file.
%%
%% 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}
%</dtx>
%<*!plain>
%<package&!plain>\NeedsTeXFormat{LaTeX2e}
%<package&!plain>\ProvidesPackage{trig}
%<driver>        \ProvidesFile{trig.drv}
% \fi
%                \ProvidesFile{trig.dtx}
                 [2016/01/03 v1.10 sin cos tan (DPC)]
%
% \iffalse
%</!plain>
%<*driver>
\documentclass{ltxdoc}
\usepackage{trig}
\begin{document}
 \DocInput{trig.dtx}
\end{document}
%</driver>
% \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}
%</package>
%    \end{macrocode}
%
% \Finale
%
\endinput