manual/luatex-math.tex

   1 % language=uk
   2
   3 \environment luatex-style
   4 \environment luatex-logos
   5
   6 \startcomponent luatex-math
   7
   8 \startchapter[reference=math,title={Math}]
   9
  10 The handling of mathematics in \LUATEX\ differs quite a bit from how \TEX82 (and
  11 therefore \PDFTEX) handles math. First, \LUATEX\ adds primitives and extends some
  12 others so that \UNICODE\ input can be used easily. Second, all of \TEX82's
  13 internal special values (for example for operator spacing) have been made
  14 accessible and changeable via control sequences. Third, there are extensions that
  15 make it easier to use \OPENTYPE\ math fonts. And finally, there are some
  16 extensions that have been proposed or considered in the past that are now added
  17 to the engine.
  18
  19 \section{The current math style}
  20
  21 It is possible to discover the math style that will be used for a formula in an
  22 expandable fashion (while the math list is still being read). To make this
  23 possible, \LUATEX\ adds the new primitive: \type {\mathstyle}. This is a \quote
  24 {convert command} like e.g. \type {\romannumeral}: its value can only be read,
  25 not set.
  26
  27 \subsection{\type {\mathstyle}}
  28
  29 The returned value is between 0 and 7 (in math mode), or $-1$ (all other modes).
  30 For easy testing, the eight math style commands have been altered so that the can
  31 be used as numeric values, so you can write code like this:
  32
  33 \starttyping
  34 \ifnum\mathstyle=\textstyle
  35     \message{normal text style}
  36 \else \ifnum\mathstyle=\crampedtextstyle
  37     \message{cramped text style}
  38 \fi \fi
  39 \stoptyping
  40
  41 Sometimes you won't get what you expect so a bit of explanation might help to
  42 understand what happens. When math is parsed and expanded it gets turned into a
  43 linked list. In a second pass the formula will be build. This has to do with the
  44 fact that in order to determine the automatically chosen sizes (in for instance
  45 fractions) following content can influence preceding sizes. A side effect of this
  46 is for instance that one cannot change the definition of a font family (and
  47 thereby reusing numbers) because the number that got used is stored and used in
  48 the second pass (so changing \type {\fam 12} mid|-|formula spoils over to
  49 preceding use of that family).
  50
  51 The style switching primitives like \type {\textstyle} are turned into nodes so
  52 the styles set there are frozen. The \type {\mathchoice} primitive results in
  53 four lists being constructed of which one is used in the second pass. The fact
  54 that some automatic styles are not yet known also means that the \type
  55 {\mathstyle} primitive expands to the current style which can of course be
  56 different from the one really used. It's a snapshot of the first pass state. As a
  57 consequence in the following example you get a style number (first pass) typeset
  58 that can actually differ from the used style (second pass). In the case of a math
  59 choice used ungrouped, the chosen style is used after the choice too, unless you
  60 group.
  61
  62 \startbuffer[1]
  63     [a:\mathstyle]\quad
  64     \bgroup
  65     \mathchoice
  66         {\bf \scriptstyle       (x:d :\mathstyle)}
  67         {\bf \scriptscriptstyle (x:t :\mathstyle)}
  68         {\bf \scriptscriptstyle (x:s :\mathstyle)}
  69         {\bf \scriptscriptstyle (x:ss:\mathstyle)}
  70     \egroup
  71     \quad[b:\mathstyle]\quad
  72     \mathchoice
  73         {\bf \scriptstyle       (y:d :\mathstyle)}
  74         {\bf \scriptscriptstyle (y:t :\mathstyle)}
  75         {\bf \scriptscriptstyle (y:s :\mathstyle)}
  76         {\bf \scriptscriptstyle (y:ss:\mathstyle)}
  77     \quad[c:\mathstyle]\quad
  78     \bgroup
  79     \mathchoice
  80         {\bf \scriptstyle       (z:d :\mathstyle)}
  81         {\bf \scriptscriptstyle (z:t :\mathstyle)}
  82         {\bf \scriptscriptstyle (z:s :\mathstyle)}
  83         {\bf \scriptscriptstyle (z:ss:\mathstyle)}
  84     \egroup
  85     \quad[d:\mathstyle]
  86 \stopbuffer
  87
  88 \startbuffer[2]
  89     [a:\mathstyle]\quad
  90     \begingroup
  91     \mathchoice
  92         {\bf \scriptstyle       (x:d :\mathstyle)}
  93         {\bf \scriptscriptstyle (x:t :\mathstyle)}
  94         {\bf \scriptscriptstyle (x:s :\mathstyle)}
  95         {\bf \scriptscriptstyle (x:ss:\mathstyle)}
  96     \endgroup
  97     \quad[b:\mathstyle]\quad
  98     \mathchoice
  99         {\bf \scriptstyle       (y:d :\mathstyle)}
 100         {\bf \scriptscriptstyle (y:t :\mathstyle)}
 101         {\bf \scriptscriptstyle (y:s :\mathstyle)}
 102         {\bf \scriptscriptstyle (y:ss:\mathstyle)}
 103     \quad[c:\mathstyle]\quad
 104     \begingroup
 105     \mathchoice
 106         {\bf \scriptstyle       (z:d :\mathstyle)}
 107         {\bf \scriptscriptstyle (z:t :\mathstyle)}
 108         {\bf \scriptscriptstyle (z:s :\mathstyle)}
 109         {\bf \scriptscriptstyle (z:ss:\mathstyle)}
 110     \endgroup
 111     \quad[d:\mathstyle]
 112 \stopbuffer
 113
 114 \typebuffer[1]
 115
 116 % \typebuffer[2]
 117
 118 This gives:
 119
 120 \blank $\displaystyle \getbuffer[1]$ \blank
 121 \blank $\textstyle    \getbuffer[1]$ \blank
 122
 123 Using \type {\begingroup} \unknown\ \type {\endgroup} instead gives:
 124
 125 \blank $\displaystyle \getbuffer[2]$ \blank
 126 \blank $\textstyle    \getbuffer[2]$ \blank
 127
 128 This might look wrong but it's just a side effect of \type {\mathstyle} expanding
 129 to the current (first pass) style and the number being injected in the list that
 130 gets converted in the second pass. It all makes sense and it illustrates the
 131 importance of grouping. In fact, the math choice style being effective afterwards
 132 has advantages. It would be hard to get it otherwise.
 133
 134 \subsection{\type {\Ustack}}
 135
 136 There are a few math commands in \TEX\ where the style that will be used is not
 137 known straight from the start. These commands (\type {\over}, \type {\atop},
 138 \type {\overwithdelims}, \type {\atopwithdelims}) would therefore normally return
 139 wrong values for \type {\mathstyle}. To fix this, \LUATEX\ introduces a special
 140 prefix command: \type {\Ustack}:
 141
 142 \starttyping
 143 $\Ustack {a \over b}$
 144 \stoptyping
 145
 146 The \type {\Ustack} command will scan the next brace and start a new math group
 147 with the correct (numerator) math style.
 148
 149 \section{Unicode math characters}
 150
 151 Character handling is now extended up to the full \UNICODE\ range (the \type {\U}
 152 prefix), which is compatible with \XETEX.
 153
 154 The math primitives from \TEX\ are kept as they are, except for the ones that
 155 convert from input to math commands: \type {mathcode}, and \type {delcode}. These
 156 two now allow for a 21-bit character argument on the left hand side of the equals
 157 sign.
 158
 159 Some of the new \LUATEX\ primitives read more than one separate value. This is
 160 shown in the tables below by a plus sign in the second column.
 161
 162 The input for such primitives would look like this:
 163
 164 \starttyping
 165 \def\overbrace{\Umathaccent 0 1 "23DE }
 166 \stoptyping
 167
 168 The altered \TEX82 primitives are:
 169
 170 \starttabulate[|l|l|r|c|l|r|]
 171 \NC \bf primitive     \NC \bf min \NC \bf max \NC \kern 2em  \NC \bf min \NC \bf max \NC \NR
 172 \NC \type {\mathcode} \NC 0       \NC 10FFFF  \NC =          \NC 0       \NC 8000    \NC \NR
 173 \NC \type {\delcode}  \NC 0       \NC 10FFFF  \NC =          \NC 0       \NC FFFFFF  \NC \NR
 174 \stoptabulate
 175
 176 The unaltered ones are:
 177
 178 \starttabulate[|l|l|r|]
 179 \NC \bf primitive        \NC \bf min \NC \bf max \NC \NR
 180 \NC \type {\mathchardef} \NC 0       \NC    8000 \NC \NR
 181 \NC \type {\mathchar}    \NC 0       \NC    7FFF \NC \NR
 182 \NC \type {\mathaccent}  \NC 0       \NC    7FFF \NC \NR
 183 \NC \type {\delimiter}   \NC 0       \NC 7FFFFFF \NC \NR
 184 \NC \type {\radical}     \NC 0       \NC 7FFFFFF \NC \NR
 185 \stoptabulate
 186
 187 For practical reasons \type {\mathchardef} will silently accept values larger
 188 that \type {0x8000} and interpret it as \type {\Umathcharnumdef}. This is needed
 189 to satisfy older macro packages.
 190
 191 The following new primitives are compatible with \XETEX:
 192
 193 % somewhat fuzzy:
 194
 195 \starttabulate[|l|l|r|c|l|r|]
 196 \NC \bf primitive                           \NC \bf min   \NC \bf max             \NC \kern 2em  \NC \bf min   \NC \bf max                    \NC \NR
 197 \NC \type {\Umathchardef}                   \NC 0+0+0     \NC 7+FF+10FFFF\rlap{\high{1}}   \NC   \NC           \NC                            \NC \NR
 198 \NC \type {\Umathcharnumdef}\rlap{\high{5}} \NC -80000000 \NC    7FFFFFFF\rlap{\high{3}}   \NC   \NC           \NC                            \NC \NR
 199 \NC \type {\Umathcode}                      \NC 0         \NC      10FFFF                  \NC = \NC 0+0+0     \NC 7+FF+10FFFF\rlap{\high{1}} \NC \NR
 200 \NC \type {\Udelcode}                       \NC 0         \NC      10FFFF                  \NC = \NC 0+0       \NC   FF+10FFFF\rlap{\high{2}} \NC \NR
 201 \NC \type {\Umathchar}                      \NC 0+0+0     \NC 7+FF+10FFFF                  \NC   \NC           \NC                            \NC \NR
 202 \NC \type {\Umathaccent}                    \NC 0+0+0     \NC 7+FF+10FFFF\rlap{\high{2,4}} \NC   \NC           \NC                            \NC \NR
 203 \NC \type {\Udelimiter}                     \NC 0+0+0     \NC 7+FF+10FFFF\rlap{\high{2}}   \NC   \NC           \NC                            \NC \NR
 204 \NC \type {\Uradical}                       \NC 0+0       \NC   FF+10FFFF\rlap{\high{2}}   \NC   \NC           \NC                            \NC \NR
 205 \NC \type {\Umathcharnum}                   \NC -80000000 \NC    7FFFFFFF\rlap{\high{3}}   \NC   \NC           \NC                            \NC \NR
 206 \NC \type {\Umathcodenum}                   \NC 0         \NC      10FFFF                  \NC = \NC -80000000 \NC    7FFFFFFF\rlap{\high{3}} \NC \NR
 207 \NC \type {\Udelcodenum}                    \NC 0         \NC      10FFFF                  \NC = \NC -80000000 \NC    7FFFFFFF\rlap{\high{3}} \NC \NR
 208 \stoptabulate
 209
 210 Specifications typically look like:
 211
 212 \starttyping
 213 \Umathchardef\xx="1"0"456
 214 \Umathcode   123="1"0"789
 215 \stoptyping
 216
 217 Note 1: The new primitives that deal with delimiter|-|style objects do not set up a
 218 \quote {large family}. Selecting a suitable size for display purposes is expected
 219 to be dealt with by the font via the \type {\Umathoperatorsize} parameter (more
 220 information can be found in a following section).
 221
 222 Note 2: For these three primitives, all information is packed into a single
 223 signed integer. For the first two (\type {\Umathcharnum} and \type
 224 {\Umathcodenum}), the lowest 21 bits are the character code, the 3 bits above
 225 that represent the math class, and the family data is kept in the topmost bits
 226 (This means that the values for math families 128--255 are actually negative).
 227 For \type {\Udelcodenum} there is no math class. The math family information is
 228 stored in the bits directly on top of the character code. Using these three
 229 commands is not as natural as using the two- and three|-|value commands, so
 230 unless you know exactly what you are doing and absolutely require the speedup
 231 resulting from the faster input scanning, it is better to use the verbose
 232 commands instead.
 233
 234 Note 3: The \type {\Umathaccent} command accepts optional keywords to control
 235 various details regarding math accents. See \in {section} [mathacc] below for
 236 details.
 237
 238 New primitives that exist in \LUATEX\ only (all of these will be explained
 239 in following sections):
 240
 241 \starttabulate[|l|l|l|l|]
 242 \NC \bf primitive         \NC \bf value range (in hex) \NC \NR
 243 \NC \type {\Uroot}           \NC 0+0--FF+10FFFF$^2$       \NC \NR
 244 \NC \type {\Uoverdelimiter}  \NC 0+0--FF+10FFFF$^2$       \NC \NR
 245 \NC \type {\Uunderdelimiter} \NC 0+0--FF+10FFFF$^2$       \NC \NR
 246 \NC \type {\Udelimiterover}  \NC 0+0--FF+10FFFF$^2$       \NC \NR
 247 \NC \type {\Udelimiterunder} \NC 0+0--FF+10FFFF$^2$       \NC \NR
 248 \stoptabulate
 249
 250 \section{Cramped math styles}
 251
 252 \LUATEX\ has four new primitives to set the cramped math styles directly:
 253
 254 \starttyping
 255 \crampeddisplaystyle
 256 \crampedtextstyle
 257 \crampedscriptstyle
 258 \crampedscriptscriptstyle
 259 \stoptyping
 260
 261 These additional commands are not all that valuable on their own, but they come
 262 in handy as arguments to the math parameter settings that will be added shortly.
 263
 264 In Eijkhouts \quotation {\TEX\ by Topic} the rules for handling styles in scripts
 265 are described as follows:
 266
 267 \startitemize
 268 \startitem
 269     In any style superscripts and subscripts are taken from the next smaller style.
 270     Exception: in display style they are in script style.
 271 \stopitem
 272 \startitem
 273     Subscripts are always in the cramped variant of the style; superscripts are only
 274     cramped if the original style was cramped.
 275 \stopitem
 276 \startitem
 277     In an \type {..\over..} formula in any style the numerator and denominator are
 278     taken from the next smaller style.
 279 \stopitem
 280 \startitem
 281     The denominator is always in cramped style; the numerator is only in cramped
 282     style if the original style was cramped.
 283 \stopitem
 284 \startitem
 285     Formulas under a \type {\sqrt} or \type {\overline} are in cramped style.
 286 \stopitem
 287 \stopitemize
 288
 289 In \LUATEX\ one can set the styles in more detail which means that you sometimes
 290 have to set both normal and cramped styles to get the effect you want. If we
 291 force styles in the script using \type {\scriptstyle} and \type {\crampedscriptstyle}
 292 we get this:
 293
 294 \startbuffer[demo]
 295 \starttabulate
 296 \NC default       \NC $b_{x=xx}^{x=xx}$ \NC \NR
 297 \NC script        \NC $b_{\scriptstyle x=xx}^{\scriptstyle x=xx}$ \NC \NR
 298 \NC crampedscript \NC $b_{\crampedscriptstyle x=xx}^{\crampedscriptstyle x=xx}$ \NC \NR
 299 \stoptabulate
 300 \stopbuffer
 301
 302 \getbuffer[demo]
 303
 304 Now we set the following parameters
 305
 306 \startbuffer[setup]
 307 \Umathordrelspacing\scriptstyle=30mu
 308 \Umathordordspacing\scriptstyle=30mu
 309 \stopbuffer
 310
 311 \typebuffer[setup]
 312
 313 This gives:
 314
 315 \start\getbuffer[setup,demo]\stop
 316
 317 But, as this is not what is expected (visually) we should say:
 318
 319 \startbuffer[setup]
 320 \Umathordrelspacing\scriptstyle=30mu
 321 \Umathordordspacing\scriptstyle=30mu
 322 \Umathordrelspacing\crampedscriptstyle=30mu
 323 \Umathordordspacing\crampedscriptstyle=30mu
 324 \stopbuffer
 325
 326 \typebuffer[setup]
 327
 328 Now we get:
 329
 330 \start\getbuffer[setup,demo]\stop
 331
 332 \section{Math parameter settings}
 333
 334 In \LUATEX, the font dimension parameters that \TEX\ used in math typesetting are
 335 now accessible via primitive commands. In fact, refactoring of the math engine
 336 has resulted in many more parameters than were accessible before.
 337
 338 \starttabulate
 339 \NC \bf primitive name               \NC \bf description \NC \NR
 340 \NC \type {\Umathquad}               \NC the width of 18 mu's \NC \NR
 341 \NC \type {\Umathaxis}               \NC height of the vertical center axis of
 342                                          the math formula above the baseline \NC \NR
 343 \NC \type {\Umathoperatorsize}       \NC minimum size of large operators in display mode \NC \NR
 344 \NC \type {\Umathoverbarkern}        \NC vertical clearance above the rule \NC \NR
 345 \NC \type {\Umathoverbarrule}        \NC the width of the rule \NC \NR
 346 \NC \type {\Umathoverbarvgap}        \NC vertical clearance below the rule \NC \NR
 347 \NC \type {\Umathunderbarkern}       \NC vertical clearance below the rule \NC \NR
 348 \NC \type {\Umathunderbarrule}       \NC the width of the rule \NC \NR
 349 \NC \type {\Umathunderbarvgap}       \NC vertical clearance above the rule \NC \NR
 350 \NC \type {\Umathradicalkern}        \NC vertical clearance above the rule \NC \NR
 351 \NC \type {\Umathradicalrule}        \NC the width of the rule \NC \NR
 352 \NC \type {\Umathradicalvgap}        \NC vertical clearance below the rule \NC \NR
 353 \NC \type {\Umathradicaldegreebefore}\NC the forward kern that takes place before placement of
 354                                          the radical degree \NC \NR
 355 \NC \type {\Umathradicaldegreeafter} \NC the backward kern that takes place after placement of
 356                                          the radical degree \NC \NR
 357 \NC \type {\Umathradicaldegreeraise} \NC this is the percentage of the total height and depth of
 358                                          the radical sign that the degree is raised by; it is
 359                                          expressed in \type {percents}, so 60\% is expressed as the
 360                                          integer $60$ \NC \NR
 361 \NC \type {\Umathstackvgap}          \NC vertical clearance between the two
 362                                          elements in a \type {\atop} stack \NC \NR
 363 \NC \type {\Umathstacknumup}         \NC numerator shift upward in \type {\atop} stack \NC \NR
 364 \NC \type {\Umathstackdenomdown}     \NC denominator shift downward in \type {\atop} stack \NC \NR
 365 \NC \type {\Umathfractionrule}       \NC the width of the rule in a \type {\over} \NC \NR
 366 \NC \type {\Umathfractionnumvgap}    \NC vertical clearance between the numerator and the rule \NC \NR
 367 \NC \type {\Umathfractionnumup}      \NC numerator shift upward in \type {\over} \NC \NR
 368 \NC \type {\Umathfractiondenomvgap}  \NC vertical clearance between the denominator and the rule \NC \NR
 369 \NC \type {\Umathfractiondenomdown}  \NC denominator shift downward in \type {\over} \NC \NR
 370 \NC \type {\Umathfractiondelsize}    \NC minimum delimiter size for \type {\...withdelims} \NC \NR
 371 \NC \type {\Umathlimitabovevgap}     \NC vertical clearance for limits above operators \NC \NR
 372 \NC \type {\Umathlimitabovebgap}     \NC vertical baseline clearance for limits above operators \NC \NR
 373 \NC \type {\Umathlimitabovekern}     \NC space reserved at the top of the limit \NC \NR
 374 \NC \type {\Umathlimitbelowvgap}     \NC vertical clearance for limits below operators \NC \NR
 375 \NC \type {\Umathlimitbelowbgap}     \NC vertical baseline clearance for limits below operators \NC \NR
 376 \NC \type {\Umathlimitbelowkern}     \NC space reserved at the bottom of the limit \NC \NR
 377 \NC \type {\Umathoverdelimitervgap}  \NC vertical clearance for limits above delimiters \NC \NR
 378 \NC \type {\Umathoverdelimiterbgap}  \NC vertical baseline clearance for limits above delimiters \NC \NR
 379 \NC \type {\Umathunderdelimitervgap} \NC vertical clearance for limits below delimiters \NC \NR
 380 \NC \type {\Umathunderdelimiterbgap} \NC vertical baseline clearance for limits below delimiters \NC \NR
 381 \NC \type {\Umathsubshiftdrop}       \NC subscript drop for boxes and subformulas \NC \NR
 382 \NC \type {\Umathsubshiftdown}       \NC subscript drop for characters \NC \NR
 383 \NC \type {\Umathsupshiftdrop}       \NC superscript drop (raise, actually) for boxes and subformulas \NC \NR
 384 \NC \type {\Umathsupshiftup}         \NC superscript raise for characters \NC \NR
 385 \NC \type {\Umathsubsupshiftdown}    \NC subscript drop in the presence of a superscript \NC \NR
 386 \NC \type {\Umathsubtopmax}          \NC the top of standalone subscripts cannot be higher than this
 387                                          above the baseline \NC \NR
 388 \NC \type {\Umathsupbottommin}       \NC the bottom of standalone superscripts cannot be less than
 389                                          this above the baseline \NC \NR
 390 \NC \type {\Umathsupsubbottommax}    \NC the bottom of the superscript of a combined super- and subscript
 391                                          be at least as high as this above the baseline \NC \NR
 392 \NC \type {\Umathsubsupvgap}         \NC vertical clearance between super- and subscript \NC \NR
 393 \NC \type {\Umathspaceafterscript}   \NC additional space added after a super- or subscript \NC \NR
 394 \NC \type {\Umathconnectoroverlapmin}\NC minimum overlap between parts in an extensible recipe \NC \NR
 395 \stoptabulate
 396
 397 Each of the parameters in this section can be set by a command like this:
 398
 399 \starttyping
 400 \Umathquad\displaystyle=1em
 401 \stoptyping
 402
 403 they obey grouping, and you can use \type {\the\Umathquad\displaystyle} if
 404 needed.
 405
 406 \section{Skips around display math}
 407
 408 The injection of \type {\abovedisplayskip} and \type {\belowdisplayskip} is not
 409 symmetrical. An above one is always inserted, also when zero, but the below is
 410 only inserted when larger than zero. Especially the later makes it sometimes hard
 411 to fully control spacing. Therefore \LUATEX\ comes with a new directive: \type
 412 {\mathdisplayskipmode}. The following values apply:
 413
 414 \starttabulate
 415 \NC 0 \NC normal \TEX\ behaviour: always above, only below when larger than zero \NC \NR
 416 \NC 1 \NC always \NC \NR
 417 \NC 2 \NC only when not zero \NC \NR
 418 \NC 3 \NC never, not even when not zero \NC \NR
 419 \stoptabulate
 420
 421 \section{Font-based Math Parameters}
 422
 423 While it is nice to have these math parameters available for tweaking, it would
 424 be tedious to have to set each of them by hand. For this reason, \LUATEX\
 425 initializes a bunch of these parameters whenever you assign a font identifier to
 426 a math family based on either the traditional math font dimensions in the font
 427 (for assignments to math family~2 and~3 using \TFM|-|based fonts like \type
 428 {cmsy} and \type {cmex}), or based on the named values in a potential \type
 429 {MathConstants} table when the font is loaded via Lua. If there is a \type
 430 {MathConstants} table, this takes precedence over font dimensions, and in that
 431 case no attention is paid to which family is being assigned to: the \type
 432 {MathConstants} tables in the last assigned family sets all parameters.
 433
 434 In the table below, the one|-|letter style abbreviations and symbolic tfm font
 435 dimension names match those using in the \TeX book. Assignments to \type
 436 {\textfont} set the values for the cramped and uncramped display and text styles,
 437 \type {\scriptfont} sets the script styles, and \type {\scriptscriptfont} sets
 438 the scriptscript styles, so we have eight parameters for three font sizes. In the
 439 \TFM\ case, assignments only happen in family~2 and family~3 (and of course only
 440 for the parameters for which there are font dimensions).
 441
 442 Besides the parameters below, \LUATEX\ also looks at the \quote {space} font
 443 dimension parameter. For math fonts, this should be set to zero.
 444
 445 \start
 446
 447 \switchtobodyfont[8pt]
 448
 449 \starttabulate[|l|l|l|p|]
 450 \NC \bf variable                      \NC \bf style             \NC \bf default value opentype               \NC \bf default value tfm \NC \NR
 451 \NC \type {\Umathaxis}                \NC --                    \NC AxisHeight                               \NC axis_height \NC \NR
 452 \NC \type {\Umathoperatorsize}        \NC D, D'                 \NC DisplayOperatorMinHeight                 \NC $^6$ \NC \NR
 453 \NC \type {\Umathfractiondelsize}     \NC D, D'                 \NC FractionDelimiterDisplayStyleSize$^9$    \NC delim1 \NC \NR
 454 \NC                                   \NC T, T', S, S', SS, SS' \NC FractionDelimiterSize$^9$                \NC delim2 \NC \NR
 455 \NC \type {\Umathfractiondenomdown}   \NC D, D'                 \NC FractionDenominatorDisplayStyleShiftDown \NC denom1 \NC \NR
 456 \NC                                   \NC T, T', S, S', SS, SS' \NC FractionDenominatorShiftDown             \NC denom2 \NC \NR
 457 \NC \type {\Umathfractiondenomvgap}   \NC D, D'                 \NC FractionDenominatorDisplayStyleGapMin    \NC 3*default_rule_thickness \NC \NR
 458 \NC                                   \NC T, T', S, S', SS, SS' \NC FractionDenominatorGapMin                \NC default_rule_thickness \NC \NR
 459 \NC \type {\Umathfractionnumup}       \NC D, D'                 \NC FractionNumeratorDisplayStyleShiftUp     \NC num1 \NC \NR
 460 \NC                                   \NC T, T', S, S', SS, SS' \NC FractionNumeratorShiftUp                 \NC num2 \NC \NR
 461 \NC \type {\Umathfractionnumvgap}     \NC D, D'                 \NC FractionNumeratorDisplayStyleGapMin      \NC 3*default_rule_thickness \NC \NR
 462 \NC                                   \NC T, T', S, S', SS, SS' \NC FractionNumeratorGapMin                  \NC default_rule_thickness \NC \NR
 463 \NC \type {\Umathfractionrule}        \NC --                    \NC FractionRuleThickness                    \NC default_rule_thickness \NC \NR
 464 \NC \type {\Umathskewedfractionhgap}  \NC --                    \NC SkewedFractionHorizontalGap              \NC math_quad/2 \NC \NR
 465 \NC \type {\Umathskewedfractionvgap}  \NC --                    \NC SkewedFractionVerticalGap                \NC math_x_height \NC \NR
 466 \NC \type {\Umathlimitabovebgap}      \NC --                    \NC UpperLimitBaselineRiseMin                \NC big_op_spacing3 \NC \NR
 467 \NC \type {\Umathlimitabovekern}      \NC --                    \NC 0$^1$                                    \NC big_op_spacing5 \NC \NR
 468 \NC \type {\Umathlimitabovevgap}      \NC --                    \NC UpperLimitGapMin                         \NC big_op_spacing1 \NC \NR
 469 \NC \type {\Umathlimitbelowbgap}      \NC --                    \NC LowerLimitBaselineDropMin                \NC big_op_spacing4 \NC \NR
 470 \NC \type {\Umathlimitbelowkern}      \NC --                    \NC 0$^1$                                    \NC big_op_spacing5 \NC \NR
 471 \NC \type {\Umathlimitbelowvgap}      \NC --                    \NC LowerLimitGapMin                         \NC big_op_spacing2 \NC \NR
 472 \NC \type {\Umathoverdelimitervgap}   \NC --                    \NC StretchStackGapBelowMin                  \NC big_op_spacing1 \NC \NR
 473 \NC \type {\Umathoverdelimiterbgap}   \NC --                    \NC StretchStackTopShiftUp                   \NC big_op_spacing3 \NC \NR
 474 \NC \type {\Umathunderdelimitervgap}  \NC--                     \NC StretchStackGapAboveMin                  \NC big_op_spacing2 \NC \NR
 475 \NC \type {\Umathunderdelimiterbgap}  \NC--                     \NC StretchStackBottomShiftDown              \NC big_op_spacing4 \NC \NR
 476 \NC \type {\Umathoverbarkern}         \NC --                    \NC OverbarExtraAscender                     \NC default_rule_thickness \NC \NR
 477 \NC \type {\Umathoverbarrule}         \NC --                    \NC OverbarRuleThickness                     \NC default_rule_thickness \NC \NR
 478 \NC \type {\Umathoverbarvgap}         \NC --                    \NC OverbarVerticalGap                       \NC 3*default_rule_thickness \NC \NR
 479 \NC \type {\Umathquad}                \NC --                    \NC <font_size(f)>$^1$                       \NC math_quad \NC \NR
 480 \NC \type {\Umathradicalkern}         \NC --                    \NC RadicalExtraAscender                     \NC default_rule_thickness \NC \NR
 481 \NC \type {\Umathradicalrule}         \NC --                    \NC RadicalRuleThickness                     \NC <not set>$^2$ \NC \NR
 482 \NC \type {\Umathradicalvgap}         \NC D, D'                 \NC RadicalDisplayStyleVerticalGap           \NC (default_rule_thickness+\crlf
 483                                                                                                                  (abs(math_x_height)/4))$^3$ \NC \NR
 484 \NC                                   \NC T, T', S, S', SS, SS' \NC RadicalVerticalGap                       \NC (default_rule_thickness+\crlf
 485                                                                                                                  (abs(default_rule_thickness)/4))$^3$ \NC \NR
 486 \NC \type {\Umathradicaldegreebefore} \NC --                    \NC RadicalKernBeforeDegree                  \NC <not set>$^2$ \NC \NR
 487 \NC \type {\Umathradicaldegreeafter}  \NC --                    \NC RadicalKernAfterDegree                   \NC <not set>$^2$ \NC \NR
 488 \NC \type {\Umathradicaldegreeraise}  \NC --                    \NC RadicalDegreeBottomRaisePercent          \NC <not set>$^{2,7}$ \NC \NR
 489 \NC \type {\Umathspaceafterscript}    \NC --                    \NC SpaceAfterScript                         \NC script_space$^4$ \NC \NR
 490 \NC \type {\Umathstackdenomdown}      \NC D, D'                 \NC StackBottomDisplayStyleShiftDown         \NC denom1 \NC \NR
 491 \NC                                   \NC T, T', S, S', SS, SS' \NC StackBottomShiftDown                     \NC denom2 \NC \NR
 492 \NC \type {\Umathstacknumup}          \NC D, D'                 \NC StackTopDisplayStyleShiftUp              \NC num1 \NC \NR
 493 \NC                                   \NC T, T', S, S', SS, SS' \NC StackTopShiftUp                          \NC num3 \NC \NR
 494 \NC \type {\Umathstackvgap}           \NC D, D'                 \NC StackDisplayStyleGapMin                  \NC 7*default_rule_thickness \NC \NR
 495 \NC                                   \NC T, T', S, S', SS, SS' \NC StackGapMin                              \NC 3*default_rule_thickness \NC \NR
 496 \NC \type {\Umathsubshiftdown}        \NC --                    \NC SubscriptShiftDown                       \NC sub1 \NC \NR
 497 \NC \type {\Umathsubshiftdrop}        \NC --                    \NC SubscriptBaselineDropMin                 \NC sub_drop \NC \NR
 498 \NC \type {\Umathsubsupshiftdown}     \NC --                    \NC SubscriptShiftDownWithSuperscript$^8$    \NC \NC \NR
 499 \NC                                   \NC                       \NC \quad\ or SubscriptShiftDown             \NC sub2 \NC \NR
 500 \NC \type {\Umathsubtopmax}           \NC --                    \NC SubscriptTopMax                          \NC (abs(math_x_height * 4) / 5) \NC \NR
 501 \NC \type {\Umathsubsupvgap}          \NC --                    \NC SubSuperscriptGapMin                     \NC 4*default_rule_thickness \NC \NR
 502 \NC \type {\Umathsupbottommin}        \NC --                    \NC SuperscriptBottomMin                     \NC (abs(math_x_height) / 4) \NC \NR
 503 \NC \type {\Umathsupshiftdrop}        \NC --                    \NC SuperscriptBaselineDropMax               \NC sup_drop \NC \NR
 504 \NC \type {\Umathsupshiftup}          \NC D                     \NC SuperscriptShiftUp                       \NC sup1 \NC \NR
 505 \NC                                   \NC T, S, SS,             \NC SuperscriptShiftUp                       \NC sup2 \NC \NR
 506 \NC                                   \NC D', T', S', SS'       \NC SuperscriptShiftUpCramped                \NC sup3 \NC \NR
 507 \NC \type {\Umathsupsubbottommax}     \NC --                    \NC SuperscriptBottomMaxWithSubscript        \NC (abs(math_x_height * 4) / 5) \NC \NR
 508 \NC \type {\Umathunderbarkern}        \NC --                    \NC UnderbarExtraDescender                   \NC default_rule_thickness \NC \NR
 509 \NC \type {\Umathunderbarrule}        \NC --                    \NC UnderbarRuleThickness                    \NC default_rule_thickness \NC \NR
 510 \NC \type {\Umathunderbarvgap}        \NC --                    \NC UnderbarVerticalGap                      \NC 3*default_rule_thickness \NC \NR
 511 \NC \type {\Umathconnectoroverlapmin} \NC --                    \NC MinConnectorOverlap                      \NC 0$^5$ \NC \NR
 512 \stoptabulate
 513
 514 \stop
 515
 516 Note 1: \OPENTYPE\ fonts set \type {\Umathlimitabovekern} and \type
 517 {\Umathlimitbelowkern} to zero and set \type {\Umathquad} to the font size of the
 518 used font, because these are not supported in the \type {MATH} table,
 519
 520 Note 2: Traditional \TFM\ fonts do not set \type {\Umathradicalrule} because
 521 \TEX82\ uses the height of the radical instead. When this parameter is indeed not
 522 set when \LUATEX\ has to typeset a radical, a backward compatibility mode will
 523 kick in that assumes that an oldstyle \TEX\ font is used. Also, they do not set
 524 \type {\Umathradicaldegreebefore}, \type {\Umathradicaldegreeafter}, and \type
 525 {\Umathradicaldegreeraise}. These are then automatically initialized to
 526 $5/18$quad, $-10/18$quad, and 60.
 527
 528 Note 3: If \TFM\ fonts are used, then the \type {\Umathradicalvgap} is not set
 529 until the first time \LUATEX\ has to typeset a formula because this needs
 530 parameters from both family~2 and family~3. This provides a partial backward
 531 compatibility with \TEX82, but that compatibility is only partial: once the \type
 532 {\Umathradicalvgap} is set, it will not be recalculated any more.
 533
 534 Note 4: When \TFM\ fonts are used a similar situation arises with respect to
 535 \type {\Umathspaceafterscript}: it is not set until the first time \LUATEX\ has
 536 to typeset a formula. This provides some backward compatibility with \TEX82. But
 537 once the \type {\Umathspaceafterscript} is set, \type {\scriptspace} will never
 538 be looked at again.
 539
 540 Note 5: Traditional \TFM\ fonts set \type {\Umathconnectoroverlapmin} to zero
 541 because \TEX82\ always stacks extensibles without any overlap.
 542
 543 Note 6: The \type {\Umathoperatorsize} is only used in \type {\displaystyle}, and
 544 is only set in \OPENTYPE\ fonts. In \TFM\ font mode, it is artificially set to
 545 one scaled point more than the initial attempt's size, so that always the \quote
 546 {first next} will be tried, just like in \TEX82.
 547
 548 Note 7: The \type {\Umathradicaldegreeraise} is a special case because it is the
 549 only parameter that is expressed in a percentage instead of as a number of scaled
 550 points.
 551
 552 Note 8: \type {SubscriptShiftDownWithSuperscript} does not actually exist in the
 553 \quote {standard} \OPENTYPE\ math font Cambria, but it is useful enough to be
 554 added.
 555
 556 Note 9: \type {FractionDelimiterDisplayStyleSize} and \type
 557 {FractionDelimiterSize} do not actually exist in the \quote {standard} \OPENTYPE\
 558 math font Cambria, but were useful enough to be added.
 559
 560 \section{Nolimit correction}
 561
 562 There are two extra math parameters \type {\Umathnolimitsupfactor} and \type
 563 {\Umathnolimitsubfactor} that were added to provide some control over how limits
 564 are spaced (for example the position of super and subscripts after integral
 565 operators). They relate to an extra parameter \type {\mathnolimitsmode}. The half
 566 corrections are what happens when scripts are placed on above and below. The
 567 problem with italic corrections is that officially that correction italic is used
 568 for above|/|below placement while advanced kerns are used for placement at the
 569 right end. The question is: how often is this implemented, and if so, does the
 570 kerns assume correction too. Anyway, with this parameter one can control it.
 571
 572 \starttabulate[|l|ck1|ck1|ck1|ck1|ck1|ck1|]
 573     \NC
 574         \NC \mathnolimitsmode0    $\displaystyle\int\nolimits^0_1$
 575         \NC \mathnolimitsmode1    $\displaystyle\int\nolimits^0_1$
 576         \NC \mathnolimitsmode2    $\displaystyle\int\nolimits^0_1$
 577         \NC \mathnolimitsmode3    $\displaystyle\int\nolimits^0_1$
 578         \NC \mathnolimitsmode4    $\displaystyle\int\nolimits^0_1$
 579         \NC \mathnolimitsmode8000 $\displaystyle\int\nolimits^0_1$
 580     \NC \NR
 581     \TB
 582     \NC \bf mode
 583         \NC \tttf 0
 584         \NC \tttf 1
 585         \NC \tttf 2
 586         \NC \tttf 3
 587         \NC \tttf 4
 588         \NC \tttf 8000
 589     \NC \NR
 590     \NC \bf superscript
 591         \NC 0
 592         \NC font
 593         \NC 0
 594         \NC 0
 595         \NC +ic/2
 596         \NC 0
 597     \NC \NR
 598     \NC \bf subscript
 599         \NC -ic
 600         \NC font
 601         \NC 0
 602         \NC -ic/2
 603         \NC -ic/2
 604         \NC 8000ic/1000
 605     \NC \NR
 606 \stoptabulate
 607
 608 When the mode is set to one, the math parameters are used. This way a macro
 609 package writer can decide what looks best. Given the current state of fonts in
 610 \CONTEXT\ we currently use mode 1 with factor 0 for the superscript and 750 for
 611 the subscripts. Positive values are used for both parameters but the subscript
 612 shifts to the left. A \type {\mathnolimitsmode} larger that 15 is considered to
 613 be a factor for the subscript correction. This feature can be handy when
 614 experimenting.
 615
 616 \section{Math spacing setting}
 617
 618 Besides the parameters mentioned in the previous sections, there are also 64 new
 619 primitives to control the math spacing table (as explained in Chapter~18 of the
 620 \TEX book). The primitive names are a simple matter of combining two math atom
 621 types, but for completeness' sake, here is the whole list:
 622
 623 \starttwocolumns
 624 \starttyping
 625 \Umathordordspacing
 626 \Umathordopspacing
 627 \Umathordbinspacing
 628 \Umathordrelspacing
 629 \Umathordopenspacing
 630 \Umathordclosespacing
 631 \Umathordpunctspacing
 632 \Umathordinnerspacing
 633 \Umathopordspacing
 634 \Umathopopspacing
 635 \Umathopbinspacing
 636 \Umathoprelspacing
 637 \Umathopopenspacing
 638 \Umathopclosespacing
 639 \Umathoppunctspacing
 640 \Umathopinnerspacing
 641 \Umathbinordspacing
 642 \Umathbinopspacing
 643 \Umathbinbinspacing
 644 \Umathbinrelspacing
 645 \Umathbinopenspacing
 646 \Umathbinclosespacing
 647 \Umathbinpunctspacing
 648 \Umathbininnerspacing
 649 \Umathrelordspacing
 650 \Umathrelopspacing
 651 \Umathrelbinspacing
 652 \Umathrelrelspacing
 653 \Umathrelopenspacing
 654 \Umathrelclosespacing
 655 \Umathrelpunctspacing
 656 \Umathrelinnerspacing
 657 \Umathopenordspacing
 658 \Umathopenopspacing
 659 \Umathopenbinspacing
 660 \Umathopenrelspacing
 661 \Umathopenopenspacing
 662 \Umathopenclosespacing
 663 \Umathopenpunctspacing
 664 \Umathopeninnerspacing
 665 \Umathcloseordspacing
 666 \Umathcloseopspacing
 667 \Umathclosebinspacing
 668 \Umathcloserelspacing
 669 \Umathcloseopenspacing
 670 \Umathcloseclosespacing
 671 \Umathclosepunctspacing
 672 \Umathcloseinnerspacing
 673 \Umathpunctordspacing
 674 \Umathpunctopspacing
 675 \Umathpunctbinspacing
 676 \Umathpunctrelspacing
 677 \Umathpunctopenspacing
 678 \Umathpunctclosespacing
 679 \Umathpunctpunctspacing
 680 \Umathpunctinnerspacing
 681 \Umathinnerordspacing
 682 \Umathinneropspacing
 683 \Umathinnerbinspacing
 684 \Umathinnerrelspacing
 685 \Umathinneropenspacing
 686 \Umathinnerclosespacing
 687 \Umathinnerpunctspacing
 688 \Umathinnerinnerspacing
 689 \stoptyping
 690 \stoptwocolumns
 691
 692 These parameters are of type \type {\muskip}, so setting a parameter can be done
 693 like this:
 694
 695 \starttyping
 696 \Umathopordspacing\displaystyle=4mu plus 2mu
 697 \stoptyping
 698
 699 They are all initialized by \type {initex} to the values mentioned in the table
 700 in Chapter~18 of the \TEX book.
 701
 702 Note 1: for ease of use as well as for backward compatibility, \type
 703 {\thinmuskip}, \type {\medmuskip} and \type {\thickmuskip} are treated
 704 especially. In their case a pointer to the corresponding internal parameter is
 705 saved, not the actual \type {\muskip} value. This means that any later changes to
 706 one of these three parameters will be taken into account.
 707
 708 Note 2: Careful readers will realise that there are also primitives for the items
 709 marked \type {*} in the \TEX book. These will not actually be used as those
 710 combinations of atoms cannot actually happen, but it seemed better not to break
 711 orthogonality. They are initialized to zero.
 712
 713 \section[mathacc]{Math accent handling}
 714
 715 \LUATEX\ supports both top accents and bottom accents in math mode, and math
 716 accents stretch automatically (if this is supported by the font the accent comes
 717 from, of course). Bottom and combined accents as well as fixed-width math accents
 718 are controlled by optional keywords following \type {\Umathaccent}.
 719
 720 The keyword \type {bottom} after \type {\Umathaccent} signals that a bottom accent
 721 is needed, and the keyword \type {both} signals that both a top and a bottom
 722 accent are needed (in this case two accents need to be specified, of course).
 723
 724 Then the set of three integers defining the accent is read. This set of integers
 725 can be prefixed by the \type {fixed} keyword to indicate that a non-stretching
 726 variant is requested (in case of both accents, this step is repeated).
 727
 728 A simple example:
 729
 730 \starttyping
 731 \Umathaccent both fixed 0 0 "20D7 fixed 0 0 "20D7 {example}
 732 \stoptyping
 733
 734 If a math top accent has to be placed and the accentee is a character and has a
 735 non-zero \type {top_accent} value, then this value will be used to place the
 736 accent instead of the \type {\skewchar} kern used by \TEX82.
 737
 738 The \type {top_accent} value represents a vertical line somewhere in the
 739 accentee. The accent will be shifted horizontally such that its own \type
 740 {top_accent} line coincides with the one from the accentee. If the \type
 741 {top_accent} value of the accent is zero, then half the width of the accent
 742 followed by its italic correction is used instead.
 743
 744 The vertical placement of a top accent depends on the \type {x_height} of the
 745 font of the accentee (as explained in the \TEX book), but if value that turns out
 746 to be zero and the font had a \type {MathConstants} table, then \type
 747 {AccentBaseHeight} is used instead.
 748
 749 The vertical placement of a bottom accent is straight below the accentee, no
 750 correction takes place.
 751
 752 Possible locations are \type {top}, \type {bottom}, \type {both} and \type
 753 {center}. When no location is given \type {top} is assumed. An additional
 754 parameter \type {fraction} can be specified followed by a number; a value of for
 755 instance 1200 means that the criterium is 1.2 times the width of the nucleus. The
 756 fraction only applies to the stepwise selected shapes and is mostly meant for the
 757 \type {overlay} location. It also works for the other locations but then it
 758 concerns the width.
 759
 760 \section{Math root extension}
 761
 762 The new primitive \type {\Uroot} allows the construction of a radical noad
 763 including a degree field. Its syntax is an extension of \type {\Uradical}:
 764
 765 \starttyping
 766 \Uradical <fam integer> <char integer> <radicand>
 767 \Uroot    <fam integer> <char integer> <degree> <radicand>
 768 \stoptyping
 769
 770 The placement of the degree is controlled by the math parameters \type
 771 {\Umathradicaldegreebefore}, \type {\Umathradicaldegreeafter}, and \type
 772 {\Umathradicaldegreeraise}. The degree will be typeset in \type
 773 {\scriptscriptstyle}.
 774
 775 \section{Math kerning in super- and subscripts}
 776
 777 The character fields in a \LUA|-|loaded \OPENTYPE\ math font can have a \quote
 778 {mathkern} table. The format of this table is the same as the \quote {mathkern}
 779 table that is returned by the \type {fontloader} library, except that all height
 780 and kern values have to be specified in actual scaled points.
 781
 782 When a super- or subscript has to be placed next to a math item, \LUATEX\ checks
 783 whether the super- or subscript and the nucleus are both simple character items.
 784 If they are, and if the fonts of both character items are \OPENTYPE\ fonts (as
 785 opposed to legacy \TEX\ fonts), then \LUATEX\ will use the \OPENTYPE\ math
 786 algorithm for deciding on the horizontal placement of the super- or subscript.
 787
 788 This works as follows:
 789
 790 \startitemize
 791     \startitem
 792         The vertical position of the script is calculated.
 793     \stopitem
 794     \startitem
 795         The default horizontal position is flat next to the base character.
 796     \stopitem
 797     \startitem
 798         For superscripts, the italic correction of the base character is added.
 799     \stopitem
 800     \startitem
 801         For a superscript, two vertical values are calculated: the bottom of the
 802         script (after shifting up), and the top of the base. For a subscript, the two
 803         values are the top of the (shifted down) script, and the bottom of the base.
 804     \stopitem
 805     \startitem
 806         For each of these two locations:
 807         \startitemize
 808             \startitem
 809                 find the math kern value at this height for the base (for a subscript
 810                 placement, this is the bottom_right corner, for a superscript
 811                 placement the top_right corner)
 812             \stopitem
 813             \startitem
 814                 find the math kern value at this height for the script (for a
 815                 subscript placement, this is the top_left corner, for a superscript
 816                 placement the bottom_left corner)
 817             \stopitem
 818             \startitem
 819                 add the found values together to get a preliminary result.
 820             \stopitem
 821         \stopitemize
 822     \stopitem
 823     \startitem
 824         The horizontal kern to be applied is the smallest of the two results from
 825         previous step.
 826     \stopitem
 827 \stopitemize
 828
 829 The math kern value at a specific height is the kern value that is specified by the
 830 next higher height and kern pair, or the highest one in the character (if there is no
 831 value high enough in the character), or simply zero (if the character has no math kern
 832 pairs at all).
 833
 834 \section{Scripts on horizontally extensible items like arrows}
 835
 836 The primitives \type {\Uunderdelimiter} and \type {\Uoverdelimiter} allow the
 837 placement of a subscript or superscript on an automatically extensible item and
 838 \type {\Udelimiterunder} and \type {\Udelimiterover} allow the placement of an
 839 automatically extensible item as a subscript or superscript on a nucleus. The
 840 input:
 841
 842 % these produce radical noads .. in fact the code base has the numbers wrong for
 843 % quite a while, so no one seems to use this
 844
 845 \startbuffer
 846 $\Uoverdelimiter  0 "2194 {\hbox{\strut  overdelimiter}}$
 847 $\Uunderdelimiter 0 "2194 {\hbox{\strut underdelimiter}}$
 848 $\Udelimiterover  0 "2194 {\hbox{\strut  delimiterover}}$
 849 $\Udelimiterunder 0 "2194 {\hbox{\strut delimiterunder}}$
 850 \stopbuffer
 851
 852 \typebuffer will render this:
 853
 854 \blank \startnarrower \getbuffer \stopnarrower \blank
 855
 856 The vertical placements are controlled by \type {\Umathunderdelimiterbgap}, \type
 857 {\Umathunderdelimitervgap}, \type {\Umathoverdelimiterbgap}, and \type
 858 {\Umathoverdelimitervgap} in a similar way as limit placements on large operators.
 859 The superscript in \type {\Uoverdelimiter} is typeset in a suitable scripted style,
 860 the subscript in \type {\Uunderdelimiter} is cramped as well.
 861
 862 These primitives accepts an option \type {width} specification. When used the
 863 also optional keywords \type {left}, \type {middle} and \type {right} will
 864 determine what happens when a requested size can't be met (which can happen when
 865 we step to successive larger variants).
 866
 867 An extra primitive \type {\Uhextensible} is available that can be used like this:
 868
 869 \startbuffer
 870 $\Uhextensible width 10cm 0 "2194$
 871 \stopbuffer
 872
 873 \typebuffer This will render this:
 874
 875 \blank \startnarrower \getbuffer \stopnarrower \blank
 876
 877 Here you can also pass options, like:
 878
 879 \startbuffer
 880 $\Uhextensible width 1pt middle 0 "2194$
 881 \stopbuffer
 882
 883 \typebuffer This gives:
 884
 885 \blank \startnarrower \getbuffer \stopnarrower \blank
 886
 887 \LUATEX\ internally uses a structure that supports \OPENTYPE\ \quote
 888 {MathVariants} as well as \TFM\ \quote {extensible recipes}. In most cases where
 889 font metrics are involved we have a different code path for traditional fonts end
 890 \OPENTYPE\ fonts.
 891
 892 \section {Extracting values}
 893
 894 You can extract the components of a math character. Say that we have defined:
 895
 896 \starttyping
 897 \Umathcode 1 2 3 4
 898 \stoptyping
 899
 900 then
 901
 902 \starttyping
 903 [\Umathcharclass1] [\Umathcharfam1] [\Umathcharslot1]
 904 \stoptyping
 905
 906 will return:
 907
 908 \starttyping
 909 [2] [3] [4]
 910 \stoptyping
 911
 912 These commands are provides as convenience. Before they came available you could
 913 do the following:
 914
 915 \starttyping
 916 \def\Umathcharclass{\directlua{tex.print(tex.getmathcode(token.scan_int())[1])}}
 917 \def\Umathcharfam  {\directlua{tex.print(tex.getmathcode(token.scan_int())[2])}}
 918 \def\Umathcharslot {\directlua{tex.print(tex.getmathcode(token.scan_int())[3])}}
 919 \stoptyping
 920
 921 \section{fractions}
 922
 923 The \type {\abovewithdelims} command accepts a keyword \type {exact}. When issued
 924 the extra space relative to the rule thickness is not added. One can of course
 925 use the \type {\Umathfraction..gap} commands to influence the spacing. Also the
 926 rule is still positioned around the math axis.
 927
 928 \starttyping
 929 $$ { {a} \abovewithdelims() exact 4pt {b} }$$
 930 \stoptyping
 931
 932 The math parameter table contains some parameters that specify a horizontal and
 933 vertical gap for skewed fractions. Of course some guessing is needed in order to
 934 implement something that uses them. And so we now provide a primitive similar to the
 935 other fraction related ones but with a few options so that one can influence the
 936 rendering. Of course a user can also mess around a bit with the parameters
 937 \type {\Umathskewedfractionhgap} and \type {\Umathskewedfractionvgap}.
 938
 939 The syntax used here is:
 940
 941 \starttyping
 942 { {1} \Uskewed / <options> {2} }
 943 { {1} \Uskewedwithdelims / () <options> {2} }
 944 \stoptyping
 945
 946 where the options can be \type {noaxis} and \type {exact}. By default we add half
 947 the axis to the shifts and by default we zero the width of the middle character.
 948 For Latin Modern The result looks as follows:
 949
 950 \def\ShowA#1#2#3{$x + { {#1} \Uskewed           /    #3 {#2} } + x$}
 951 \def\ShowB#1#2#3{$x + { {#1} \Uskewedwithdelims / () #3 {#2} } + x$}
 952
 953 \start
 954     \switchtobodyfont[modern]
 955     \starttabulate[||||||]
 956         \NC \NC
 957             \ShowA{a}{b}{} \NC
 958             \ShowA{1}{2}{} \NC
 959             \ShowB{a}{b}{} \NC
 960             \ShowB{1}{2}{} \NC
 961         \NR
 962         \NC \type{exact} \NC
 963             \ShowA{a}{b}{exact} \NC
 964             \ShowA{1}{2}{exact} \NC
 965             \ShowB{a}{b}{exact} \NC
 966             \ShowB{1}{2}{exact} \NC
 967         \NR
 968         \NC \type{noaxis} \NC
 969             \ShowA{a}{b}{noaxis} \NC
 970             \ShowA{1}{2}{noaxis} \NC
 971             \ShowB{a}{b}{noaxis} \NC
 972             \ShowB{1}{2}{noaxis} \NC
 973         \NR
 974         \NC \type{exact noaxis} \NC
 975             \ShowA{a}{b}{exact noaxis} \NC
 976             \ShowA{1}{2}{exact noaxis} \NC
 977             \ShowB{a}{b}{exact noaxis} \NC
 978             \ShowB{1}{2}{exact noaxis} \NC
 979         \NR
 980     \stoptabulate
 981 \stop
 982
 983 \section {Last lines}
 984
 985 There is a new primitive to control the overshoot in the calculation of the
 986 previous line in mid|-|paragraph display math. The default value is 2 times
 987 the em width of the current font:
 988
 989 \starttyping
 990 \predisplaygapfactor=2000
 991 \stoptyping
 992
 993 If you want to have the length of the last line independent of math i.e.\ you don't
 994 want to revert to a hack where you insert a fake display math formula in order to
 995 get the length of the last line, the following will often work too:
 996
 997 \starttyping
 998 \def\lastlinelength{\dimexpr
 999     \directlua {tex.sprint (
1000         (nodes.dimensions(node.tail(tex.lists.page_head).list))
1001     )}sp
1002 \relax}
1003 \stoptyping
1004
1005 \section {Other Math changes}
1006
1007 \subsection {Verbose versions of single-character math commands}
1008
1009 \LUATEX\ defines six new primitives that have the same function as
1010 \type {^}, \type {_}, \type {$}, and \type {$$}:
1011
1012 \starttabulate[|l|l|l|l|]
1013 \NC \bf primitive           \NC \bf explanation \NC \NR
1014 \NC \type {\Usuperscript}      \NC Duplicates the functionality of \type {^} \NC \NR
1015 \NC \type {\Usubscript}        \NC Duplicates the functionality of \type {_} \NC \NR
1016 \NC \type {\Ustartmath}        \NC Duplicates the functionality of \type {$}, % $
1017                                    when used in non-math mode. \NC \NR
1018 \NC \type {\Ustopmath}         \NC Duplicates the functionality of \type {$}, % $
1019                                    when used in inline math mode. \NC \NR
1020 \NC \type {\Ustartdisplaymath} \NC Duplicates the functionality of \type {$$}, % $$
1021                                    when used in non-math mode. \NC \NR
1022 \NC \type {\Ustopdisplaymath}  \NC Duplicates the functionality of \type {$$}, % $$
1023                                    when used in display math mode. \NC \NR
1024 \stoptabulate
1025
1026 The \type {\Ustopmath} and \type {\Ustopdisplaymath} primitives check if the current
1027 math mode is the correct one (inline vs.\ displayed), but you can freely intermix
1028 the four mathon|/|mathoff commands with explicit dollar sign(s).
1029
1030 \subsection{Allowed math commands in non-math modes}
1031
1032 The commands \type {\mathchar}, and \type {\Umathchar} and control sequences that
1033 are the result of \type {\mathchardef} or \type {\Umathchardef} are also
1034 acceptable in the horizontal and vertical modes. In those cases, the \type
1035 {\textfont} from the requested math family is used.
1036
1037 \section{Math surrounding skips}
1038
1039 Inline math is surrounded by (optional) \type {\mathsurround} spacing but that is fixed
1040 dimension. There is now an additional parameter \type {\mathsurroundskip}. When set to a
1041 non|-|zero value (or zero with some stretch or shrink) this parameter will replace
1042 \type {\mathsurround}. By using an additional parameter instead of changing the nature
1043 of \type {\mathsurround}, we can remain compatible. In the meantime a bit more
1044 control has been added via \type {\mathsurroundmode}. This directive can take 6 values
1045 with zero being the default behaviour.
1046
1047 \start
1048
1049 \def\OneLiner#1#2%
1050   {\NC #1
1051 %    \NC \ruledhbox{\mathsurroundmode#1\relax\hsize 100pt   x$x$x}
1052 %    \NC \ruledhbox{\mathsurroundmode#1\relax\hsize 100pt x $x$ x}
1053    \NC \dontleavehmode\inframed[align=normal,offset=0pt,frame=off]{\mathsurroundmode#1\relax\hsize 100pt   x$x$x}
1054    \NC \dontleavehmode\inframed[align=normal,offset=0pt,frame=off]{\mathsurroundmode#1\relax\hsize 100pt x $x$ x}
1055    \NC #2
1056    \NC \NR}
1057
1058 \startbuffer
1059 \mathsurround    10pt
1060 \mathsurroundskip20pt
1061 \stopbuffer
1062
1063 \typebuffer \getbuffer
1064
1065 \starttabulate[|Tc|c|c|pl|]
1066 \HL
1067 \BC mode \BC \type {x$x$x} \BC \type {x $x$ x} \BC effect \NC \NR
1068 \HL
1069 \OneLiner{0}{obey \type {\mathsurround} when \type {\mathsurroundskip} is 0pt}
1070 \OneLiner{1}{only add skip to the left}
1071 \OneLiner{2}{only add skip to the right}
1072 \OneLiner{3}{add skip to the left and right}
1073 \OneLiner{4}{ignore the skip setting, obey \type {\mathsurround}}
1074 \OneLiner{5}{disable all spacing around math}
1075 \OneLiner{6}{only apply \type {\mathsurroundskip} when also spacing}
1076 \OneLiner{7}{only apply \type {\mathsurroundskip} when no spacing}
1077 \HL
1078 \stoptabulate
1079
1080 \stop
1081
1082 Method six omits the surround glue when there is (x)spacing glue present while
1083 method seven does the opposite, the glue is only applied when there is (x)space
1084 glue present too. Anything more fance, like checking the begining or end of a
1085 paragraph (or edges of a box) would not be robust anyway. If you want that you
1086 can write a callback that runs over a list and analyzes a paragraph. Actually, in
1087 that case you could also inject glue (or set the properties of a math node)
1088 explicitly. So, these modes are in practice mostly useful for special purposes
1089 and experiments (they originate in a tracker item). Keep in mind that this glue
1090 is part of the math node and not always treated as normal glue: it travels with
1091 the begin and end math nodes. Also, method 6 and 7 will zero the skip related
1092 fields in a node when applicable in the first occasion that checks them
1093 (linebreaking or packaging).
1094
1095 % \section{Math todo}
1096 %
1097 % The following items are still todo.
1098 %
1099 % \startitemize
1100 % \startitem
1101 %     Pre-scripts.
1102 % \stopitem
1103 % \startitem
1104 %     Multi-story stacks.
1105 % \stopitem
1106 % \startitem
1107 %     Flattened accents for high characters (maybe).
1108 % \stopitem
1109 % \startitem
1110 %     Better control over the spacing around displays and handling of equation numbers.
1111 % \stopitem
1112 % \startitem
1113 %     Support for multi|-|line displays using \MATHML\ style alignment points.
1114 % \stopitem
1115 % \stopitemize
1116
1117 \subsection {Delimiters: \type{\Uleft}, \type {\Umiddle} and \type {\Uright}}
1118
1119 Normally you will force delimiters to certain sizes by putting an empty box or
1120 rule next to it. The resulting delimiter will either be a character from the
1121 stepwise size range or an extensible. The latter can be quite differently
1122 positioned that the characters as it depends on the fit as well as the fact if
1123 the used characters in the font have depth or height. Commands like (plain \TEX
1124 s) \type {\big} need use this feature. In \LUATEX\ we provide a bit more control
1125 by three variants that supporting optional parameters \type {height}, \type
1126 {depth} and \type {axis}. The following example uses this:
1127
1128 \startbuffer
1129 \Uleft   height 30pt depth 10pt      \Udelimiter "0 "0 "000028
1130 \quad x\quad
1131 \Umiddle height 40pt depth 15pt      \Udelimiter "0 "0 "002016
1132 \quad x\quad
1133 \Uright  height 30pt depth 10pt      \Udelimiter "0 "0 "000029
1134 \quad \quad \quad
1135 \Uleft   height 30pt depth 10pt axis \Udelimiter "0 "0 "000028
1136 \quad x\quad
1137 \Umiddle height 40pt depth 15pt axis \Udelimiter "0 "0 "002016
1138 \quad x\quad
1139 \Uright  height 30pt depth 10pt axis \Udelimiter "0 "0 "000029
1140 \stopbuffer
1141
1142 \typebuffer
1143
1144 \startlinecorrection
1145 \ruledhbox{\mathematics{\getbuffer}}
1146 \stoplinecorrection
1147
1148 The keyword \type {exact} can be used as directive that the real dimensions
1149 should be applied when the criteria can't be met which can happen when we're
1150 still stepping through the successively larger variants. When no dimensions are
1151 given the \type {noaxis} command can be used to prevent shifting over the axis.
1152
1153 You can influence the final class with the keyword \type {class} which will
1154 influence the spacing.
1155
1156 \subsection{Fixed scripts}
1157
1158 We have three parameters that are used for this fixed anchoring:
1159
1160 \starttabulate[|l|l|]
1161 \NC $d$ \NC \type {\Umathsubshiftdown}    \NC \NR
1162 \NC $u$ \NC \type {\Umathsupshiftup}      \NC \NR
1163 \NC $s$ \NC \type {\Umathsubsupshiftdown} \NC \NR
1164 \stoptabulate
1165
1166 When we set \type {\mathscriptsmode} to a value other than zero these are used
1167 for calculating fixed positions. This is something that is needed for instance
1168 for chemistry. You can manipulate the mentioned variables to achive different
1169 effects.
1170
1171 \def\SampleMath#1%
1172   {$\mathscriptsmode#1\mathupright CH_2 + CH^+_2 + CH^2_2$}
1173
1174 \starttabulate[|c|c|c|l|]
1175 \NC \bf mode \NC \bf down      \NC \bf up        \NC                \NC \NR
1176 \NC 0        \NC dynamic       \NC dynamic       \NC \SampleMath{0} \NC \NR
1177 \NC 1        \NC $d$           \NC $u$           \NC \SampleMath{1} \NC \NR
1178 \NC 2        \NC $s$           \NC $u$           \NC \SampleMath{2} \NC \NR
1179 \NC 3        \NC $s$           \NC $u + s - d$   \NC \SampleMath{3} \NC \NR
1180 \NC 4        \NC $d + (s-d)/2$ \NC $u + (s-d)/2$ \NC \SampleMath{4} \NC \NR
1181 \NC 5        \NC $d$           \NC $u + s - d$   \NC \SampleMath{5} \NC \NR
1182 \stoptabulate
1183
1184 The value of this parameter obeys grouping but applies to the whole current
1185 formula.
1186
1187 % if needed we can put the value in stylenodes but maybe more should go there
1188
1189 \subsection {Tracing}
1190
1191 Because there are quite some math related parameters and values, it is possible
1192 to limit tracing. Only when \type {tracingassigns} and|/|or \type
1193 {tracingrestores} are set to~2 or more they will be traced.
1194
1195 \subsection {Math options}
1196
1197 The logic in the math engine is rather complex and there are often no universal
1198 solutions (read: what works out well for one font, fails for another). Therefore
1199 some variations in the implementation will be driven by options for which a new
1200 primitive \type {\mathoption} has been introduced (so that we don't end up with
1201 many new commands). The approach of options also permits us to see what effect a
1202 specific solution has.
1203
1204 \subsubsection {\type {\mathoption old}}
1205
1206 This option was introduced for testing purposes when the math engine got split
1207 code paths and it forces the engine to treat new fonts as old ones with respect
1208 to italic correction etc. There are no guarantees given with respect to the final
1209 result and unexpected side effects are not seens as bugs as they relate to font
1210 properties.
1211
1212 \startbuffer
1213 \mathoption old 1
1214 \stopbuffer
1215
1216 The \type {oldmath} boolean flag in the \LUA\ font table is the official way to
1217 force old treatment as it's bound to fonts.
1218
1219 \subsubsection {\type {\mathoption noitaliccompensation}}
1220
1221 This option compensates placement for characters with a built|-|in italic
1222 correction.
1223
1224 \startbuffer
1225 {\showboxes\int}\quad
1226 {\showboxes\int_{|}^{|}}\quad
1227 {\showboxes\int\limits_{|}^{|}}
1228 \stopbuffer
1229
1230 \typebuffer
1231
1232 Gives (with computer modern that has such italics):
1233
1234 \startlinecorrection[blank]
1235     \switchtobodyfont[modern]
1236     \startcombination[nx=2,ny=2,distance=5em]
1237         {\mathoption noitaliccompensation 0\relax \mathematics{\getbuffer}}
1238             {\nohyphens\type{0:inline}}
1239         {\mathoption noitaliccompensation 0\relax \mathematics{\displaymath\getbuffer}}
1240             {\nohyphens\type{0:display}}
1241         {\mathoption noitaliccompensation 1\relax \mathematics{\getbuffer}}
1242             {\nohyphens\type{1:inline}}
1243         {\mathoption noitaliccompensation 1\relax \mathematics{\displaymath\getbuffer}}
1244             {\nohyphens\type{1:display}}
1245     \stopcombination
1246 \stoplinecorrection
1247
1248 \subsubsection {\type {\mathoption nocharitalic}}
1249
1250 When two characters follow each other italic correction can interfere. The
1251 following example shows what this option does:
1252
1253 \startbuffer
1254 \catcode"1D443=11
1255 \catcode"1D444=11
1256 \catcode"1D445=11
1257 P( PP PQR
1258 \stopbuffer
1259
1260 \typebuffer
1261
1262 Gives (with computer modern that has such italics):
1263
1264 \startlinecorrection[blank]
1265     \switchtobodyfont[modern]
1266     \startcombination[nx=2,ny=2,distance=5em]
1267         {\mathoption nocharitalic 0\relax \mathematics{\getbuffer}}
1268             {\nohyphens\type{0:inline}}
1269         {\mathoption nocharitalic 0\relax \mathematics{\displaymath\getbuffer}}
1270             {\nohyphens\type{0:display}}
1271         {\mathoption nocharitalic 1\relax \mathematics{\getbuffer}}
1272             {\nohyphens\type{1:inline}}
1273         {\mathoption nocharitalic 1\relax \mathematics{\displaymath\getbuffer}}
1274             {\nohyphens\type{1:display}}
1275     \stopcombination
1276 \stoplinecorrection
1277
1278 \subsubsection {\type {\mathoption useoldfractionscaling}}
1279
1280 This option has been introduced as solution for tracker item 604 for fuzzy cases
1281 around either or not present fraction related settings for new fonts.
1282
1283 \stopchapter
1284
1285 \stopcomponent