manual/luatex-math.tex

   1 \environment luatex-style
   2 \environment luatex-logos
   3
   4 \startcomponent luatex-math
   5
   6 \startchapter[reference=math,title={Math}]
   7
   8 The handling of mathematics in \LUATEX\ differs quite a bit from how \TEX82 (and
   9 therefore \PDFTEX) handles math. First, \LUATEX\ adds primitives and extends some
  10 others so that \UNICODE\ input can be used easily. Second, all of \TEX82's
  11 internal special values (for example for operator spacing) have been made
  12 accessible and changeable via control sequences. Third, there are extensions that
  13 make it easier to use \OPENTYPE\ math fonts. And finally, there are some
  14 extensions that have been proposed in the past that are now added to the engine.
  15
  16 \section{The current math style}
  17
  18 It is possible to discover the math style that will be used for a formula in an
  19 expandable fashion (while the math list is still being read). To make this
  20 possible, \LUATEX\ adds the new primitive: \type {\mathstyle}. This is a \quote
  21 {convert command} like e.g. \type {\romannumeral}: its value can only be read,
  22 not set.
  23
  24 \subsection{\type {\mathstyle}}
  25
  26 The returned value is between 0 and 7 (in math mode), or $-1$ (all other modes).
  27 For easy testing, the eight math style commands have been altered so that the can
  28 be used as numeric values, so you can write code like this:
  29
  30 \starttyping
  31 \ifnum\mathstyle=\textstyle
  32     \message{normal text style}
  33 \else \ifnum\mathstyle=\crampedtextstyle
  34     \message{cramped text style}
  35 \fi \fi
  36 \stoptyping
  37
  38 \subsection{\type {\Ustack}}
  39
  40 There are a few math commands in \TEX\ where the style that will be used is not
  41 known straight from the start. These commands (\type {\over}, \type {\atop},
  42 \type {\overwithdelims}, \type {\atopwithdelims}) would therefore normally return
  43 wrong values for \type {\mathstyle}. To fix this, \LUATEX\ introduces a special
  44 prefix command: \type {\Ustack}:
  45
  46 \starttyping
  47 $\Ustack {a \over b}$
  48 \stoptyping
  49
  50 The \type {\Ustack} command will scan the next brace and start a new math group
  51 with the correct (numerator) math style.
  52
  53 \section{Unicode math characters}
  54
  55 Character handling is now extended up to the full \UNICODE\ range (the \type {\U}
  56 prefix), which is compatible with \XETEX.
  57
  58 The math primitives from \TEX\ are kept as they are, except for the ones that
  59 convert from input to math commands: \type {mathcode}, and \type {delcode}. These
  60 two now allow for a 21-bit character argument on the left hand side of the equals
  61 sign.
  62
  63 Some of the new \LUATEX\ primitives read more than one separate value. This is
  64 shown in the tables below by a plus sign in the second column.
  65
  66 The input for such primitives would look like this:
  67
  68 \starttyping
  69 \def\overbrace{\Umathaccent 0 1 "23DE }
  70 \stoptyping
  71
  72 Altered \TEX82 primitives:
  73
  74 \starttabulate[|l|l|l|]
  75 \NC \bf primitive  \NC \bf value range (in hex) \NC \NR
  76 \NC \type {\mathcode} \NC 0--10FFFF = 0--8000      \NC \NR
  77 \NC \type {\delcode}  \NC 0--10FFFF = 0--FFFFFF    \NC \NR
  78 \stoptabulate
  79
  80 Unaltered:
  81
  82 \starttabulate[|l|l|l|]
  83 \NC \bf primitive     \NC \bf value range (in hex) \NC \NR
  84 \NC \type {\mathchardef} \NC 0--8000                  \NC \NR
  85 \NC \type {\mathchar}    \NC 0--7FFF                  \NC \NR
  86 \NC \type {\mathaccent}  \NC 0--7FFF                  \NC \NR
  87 \NC \type {\delimiter}   \NC 0--7FFFFFF               \NC \NR
  88 \NC \type {\radical}     \NC 0--7FFFFFF               \NC \NR
  89 \stoptabulate
  90
  91 New primitives that are compatible with \XETEX:
  92
  93 \starttabulate[|l|l|l|l|]
  94 \NC \bf primitive             \NC \bf value range (in hex)             \NC \NR
  95 \NC \type {\Umathchardef}        \NC 0+0+0--7+FF+10FFFF$^1$               \NC \NR
  96 \NC \type {\Umathcharnumdef}$^5$ \NC -80000000--7FFFFFFF$^3$              \NC \NR
  97 \NC \type {\Umathcode}           \NC 0--10FFFF = 0+0+0--7+FF+10FFFF$^1$   \NC \NR
  98 \NC \type {\Udelcode}            \NC 0--10FFFF = 0+0--FF+10FFFF$^2$       \NC \NR
  99 \NC \type {\Umathchar}           \NC 0+0+0--7+FF+10FFFF                   \NC \NR
 100 \NC \type {\Umathaccent}         \NC 0+0+0--7+FF+10FFFF$^{2,4}$           \NC \NR
 101 \NC \type {\Udelimiter}          \NC 0+0+0--7+FF+10FFFF$^2$               \NC \NR
 102 \NC \type {\Uradical}            \NC 0+0--FF+10FFFF$^2$                   \NC \NR
 103 \NC \type {\Umathcharnum}        \NC -80000000--7FFFFFFF$^3$              \NC \NR
 104 \NC \type {\Umathcodenum}        \NC 0--10FFFF = -80000000--7FFFFFFF$^3$  \NC \NR
 105 \NC \type {\Udelcodenum}         \NC 0--10FFFF = -80000000--7FFFFFFF$^3$  \NC \NR
 106 \stoptabulate
 107
 108 Note 1: \type {\Umathchardef<csname>="8"0"0} and \type
 109 {\Umathchardef<number>="8"0"0} are also accepted.
 110
 111 Note 2: The new primitives that deal with delimiter-style objects do not set up a
 112 \quote {large family}. Selecting a suitable size for display purposes is expected
 113 to be dealt with by the font via the \type {\Umathoperatorsize} parameter (more
 114 information can be found in a following section).
 115
 116 Note 3: For these three primitives, all information is packed into a single
 117 signed integer. For the first two (\type {\Umathcharnum} and \type {\Umathcodenum}),
 118 the lowest 21 bits are the character code, the 3 bits above that represent the
 119 math class, and the family data is kept in the topmost bits (This means that the
 120 values for math families 128--255 are actually negative). For \type {\Udelcodenum}
 121 there is no math class; the math family information is stored in the bits
 122 directly on top of the character code. Using these three commands is not as
 123 natural as using the two- and three-value commands, so unless you know exactly
 124 what you are doing and absolutely require the speedup resulting from the faster
 125 input scanning, it is better to use the verbose commands instead.
 126
 127 Note 4: The \type {\Umathaccent} command accepts optional keywords to control
 128 various details regarding math accents. See \in {section} [mathacc] below for
 129 details.
 130
 131 New primitives that exist in \LUATEX\ only (all of these will be explained
 132 in following sections):
 133
 134 \starttabulate[|l|l|l|l|]
 135 \NC \bf primitive         \NC \bf value range (in hex) \NC \NR
 136 \NC \type {\Uroot}           \NC 0+0--FF+10FFFF$^2$       \NC \NR
 137 \NC \type {\Uoverdelimiter}  \NC 0+0--FF+10FFFF$^2$       \NC \NR
 138 \NC \type {\Uunderdelimiter} \NC 0+0--FF+10FFFF$^2$       \NC \NR
 139 \NC \type {\Udelimiterover}  \NC 0+0--FF+10FFFF$^2$       \NC \NR
 140 \NC \type {\Udelimiterunder} \NC 0+0--FF+10FFFF$^2$       \NC \NR
 141 \stoptabulate
 142
 143 \section{Cramped math styles}
 144
 145 \LUATEX\ has four new primitives to set the cramped math styles directly:
 146
 147 \starttyping
 148 \crampeddisplaystyle
 149 \crampedtextstyle
 150 \crampedscriptstyle
 151 \crampedscriptscriptstyle
 152 \stoptyping
 153
 154 These additional commands are not all that valuable on their own, but they come
 155 in handy as arguments to the math parameter settings that will be added shortly.
 156
 157 \section{Math parameter settings}
 158
 159 In \LUATEX, the font dimension parameters that \TEX\ used in math typesetting are
 160 now accessible via primitive commands. In fact, refactoring of the math engine
 161 has resulted in many more parameters than were accessible before.
 162
 163 \starttabulate
 164 \NC \bf primitive name              \NC \bf description \NC \NR
 165 \NC \type {\Umathquad}               \NC the width of 18mu's \NC \NR
 166 \NC \type {\Umathaxis}               \NC height of the vertical center axis of
 167                                         the math formula above the baseline \NC \NR
 168 \NC \type {\Umathoperatorsize}       \NC minimum size of large operators in display mode \NC \NR
 169 \NC \type {\Umathoverbarkern}        \NC vertical clearance above the rule \NC \NR
 170 \NC \type {\Umathoverbarrule}        \NC the width of the rule \NC \NR
 171 \NC \type {\Umathoverbarvgap}        \NC vertical clearance below the rule \NC \NR
 172 \NC \type {\Umathunderbarkern}       \NC vertical clearance below the rule \NC \NR
 173 \NC \type {\Umathunderbarrule}       \NC the width of the rule \NC \NR
 174 \NC \type {\Umathunderbarvgap}       \NC vertical clearance above the rule \NC \NR
 175 \NC \type {\Umathradicalkern}        \NC vertical clearance above the rule \NC \NR
 176 \NC \type {\Umathradicalrule}        \NC the width of the rule \NC \NR
 177 \NC \type {\Umathradicalvgap}        \NC vertical clearance below the rule \NC \NR
 178 \NC \type {\Umathradicaldegreebefore}\NC the forward kern that takes place before placement of
 179                                         the radical degree \NC \NR
 180 \NC \type {\Umathradicaldegreeafter} \NC the backward kern that takes place after placement of
 181                                         the radical degree \NC \NR
 182 \NC \type {\Umathradicaldegreeraise} \NC this is the percentage of the total height and depth of
 183                                         the radical sign that the degree is raised by. It is
 184                                         expressed in \type {percents}, so 60\% is expressed as the
 185                                         integer $60$. \NC \NR
 186 \NC \type {\Umathstackvgap}          \NC vertical clearance between the two
 187                                         elements in a \type {\atop} stack \NC \NR
 188 \NC \type {\Umathstacknumup}         \NC numerator shift upward in \type {\atop} stack \NC \NR
 189 \NC \type {\Umathstackdenomdown}     \NC denominator shift downward in \type {\atop} stack \NC \NR
 190 \NC \type {\Umathfractionrule}       \NC the width of the rule in a \type {\over} \NC \NR
 191 \NC \type {\Umathfractionnumvgap}    \NC vertical clearance between the numerator and the rule \NC \NR
 192 \NC \type {\Umathfractionnumup}      \NC numerator shift upward in \type {\over} \NC \NR
 193 \NC \type {\Umathfractiondenomvgap}  \NC vertical clearance between the denominator and the rule \NC \NR
 194 \NC \type {\Umathfractiondenomdown}  \NC denominator shift downward in \type {\over} \NC \NR
 195 \NC \type {\Umathfractiondelsize}    \NC minimum delimiter size for \type {\...withdelims} \NC \NR
 196 \NC \type {\Umathlimitabovevgap}     \NC vertical clearance for limits above operators \NC \NR
 197 \NC \type {\Umathlimitabovebgap}     \NC vertical baseline clearance for limits above operators \NC \NR
 198 \NC \type {\Umathlimitabovekern}     \NC space reserved at the top of the limit \NC \NR
 199 \NC \type {\Umathlimitbelowvgap}     \NC vertical clearance for limits below operators \NC \NR
 200 \NC \type {\Umathlimitbelowbgap}     \NC vertical baseline clearance for limits below operators \NC \NR
 201 \NC \type {\Umathlimitbelowkern}     \NC space reserved at the bottom of the limit \NC \NR
 202 \NC \type {\Umathoverdelimitervgap}  \NC vertical clearance for limits above delimiters \NC \NR
 203 \NC \type {\Umathoverdelimiterbgap}  \NC vertical baseline clearance for limits above delimiters \NC \NR
 204 \NC \type {\Umathunderdelimitervgap} \NC vertical clearance for limits below delimiters \NC \NR
 205 \NC \type {\Umathunderdelimiterbgap} \NC vertical baseline clearance for limits below delimiters \NC \NR
 206 \NC \type {\Umathsubshiftdrop}       \NC subscript drop for boxes and subformulas \NC \NR
 207 \NC \type {\Umathsubshiftdown}       \NC subscript drop for characters \NC \NR
 208 \NC \type {\Umathsupshiftdrop}       \NC superscript drop (raise, actually) for boxes and subformulas \NC \NR
 209 \NC \type {\Umathsupshiftup}         \NC superscript raise for characters \NC \NR
 210 \NC \type {\Umathsubsupshiftdown}    \NC subscript drop in the presence of a superscript \NC \NR
 211 \NC \type {\Umathsubtopmax}          \NC the top of standalone subscripts cannot be higher than this
 212                                         above the baseline \NC \NR
 213 \NC \type {\Umathsupbottommin}       \NC the bottom of standalone superscripts cannot be less than
 214                                         this above the baseline \NC \NR
 215 \NC \type {\Umathsupsubbottommax}    \NC the bottom of the superscript of a combined super- and subscript
 216                                         be at least as high as this above the baseline \NC \NR
 217 \NC \type {\Umathsubsupvgap}         \NC vertical clearance between super- and subscript \NC \NR
 218 \NC \type {\Umathspaceafterscript}   \NC additional space added after a super- or subscript \NC \NR
 219 \NC \type {\Umathconnectoroverlapmin}\NC minimum overlap between parts in an extensible recipe \NC \NR
 220 \stoptabulate
 221
 222 Each of the parameters in this section can be set by a command like this:
 223
 224 \starttyping
 225 \Umathquad\displaystyle=1em
 226 \stoptyping
 227
 228 they obey grouping, and you can use \type {\the\Umathquad\displaystyle} if
 229 needed.
 230
 231 \section{Font-based Math Parameters}
 232
 233 While it is nice to have these math parameters available for tweaking, it would
 234 be tedious to have to set each of them by hand. For this reason, \LUATEX\
 235 initializes a bunch of these parameters whenever you assign a font identifier to
 236 a math family based on either the traditional math font dimensions in the font
 237 (for assignments to math family~2 and~3 using \TFM|-|based fonts like \type
 238 {cmsy} and \type {cmex}), or based on the named values in a potential \type
 239 {MathConstants} table when the font is loaded via Lua. If there is a \type
 240 {MathConstants} table, this takes precedence over font dimensions, and in that
 241 case no attention is paid to which family is being assigned to: the \type
 242 {MathConstants} tables in the last assigned family sets all parameters.
 243
 244 In the table below, the one|-|letter style abbreviations and symbolic tfm font
 245 dimension names match those using in the \TeX book. Assignments to \type
 246 {\textfont} set the values for the cramped and uncramped display and text styles.
 247 Use \type {\scriptfont} for the script styles, and \type {\scriptscriptfont} for the
 248 scriptscript styles (totalling eight parameters for three font sizes). In the
 249 \TFM\ case, assignments only happen in family~2 and family~3 (and of course only
 250 for the parameters for which there are font dimensions).
 251
 252 Besides the parameters below, \LUATEX\ also looks at the \quote {space} font
 253 dimension parameter. For math fonts, this should be set to zero.
 254
 255 \start
 256
 257 \switchtobodyfont[8pt]
 258
 259 \starttabulate[|l|l|l|p|]
 260 \NC \bf variable                   \NC \bf style             \NC \bf default value opentype               \NC \bf default value tfm \NC \NR
 261 \NC \type {\Umathaxis}                \NC --                    \NC AxisHeight                               \NC axis_height \NC \NR
 262 \NC \type {\Umathoperatorsize}        \NC D, D'                 \NC DisplayOperatorMinHeight                 \NC $^6$ \NC \NR
 263 \NC \type {\Umathfractiondelsize}     \NC D, D'                 \NC FractionDelimiterDisplayStyleSize$^9$    \NC delim1 \NC \NR
 264 \NC "                              \NC T, T', S, S', SS, SS' \NC FractionDelimiterSize$^9$                \NC delim2 \NC \NR
 265 \NC \type {\Umathfractiondenomdown}   \NC D, D'                 \NC FractionDenominatorDisplayStyleShiftDown \NC denom1 \NC \NR
 266 \NC "                              \NC T, T', S, S', SS, SS' \NC FractionDenominatorShiftDown             \NC denom2 \NC \NR
 267 \NC \type {\Umathfractiondenomvgap}   \NC D, D'                 \NC FractionDenominatorDisplayStyleGapMin    \NC 3*default_rule_thickness \NC \NR
 268 \NC "                              \NC T, T', S, S', SS, SS' \NC FractionDenominatorGapMin                \NC default_rule_thickness \NC \NR
 269 \NC \type {\Umathfractionnumup}       \NC D, D'                 \NC FractionNumeratorDisplayStyleShiftUp     \NC num1 \NC \NR
 270 \NC "                              \NC T, T', S, S', SS, SS' \NC FractionNumeratorShiftUp                 \NC num2 \NC \NR
 271 \NC \type {\Umathfractionnumvgap}     \NC D, D'                 \NC FractionNumeratorDisplayStyleGapMin      \NC 3*default_rule_thickness \NC \NR
 272 \NC "                              \NC T, T', S, S', SS, SS' \NC FractionNumeratorGapMin                  \NC default_rule_thickness \NC \NR
 273 \NC \type {\Umathfractionrule}        \NC --                    \NC FractionRuleThickness                    \NC default_rule_thickness \NC \NR
 274 \NC \type {\Umathlimitabovebgap}      \NC --                    \NC UpperLimitBaselineRiseMin                \NC big_op_spacing3 \NC \NR
 275 \NC \type {\Umathlimitabovekern}      \NC --                    \NC 0$^1$                                    \NC big_op_spacing5 \NC \NR
 276 \NC \type {\Umathlimitabovevgap}      \NC --                    \NC UpperLimitGapMin                         \NC big_op_spacing1 \NC \NR
 277 \NC \type {\Umathlimitbelowbgap}      \NC --                    \NC LowerLimitBaselineDropMin                \NC big_op_spacing4 \NC \NR
 278 \NC \type {\Umathlimitbelowkern}      \NC --                    \NC 0$^1$                                    \NC big_op_spacing5 \NC \NR
 279 \NC \type {\Umathlimitbelowvgap}      \NC --                    \NC LowerLimitGapMin                         \NC big_op_spacing2 \NC \NR
 280 \NC \type {\Umathoverdelimitervgap}   \NC --                    \NC StretchStackGapBelowMin                  \NC big_op_spacing1 \NC \NR
 281 \NC \type {\Umathoverdelimiterbgap}   \NC --                    \NC StretchStackTopShiftUp                   \NC big_op_spacing3 \NC \NR
 282 \NC \type {\Umathunderdelimitervgap}  \NC--                     \NC StretchStackGapAboveMin                  \NC big_op_spacing2 \NC \NR
 283 \NC \type {\Umathunderdelimiterbgap}  \NC--                     \NC StretchStackBottomShiftDown              \NC big_op_spacing4 \NC \NR
 284 \NC \type {\Umathoverbarkern}         \NC --                    \NC OverbarExtraAscender                     \NC default_rule_thickness \NC \NR
 285 \NC \type {\Umathoverbarrule}         \NC --                    \NC OverbarRuleThickness                     \NC default_rule_thickness \NC \NR
 286 \NC \type {\Umathoverbarvgap}         \NC --                    \NC OverbarVerticalGap                       \NC 3*default_rule_thickness \NC \NR
 287 \NC \type {\Umathquad}                \NC --                    \NC <font_size(f)>$^1$                       \NC math_quad \NC \NR
 288 \NC \type {\Umathradicalkern}         \NC --                    \NC RadicalExtraAscender                     \NC default_rule_thickness \NC \NR
 289 \NC \type {\Umathradicalrule}         \NC --                    \NC RadicalRuleThickness                     \NC <not set>$^2$ \NC \NR
 290 \NC \type {\Umathradicalvgap}         \NC D, D'                 \NC RadicalDisplayStyleVerticalGap           \NC (default_rule_thickness+\crlf
 291                                                                                                               (abs(math_x_height)/4))$^3$ \NC \NR
 292 \NC "                              \NC T, T', S, S', SS, SS' \NC RadicalVerticalGap                       \NC (default_rule_thickness+\crlf
 293                                                                                                               (abs(default_rule_thickness)/4))$^3$ \NC \NR
 294 \NC \type {\Umathradicaldegreebefore} \NC --                    \NC RadicalKernBeforeDegree                  \NC <not set>$^2$ \NC \NR
 295 \NC \type {\Umathradicaldegreeafter}  \NC --                    \NC RadicalKernAfterDegree                   \NC <not set>$^2$ \NC \NR
 296 \NC \type {\Umathradicaldegreeraise}  \NC --                    \NC RadicalDegreeBottomRaisePercent          \NC <not set>$^{2,7}$ \NC \NR
 297 \NC \type {\Umathspaceafterscript}    \NC --                    \NC SpaceAfterScript                         \NC script_space$^4$ \NC \NR
 298 \NC \type {\Umathstackdenomdown}      \NC D, D'                 \NC StackBottomDisplayStyleShiftDown         \NC denom1 \NC \NR
 299 \NC "                              \NC T, T', S, S', SS, SS' \NC StackBottomShiftDown                     \NC denom2 \NC \NR
 300 \NC \type {\Umathstacknumup}          \NC D, D'                 \NC StackTopDisplayStyleShiftUp              \NC num1 \NC \NR
 301 \NC "                              \NC T, T', S, S', SS, SS' \NC StackTopShiftUp                          \NC num3 \NC \NR
 302 \NC \type {\Umathstackvgap}           \NC D, D'                 \NC StackDisplayStyleGapMin                  \NC 7*default_rule_thickness \NC \NR
 303 \NC "                              \NC T, T', S, S', SS, SS' \NC StackGapMin                              \NC 3*default_rule_thickness \NC \NR
 304 \NC \type {\Umathsubshiftdown}        \NC --                    \NC SubscriptShiftDown                       \NC sub1 \NC \NR
 305 \NC \type {\Umathsubshiftdrop}        \NC --                    \NC SubscriptBaselineDropMin                 \NC sub_drop \NC \NR
 306 \NC \type {\Umathsubsupshiftdown}     \NC --                    \NC SubscriptShiftDownWithSuperscript$^8$    \NC \NC \NR
 307 \NC                                \NC                       \NC \quad\ or SubscriptShiftDown             \NC sub2 \NC \NR
 308 \NC \type {\Umathsubtopmax}           \NC --                    \NC SubscriptTopMax                          \NC (abs(math_x_height * 4) / 5) \NC \NR
 309 \NC \type {\Umathsubsupvgap}          \NC --                    \NC SubSuperscriptGapMin                     \NC 4*default_rule_thickness \NC \NR
 310 \NC \type {\Umathsupbottommin}        \NC --                    \NC SuperscriptBottomMin                     \NC (abs(math_x_height) / 4) \NC \NR
 311 \NC \type {\Umathsupshiftdrop}        \NC --                    \NC SuperscriptBaselineDropMax               \NC sup_drop \NC \NR
 312 \NC \type {\Umathsupshiftup}          \NC D                     \NC SuperscriptShiftUp                       \NC sup1 \NC \NR
 313 \NC "                              \NC T, S, SS,             \NC SuperscriptShiftUp                       \NC sup2 \NC \NR
 314 \NC "                              \NC D', T', S', SS'       \NC SuperscriptShiftUpCramped                \NC sup3 \NC \NR
 315 \NC \type {\Umathsupsubbottommax}     \NC --                    \NC SuperscriptBottomMaxWithSubscript        \NC (abs(math_x_height * 4) / 5) \NC \NR
 316 \NC \type {\Umathunderbarkern}        \NC --                    \NC UnderbarExtraDescender                   \NC default_rule_thickness \NC \NR
 317 \NC \type {\Umathunderbarrule}        \NC --                    \NC UnderbarRuleThickness                    \NC default_rule_thickness \NC \NR
 318 \NC \type {\Umathunderbarvgap}        \NC --                    \NC UnderbarVerticalGap                      \NC 3*default_rule_thickness \NC \NR
 319 \NC \type {\Umathconnectoroverlapmin} \NC --                    \NC MinConnectorOverlap                      \NC 0$^5$ \NC \NR
 320 \stoptabulate
 321
 322 \stop
 323
 324 Note 1: \OPENTYPE\ fonts set \type {\Umathlimitabovekern} and \type
 325 {\Umathlimitbelowkern} to zero and set \type {\Umathquad} to the font size of the
 326 used font, because these are not supported in the \type {MATH} table,
 327
 328 Note 2: \TFM\ fonts do not set \type {\Umathradicalrule} because \TEX82\ uses the
 329 height of the radical instead. When this parameter is indeed not set when
 330 \LUATEX\ has to typeset a radical, a backward compatibility mode will kick in
 331 that assumes that an oldstyle \TEX\ font is used. Also, they do not set \type
 332 {\Umathradicaldegreebefore}, \type {\Umathradicaldegreeafter}, and \type
 333 {\Umathradicaldegreeraise}. These are then automatically initialized to
 334 $5/18$quad, $-10/18$quad, and 60.
 335
 336 Note 3: If tfm fonts are used, then the \type {\Umathradicalvgap} is not set until
 337 the first time \LUATEX\ has to typeset a formula because this needs parameters
 338 from both family2 and family3. This provides a partial backward compatibility
 339 with \TEX82, but that compatibility is only partial: once the \type
 340 {\Umathradicalvgap} is set, it will not be recalculated any more.
 341
 342 Note 4: (also if tfm fonts are used) A similar situation arises wrt. \type
 343 {\Umathspaceafterscript}: it is not set until the first time \LUATEX\ has to
 344 typeset a formula. This provides some backward compatibility with \TEX82. But
 345 once the \type {\Umathspaceafterscript} is set, \type {\scriptspace} will never be
 346 looked at again.
 347
 348 Note 5: Tfm fonts set \type {\Umathconnectoroverlapmin} to zero because \TEX82\
 349 always stacks extensibles without any overlap.
 350
 351 Note 6: The \type {\Umathoperatorsize} is only used in \type {\displaystyle}, and is
 352 only set in \OPENTYPE\ fonts. In \TFM\ font mode, it is artificially set to one
 353 scaled point more than the initial attempt's size, so that always the \quote
 354 {first next} will be tried, just like in \TEX82.
 355
 356 Note 7: The \type {\Umathradicaldegreeraise} is a special case because it is the
 357 only parameter that is expressed in a percentage instead of as a number of scaled
 358 points.
 359
 360 Note 8: \type {SubscriptShiftDownWithSuperscript} does not actually exist in the
 361 \quote {standard} Opentype Math font Cambria, but it is useful enough to be
 362 added.
 363
 364 Note 9: \type {FractionDelimiterDisplayStyleSize} and \type
 365 {FractionDelimiterSize} do not actually exist in the \quote {standard} Opentype
 366 Math font Cambria, but were useful enough to be added.
 367
 368 \section{Math spacing setting}
 369
 370 Besides the parameters mentioned in the previous sections, there are also 64 new
 371 primitives to control the math spacing table (as explained in Chapter~18 of the
 372 \TEX book). The primitive names are a simple matter of combining two math atom
 373 types, but for completeness' sake, here is the whole list:
 374
 375 \starttwocolumns
 376 \starttyping
 377 \Umathordordspacing
 378 \Umathordopspacing
 379 \Umathordbinspacing
 380 \Umathordrelspacing
 381 \Umathordopenspacing
 382 \Umathordclosespacing
 383 \Umathordpunctspacing
 384 \Umathordinnerspacing
 385 \Umathopordspacing
 386 \Umathopopspacing
 387 \Umathopbinspacing
 388 \Umathoprelspacing
 389 \Umathopopenspacing
 390 \Umathopclosespacing
 391 \Umathoppunctspacing
 392 \Umathopinnerspacing
 393 \Umathbinordspacing
 394 \Umathbinopspacing
 395 \Umathbinbinspacing
 396 \Umathbinrelspacing
 397 \Umathbinopenspacing
 398 \Umathbinclosespacing
 399 \Umathbinpunctspacing
 400 \Umathbininnerspacing
 401 \Umathrelordspacing
 402 \Umathrelopspacing
 403 \Umathrelbinspacing
 404 \Umathrelrelspacing
 405 \Umathrelopenspacing
 406 \Umathrelclosespacing
 407 \Umathrelpunctspacing
 408 \Umathrelinnerspacing
 409 \Umathopenordspacing
 410 \Umathopenopspacing
 411 \Umathopenbinspacing
 412 \Umathopenrelspacing
 413 \Umathopenopenspacing
 414 \Umathopenclosespacing
 415 \Umathopenpunctspacing
 416 \Umathopeninnerspacing
 417 \Umathcloseordspacing
 418 \Umathcloseopspacing
 419 \Umathclosebinspacing
 420 \Umathcloserelspacing
 421 \Umathcloseopenspacing
 422 \Umathcloseclosespacing
 423 \Umathclosepunctspacing
 424 \Umathcloseinnerspacing
 425 \Umathpunctordspacing
 426 \Umathpunctopspacing
 427 \Umathpunctbinspacing
 428 \Umathpunctrelspacing
 429 \Umathpunctopenspacing
 430 \Umathpunctclosespacing
 431 \Umathpunctpunctspacing
 432 \Umathpunctinnerspacing
 433 \Umathinnerordspacing
 434 \Umathinneropspacing
 435 \Umathinnerbinspacing
 436 \Umathinnerrelspacing
 437 \Umathinneropenspacing
 438 \Umathinnerclosespacing
 439 \Umathinnerpunctspacing
 440 \Umathinnerinnerspacing
 441 \stoptyping
 442 \stoptwocolumns
 443
 444 These parameters are of type \type {\muskip}, so setting a parameter can be done
 445 like this:
 446
 447 \starttyping
 448 \Umathopordspacing\displaystyle=4mu plus 2mu
 449 \stoptyping
 450
 451 They are all initialized by initex to the values mentioned in the table in
 452 Chapter~18 of the \TEX book.
 453
 454 Note 1: for ease of use as well as for backward compatibility, \type
 455 {\thinmuskip}, \type {\medmuskip} and \type {\thickmuskip} are treated
 456 especially. In their case a pointer to the corresponding internal parameter is
 457 saved, not the actual \type {\muskip} value. This means that any later changes to
 458 one of these three parameters will be taken into account.
 459
 460 Note 2: Careful readers will realise that there are also primitives for the items
 461 marked \type {*} in the \TEX book. These will not actually be used as those
 462 combinations of atoms cannot actually happen, but it seemed better not to break
 463 orthogonality. They are initialized to zero.
 464
 465 \section[mathacc]{Math accent handling}
 466
 467 \LUATEX\ supports both top accents and bottom accents in math mode, and math
 468 accents stretch automatically (if this is supported by the font the accent comes
 469 from, of course). Bottom and combined accents as well as fixed-width math accents
 470 are controlled by optional keywords following \type {\Umathaccent}.
 471
 472 The keyword \type {bottom} after \type {\Umathaccent} signals that a bottom accent
 473 is needed, and the keyword \type {both} signals that both a top and a bottom
 474 accent are needed (in this case two accents need to be specified, of course).
 475
 476 Then the set of three integers defining the accent is read. This set of integers
 477 can be prefixed by the \type {fixed} keyword to indicate that a non-stretching
 478 variant is requested (in case of both accents, this step is repeated).
 479
 480 A simple example:
 481
 482 \starttyping
 483 \Umathaccent both fixed 0 0 "20D7 fixed 0 0 "20D7 {example}
 484 \stoptyping
 485
 486 If a math top accent has to be placed and the accentee is a character and has a
 487 non-zero \type {top_accent} value, then this value will be used to place the
 488 accent instead of the \type {\skewchar} kern used by \TEX82.
 489
 490 The \type {top_accent} value represents a vertical line somewhere in the
 491 accentee. The accent will be shifted horizontally such that its own \type
 492 {top_accent} line coincides with the one from the accentee. If the \type
 493 {top_accent} value of the accent is zero, then half the width of the accent
 494 followed by its italic correction is used instead.
 495
 496 The vertical placement of a top accent depends on the \type {x_height} of the
 497 font of the accentee (as explained in the \TEX book), but if value that turns out
 498 to be zero and the font had a MathConstants table, then \type {AccentBaseHeight}
 499 is used instead.
 500
 501 If a math bottom accent has to be placed, the \type {bot_accent} value is checked
 502 instead of \type {top_accent}. Because bottom accents do not exist in \TEX82, the
 503 \type {\skewchar} kern is ignored.
 504
 505 The vertical placement of a bottom accent is straight below the accentee, no
 506 correction takes place.
 507
 508 \section{Math root extension}
 509
 510 The new primitive \type {\Uroot} allows the construction of a radical noad
 511 including a degree field. Its syntax is an extension of \type {\Uradical}:
 512
 513 \starttyping
 514 \Uradical <fam integer> <char integer> <radicand>
 515 \Uroot    <fam integer> <char integer> <degree> <radicand>
 516 \stoptyping
 517
 518 The placement of the degree is controlled by the math parameters \type
 519 {\Umathradicaldegreebefore}, \type {\Umathradicaldegreeafter}, and \type
 520 {\Umathradicaldegreeraise}. The degree will be typeset in \type
 521 {\scriptscriptstyle}.
 522
 523 \section{Math kerning in super- and subscripts}
 524
 525 The character fields in a lua-loaded OpenType math font can have a \quote
 526 {mathkern} table. The format of this table is the same as the \quote {mathkern}
 527 table that is returned by the \type {fontloader} library, except that all height
 528 and kern values have to be specified in actual scaled points.
 529
 530 When a super- or subscript has to be placed next to a math item, \LUATEX\ checks
 531 whether the super- or subscript and the nucleus are both simple character items.
 532 If they are, and if the fonts of both character imtes are OpenType fonts (as
 533 opposed to legacy \TEX\ fonts), then \LUATEX\ will use the OpenType MATH
 534 algorithm for deciding on the horizontal placement of the super- or subscript.
 535
 536 This works as follows:
 537
 538 \startitemize
 539     \startitem
 540         The vertical position of the script is calculated.
 541     \stopitem
 542     \startitem
 543         The default horizontal position is flat next to the base character.
 544     \stopitem
 545     \startitem
 546         For superscripts, the italic correction of the base character is added.
 547     \stopitem
 548     \startitem
 549         For a superscript, two vertical values are calculated: the bottom of the
 550         script (after shifting up), and the top of the base. For a subscript, the two
 551         values are the top of the (shifted down) script, and the bottom of the base.
 552     \stopitem
 553     \startitem
 554         For each of these two locations:
 555         \startitemize
 556             \startitem
 557                 find the mathkern value at this height for the base (for a subscript
 558                 placement, this is the bottom_right corner, for a superscript
 559                 placement the top_right corner)
 560             \stopitem
 561             \startitem
 562                 find the mathkern value at this height for the script (for a
 563                 subscript placement, this is the top_left corner, for a superscript
 564                 placement the bottom_left corner)
 565             \stopitem
 566             \startitem
 567                 add the found values together to get a preliminary result.
 568             \stopitem
 569         \stopitemize
 570     \stopitem
 571     \startitem
 572         The horizontal kern to be applied is the smallest of the two results from
 573         previous step.
 574     \stopitem
 575 \stopitemize
 576
 577 The mathkern value at a specific height is the kern value that is specified by the
 578 next higher height and kern pair, or the highest one in the character (if there is no
 579 value high enough in the character), or simply zero (if the character has no mathkern
 580 pairs at all).
 581
 582 \section{Scripts on horizontally extensible items like arrows}
 583
 584 The primitives \type {\Uunderdelimiter} and \type {\Uoverdelimiter} allow the
 585 placement of a subscript or superscript on an automatically extensible item and
 586 \type {\Udelimiterunder} and \type {\Udelimiterover} allow the placement of an
 587 automatically extensible item as a subscript or superscript on a nucleus. The
 588 input:
 589
 590 % these produce radical noads .. in fact the code base has the numbers wrong for
 591 % quite a while, so no one seems to use this
 592
 593 \startbuffer
 594 $\Uoverdelimiter  0 "2194 {\hbox{\strut  overdelimiter}}$
 595 $\Uunderdelimiter 0 "2194 {\hbox{\strut underdelimiter}}$
 596 $\Udelimiterover  0 "2194 {\hbox{\strut  delimiterover}}$
 597 $\Udelimiterunder 0 "2194 {\hbox{\strut delimiterunder}}$
 598 \stopbuffer
 599
 600 \typebuffer will render this:
 601
 602 \blank \startnarrower \getbuffer \stopnarrower \blank
 603
 604 The vertical placements are controlled by \type {\Umathunderdelimiterbgap}, \type
 605 {\Umathunderdelimitervgap}, \type {\Umathoverdelimiterbgap}, and \type
 606 {\Umathoverdelimitervgap} in a similar way as limit placements on large operators.
 607 The superscript in \type {\Uoverdelimiter} is typeset in a suitable scripted style,
 608 the subscript in \type {\Uunderdelimiter} is cramped as well.
 609
 610 \section {Extensible delimiters}
 611
 612 \LUATEX\ internally uses a structure that supports \OPENTYPE\ \quote
 613 {MathVariants} as well as \TFM\ \quote {extensible recipes}.
 614
 615 \section{Other Math changes}
 616
 617 \subsection {Verbose versions of single-character math commands}
 618
 619 \LUATEX\ defines six new primitives that have the same function as
 620 \type {^}, \type {_}, \type {$}, and \type {$$}. %$
 621
 622 \starttabulate[|l|l|l|l|]
 623 \NC \bf primitive           \NC \bf explanation \NC \NR
 624 \NC \type {\Usuperscript}      \NC Duplicates the functionality of \type {^} \NC \NR
 625 \NC \type {\Usubscript}        \NC Duplicates the functionality of \type {_} \NC \NR
 626 \NC \type {\Ustartmath}        \NC Duplicates the functionality of \type {$}, % $
 627                                    when used in non-math mode. \NC \NR
 628 \NC \type {\Ustopmath}         \NC Duplicates the functionality of \type {$}, % $
 629                                    when used in inline math mode. \NC \NR
 630 \NC \type {\Ustartdisplaymath} \NC Duplicates the functionality of \type {$$}, % $$
 631                                    when used in non-math mode. \NC \NR
 632 \NC \type {\Ustopdisplaymath}  \NC Duplicates the functionality of \type {$$}, % $$
 633                                    when used in display math mode. \NC \NR
 634 \stoptabulate
 635
 636 The \type {\Ustopmath} and \type {\Ustopdisplaymath} primitives check if the current
 637 math mode is the correct one (inline vs.\ displayed), but you can freely intermix
 638 the four mathon|/|mathoff commands with explicit dollar sign(s).
 639
 640 \subsection{Allowed math commands in non-math modes}
 641
 642 The commands \type {\mathchar}, and \type {\Umathchar} and control sequences that
 643 are the result of \type {\mathchardef} or \type {\Umathchardef} are also
 644 acceptable in the horizontal and vertical modes. In those cases, the \type
 645 {\textfont} from the requested math family is used.
 646
 647 \section{Math todo}
 648
 649 The following items are still todo.
 650
 651 \startitemize
 652 \startitem
 653     Pre-scripts.
 654 \stopitem
 655 \startitem
 656     Multi-story stacks.
 657 \stopitem
 658 \startitem
 659     Flattened accents for high characters (maybe).
 660 \stopitem
 661 \startitem
 662     Better control over the spacing around displays and handling of equation numbers.
 663 \stopitem
 664 \startitem
 665     Support for multi|-|line displays using \MATHML\ style alignment points.
 666 \stopitem
 667 \stopitemize
 668
 669 \stopchapter
 670
 671 \stopcomponent