12 MAPS 38 Ulrik Vieth
OpenType Math Illuminated
Abstract dressing the very complex and challenging require- In recent years, we have seen the development of new ments of Arabic typography. TEX engines, X TE EX and LuaTEX, adopting OpenType However, when it comes to math typesetting, one font technology for providing Unicode typesetting of the traditional strongholds of TEX, support for Uni- support. While there are already plenty of OpenType code and OpenType math is only just beginning to take text fonts available for use, both from the T X E shape. community and from commercial font suppliers, there is little support for OpenType math fonts so far. Ironically, it was left to Microsoft to develop the first Ironically, it was left to Microsoft to develop a de facto system to offer support for Unicode math. When Mi- standard for OpenType math font information and to crosoft introduced support for math typesetting in Of- provide the first reference implementation of a fice 2007 [1, 2], they extended the OpenType font for- full-featured OpenType math font. mat and commissioned the design of Cambria Math [3] as a reference implementation of a full-featured Open- In order to develop the much-needed math support for Type math font. Latin Modern and TEX Gyre fonts, it will be crucially Fortunately for us, Microsoft was smart enough to important to develop a good understanding of the borrow from the best examples of math typesetting internals of OpenType math tables, much as it is technology, thus many concepts of OpenType math are necessary to develop a good understanding of not only derived from the model of TEX, but also go be- Appendix G and TEX’s \fontdimen parameters to yond T X and introduce extensions or generalizations develop math support for traditional T X fonts. In this E E of familiar concepts. paper, we try to help improve the understanding of OpenType math internals, summarizing the parameters While OpenType math is officially still considered of OpenType math fonts as well as illustrating the experimental, it is quickly becoming a de facto stan- similarities and differences between traditional TEX dard, as it has already been widely deployed to mil- math fonts and OpenType math fonts. lions of installations of Microsoft Office 2007 and it is also being been adopted by other projects such as the FontForge [4] font editor or independent font designs Background on OpenType math such as Asana Math [5]. Most importantly, support for OpenType math has In recent years, the TEX community has been go- already been implemented or is currently being imple- ing through a phase of very significant developments. mented in the new TEX engines, thus adopting Open- Among the most important achievements, we have Type math for the development of the much-needed seen the development of new TEX engines, X TE EX and Unicode math support for Latin Modern and TEX Gyre LuaTEX, providing support for Unicode and OpenType obviously seems to be most promising choice of tech- font technology. At about the same time we have nology. also seen the development of new font distributions, Latin Modern and TEX Gyre, provided simultaneously in Type 1 format as a set of 8-bit font encodings as well Design and quality of math fonts as in OpenType format. Together these developments have enabled TEX When it comes to developing math fonts, designing users to keep up with current trends in the publish- the glyph shapes is only part of the job. Another part, ing industry, providing users of the new TEX engines which is equally important, is to adjust the glyph met- with a comprehensive set of free OpenType fonts and rics of individual glyphs and to set up the global pa- enabling them to take advantage of the many offerings rameters affecting various aspects of glyph positioning by commercial font suppliers. in math typesetting. As far as text typesetting is concerned, support for As we have discussed at previous conferences, the OpenType font technology in the new TEX engines is al- quality of math typesetting crucially depends on the ready very advanced, supporting not only traditional fine-tuning of these parameters. Developing a good typographic features of Latin alphabets, but also ad- understanding of these parameters will therefore be- OpenType Math Illuminated VOORJAAR 2009 13 come an important prerequisite to support the devel- placement of limits on big operators, the placement opment of new math fonts. of numerators and denominators in fractions, or the In the case of traditional TEX math fonts, we have placement of superscripts and subscripts. to deal with the many \fontdimen parameters which While a number of parameters are specified in TEX have been analyzed in Bogusław Jackowski’s paper through the \fontdimen parameters of math fonts, Appendix G Illuminated and a follow-up paper by the there are other parameters which are defined by built- present author [6, 7]. in rules of TEX’s math typesetting engine. In many such In the case of OpenType math fonts, we need to de- cases, additional parameters have been introduced in velop a similar understanding of the various tables and the OpenType MATH table, making it possible to specify parameters and how the concepts of OpenType math all the relevant parameters in the math font without relate to the concepts of TEX. relying on built-in rules of any particular typesetting engine. Overview of the OpenType font format In view of the conference motto, it is interesting to note that the two new TEX engines, X TE EX and The OpenType font format [8] was developed jointly LuaTEX, have taken a very different approach how to by Adobe and Microsoft, based on elements of the ear- support the additional parameters of OpenType math lier PostScript and TrueType font formats by the same fonts: While X TE EX has retained TEX’s original math vendors. The overall structure of OpenType fonts con- typesetting engine and uses an internal mapping to sists of a number of tables, some of which are required set up \fontdimen parameters from OpenType para- while others are optional [9]. meters [10], LuaTEX has introduced an extension of In the case of OpenType math, the extension of the TEX’s math typesetting engine [11], which will allow font format essentially consists of adding another op- to take full advantage of most of the additional Open- tional table, the so-called MATH table, containing all Type parameters. (More precisely, while X TE EX only the information related to math typesetting. Since it provides access to the OpenType parameters as ad- is an optional table, it would be interpreted only by ditional \fontdimens, LuaTEX uses an internal data software which knows about it (such as the new TEX structure based on the combined set of OpenType and engines or Microsoft Office 2007), while it would be TEX parameters, making it possible to supply missing ignored by other software. values which are not supported in either OpenType Unlike a database table, which has a very rigid for- math fonts or traditional TEX math fonts.) mat, an OpenType font table can have a fairly com- For font designers developing OpenType math fonts, plex structure, combining a variety of different kinds it may be best to supply all of the additional OpenType of information in the same table. In the case of the parameters in order to make their fonts as widely us- OpenType MATH table, we have the following kinds of able as possible with any typesetting engine, not nec- information: essarily limited to any specific one of the new TEX en- gines. a number of global parameters specific to math In the following sections, we will take a closer look 2 typesetting (similar to TEX’s many \fontdimen at the various groups of OpenType parameters, orga- parameters of Appendix G) nized in a similar way as they are presented to font instructions for vertical and horizontal variants designers in the FontForge font editor, but not neces- 2 and/or constructions (similar to TEX’s charlists and sarily in the same order. extensible recipes) We will use the figures from [6, 7] as a visual clue additional glyph metric information specific to to illustrate how the various parameters are defined 2 math mode (such as italic corrections, accent in TEX, while summarizing the similarities and differ- placement, or kerning) ences between OpenType parameters and TEX parame- ters in tabular form. In the following sections, we will discuss some of these parameters in more detail, illustrating the simi- Limits on big operators larities and differences between traditional TEX math In TEX math fonts, there are five parameters controlling fonts and OpenType math fonts. the placement of limits on big operators (see figure 1), which are denoted as ξ9 to ξ13 using the notation of Appendix G. Parameters of OpenType math fonts Two of them control the default position of the limits The parameters of the OpenType MATH table play a (ξ10 and ξ12), two of them control the inside gap (ξ9 similar role as TEX’s \fontdimen parameters, control- and ξ11), while the final one controls the outside gap ling various aspects of math typesetting, such as the above and below the limits (ξ13). 14 MAPS 38 Ulrik Vieth
ξ 13 OpenType parameter T X parameter δ/2 E Q StretchStackTopShiftUp ξ11 ξ11 StretchStackGapAboveMin ξ9 ≥ξ9 StretchStackGapBelowMin ξ10 Z StretchStackBottomShiftDown ξ12 Table 2. Correspondence of font metric parameters between OpenType and TEX related to stretch stacks.
Stretch Stacks Stretch stacks are a new feature in OpenType math fonts, which do not have a direct correspondence in TEX. They can be understood in terms of material stacked above or below stretchable elements such as overbraces, underbraces or long arrows. ≥ξ In TEX, such elements were typically handled at the 10 macro level and effectively treated in the same way as δ/2 ξ12 limits on big operators. In LuaTEX, such elements will be implemented by M=1 ξ 13 new primitives using either the new OpenType para- meters for stretch stacks (as shown in table 2) or the Figure 1. TEX font metric parameters affecting the parameters for limits on big operators when using tra- placement of limits above or below big operators. ditional TEX math fonts.
In OpenType math fonts, the MATH table contains Overbars and Underbars only four parameters controlling the placement of lim- In TEX math fonts, there are no specific parameters re- its on big operators. Those four parameters have a lated to the placement of overlines and underlines. In- direct correspondence to TEX’s parameters (as shown stead, there is only one parameter controlling the de- in table 1), while the remaining one has no corre- fault rule thickness (ξ8), which is used in a number spondence and is effectively set to zero. (Consider- of different situations where other parameters are ex- ing the approach taken in other circumstances, it is pressed in multiples of the rule thickness. very likely that if there were any such correspondence, In OpenType math fonts, a different approach was there might actually be two parameters in OpenType taken, introducing extra parameters for each purpose, instead of only one, such as UpperLimitExtraAscender and even supporting different sets of parameters for over- LowerLimitExtraDescender. In LuaTEX’s internal data struc- lines and underlines. Thus the MATH table contains the tures, there are actually two parameters for this pur- following parameters related to overlines and under- pose, which are either initialized from TEX’s parameter lines (as shown in table 3), which only have an indirect ξ13 when using TEX math fonts or set to zero when us- correspondence in TEX. ing OpenType math fonts.)
OpenType parameter TEX parameter OpenType parameter TEX parameter OverbarExtraAscender (= ξ8) UpperLimitBaselineRiseMin ξ11 OverbarRuleThickness (= ξ8) UpperLimitGapMin ξ9 OverbarVerticalGap (= 3 ξ8) LowerLimitGapMin ξ10 UnderbarVerticalGap (= 3 ξ8) LowerLimitBaselineDropMin ξ12 UnderbarRuleThickness (= ξ8) (= ξ ) (no correspondence) ξ13 UnderbarExtraDescender 8
Table 1. Correspondence of font metric parameters Table 3. Correspondence of font metric parameters between OpenType and TEX affecting the placement of between OpenType and TEX affecting the placement of limits above or below big operators. overlines and underlines. OpenType Math Illuminated VOORJAAR 2009 15
0 0 σ8 styles D,D σ8 styles D,D ϕ 0 0 σ9 other styles σ10 other styles 3ξ8 styles D,D ϕ 7ξ8 styles D,D ϕ ξ8 other styles 3ξ8 other styles 0 0 σ11 styles D,D σ11 styles D,D σ12 other styles σ12 other styles
Figure 2. TEX font metric parameters affecting the Figure 3. TEX’s boundary conditions affecting the placement of numerators and denominators in regular and placement of numerators and denominators in regular and generalized fractions. generalized fractions.
It is interesting to note that the introduction of ad- As shown in table 4, there is a correspondence for ditional parameters in OpenType math fonts provides all TEX parameters, but this correspondence isn’t neces- for greater flexibility of the font designer to adjust the sarily unique when the same TEX parameter is used for values for best results. multiple purposes in fractions and stacks. Obviously, While TEX’s built-in rules always use a fixed multi- font designers of OpenType math fonts should be care- plier of the rule thickness regardless of its size, Open- ful about choosing the values of OpenType parameters Type math fonts can compensate for a larger rule thick- in a consistent way. ness by using a smaller multiplier. Analyzing the font parameters of Cambria Math An example can be found when inspecting the pa- once again shows how the introduction of additional rameter values of Cambria Math: In relative terms the parameters increases the flexibility of the designer inside gap is only about 2.5 times rather than 3 times to adjust the parameters for best results: In relative the rule thickness, while the latter (at about 0.65 pt terms, FractionDisplayStyleGapMin is only about 2 times compared to 0.4 pt) is quite a bit larger than in typical rather than 3 times the rule thickness. Similarly, TEX fonts. StackDisplayStyleGapMin is only about 4.5 times rather than Obviously, making use of the individual OpenType 7 times the rule thickness. In absolute terms, however, parameters (as in LuaTEX) instead of relying on TEX’s both parameters are about the same order of magni- built-in rules (as in X TE EX) would more closely reflect tude as in typical TEX fonts. the intention of the font designer.
Fractions and Stacks OpenType parameter TEX parameter In TEX math fonts, there are five parameters controlling the placement of numerators and denominators (see FractionNumeratorDisplayStyleShiftUp σ8 σ figure 2), which are denoted as σ8 to σ12 using the FractionNumeratorShiftUp 9 notation of Appendix G. FractionNumeratorDisplayStyleGapMin (= 3 ξ8) Four of them apply to regular fractions, either in dis- FractionNumeratorGapMin (= ξ8) play style (σ8 and σ11) or in text style and below (σ9 FractionRuleThickness (= ξ8) and σ12), while the remaining one applies to the spe- FractionDenominatorDisplayStyleGapMin (= 3 ξ8) cial case of generalized fractions when the fraction bar FractionDenominatorGapMin (= ξ8) is absent (σ10). FractionDenominatorDisplayStyleShiftDown σ Besides those specific parameters, there are also 11 FractionDenominatorShiftDown σ a number of parameters which are based on built-in 12 rules of TEX’s math typesetting engine, expressed in StackTopDisplayStyleShiftUp σ8 multiples of the rule thickness (ξ8), such as the thick- StackTopShiftUp σ10 ness of the fraction rule or the inside gap above and StackDisplayStyleGapMin (= 7 ξ8) below the fraction rule (see figure 3). StackGapMin (= 3 ξ8) In OpenType math fonts, a different approach was StackBottomDisplayStyleShiftDown σ once again taken, introducing a considerable number 11 StackBottomShiftDown σ of additional parameters for each purpose. Thus the 12 MATH table contains 9 parameters related to regular Table 4. Correspondence of font metric parameters fractions and 6 more parameters related to generalized between OpenType and TEX affecting the placement of fractions (also known as stacks). numerators and denominators. 16 MAPS 38 Ulrik Vieth
σ13 σ14 σ15 4 5 σ5 σ16 1 σ17 4 σ5
σ↑18
4ξ8 4 4ξ8 5 σ5
σ↓19
Figure 4. TEX font metric parameters affecting the placement of superscripts and subscripts on a simple character or a boxed subformula. Figure 5. TEX font metric parameters affecting the placement of superscripts and subscripts in cases of Superscripts and Subscripts resolving collisions. In TEX math fonts, there are seven parameters control- ling the placement of superscripts and subscripts (see figure 4), which are denoted as σ13 to σ19 using the OpenType parameter TEX parameter notation of Appendix G. Three of them apply to superscripts, either in dis- SuperscriptShiftUp σ13, σ14 play style (σ13), in text style and below (σ14), or in SuperscriptShiftUpCramped σ15 cramped style (σ ), while the other two apply to the 15 SubscriptShiftDown σ16, σ17 placement of subscripts, either with or without a su- σ perscript (σ16 and σ17). SuperscriptBaselineDropMax 18 Finally, there are two more parameters which apply SubscriptBaselineDropMin σ19 to superscripts and subscripts on a boxed subformula SuperscriptBottomMin (= 1 σ ) (σ and σ ), which also apply to limits attached to 4 5 18 19 SubscriptTopMax (= 4 σ ) big operators with \nolimits. 5 5 Besides those specific parameters, there are also a SubSuperscriptGapMin (= 4 ξ8) SuperscriptBottomMaxWithSubscript (= 4 σ ) number of parameters which are based on TEX’s built- 5 5 in rules, expressed in multiples of the x-height (σ5) or SpaceAfterScript \scriptspace the rule thickness (ξ8), most of them related to resolv- ing collisions between superscripts and subscripts or adjusting the position when a superscript or subscript Table 5. Correspondence of font metric parameters becomes too big (see figure 5). between OpenType and TEX affecting the placement of In OpenType math fonts, we once again find a num- superscripts and subscripts. ber of additional parameters for each specific purpose, as shown in table 5. Radicals It is interesting to note that some of the usual dis- In TEX math fonts, there are no specific parameters tinctions made in TEX were apparently omitted in the related to typesetting radicals. Instead, the relevant OpenType MATH table, as there is no specific value for parameters are based on built-in rules of TEX’s math the superscript position in display style, nor are there typesetting engine, expressed in multiples of the rule any differences in subscript position in the presence or thickness (ξ8) or the x-height (σ5). absence of superscripts. To be precise, there are even more complications in- While it is not clear why there is no correspondence volved [6], as the height of the fraction rule is actually for these parameters, it is quite possible that there was taken from the height of the fraction glyph rather than a conscious design decision to omit them, perhaps to the default rule thickness to account for effects of pixel avoid inconsistencies in alignment. rounding in bitmap fonts. OpenType Math Illuminated VOORJAAR 2009 17
OpenType parameter TEX parameter OpenType parameter TEX parameter
RadicalExtraAscender (= ξ8) ScriptPercentScaleDown e. g. 70–80 % RadicalRuleThickness (= ξ8) ScriptScriptPercentScaleDown e. g. 50–60 % (= ξ + 1 σ ) RadicalDisplayStyleVerticalGap 8 4 5 ?? (e. g. 12–15 pt) 1 DisplayOperatorMinHeight RadicalVerticalGap (= ξ8 + ξ8) 4 (no correspondence) σ20 (e. g. 20–24 pt) 5 RadicalKernBeforeDegree e. g. 18 em DelimitedSubFormulaMinHeight σ21 (e. g. 10–12 pt) RadicalKernAfterDegree e. g. 10 em 18 AxisHeight σ22 (axis height) RadicalDegreeBottomRaisePercent e. g. 60 % AccentBaseHeight σ5 (x-height) Table 6. Correspondence of font metric parameters FlattenedAccentBaseHeight ?? (capital height) between OpenType and TEX affecting the placement of radicals. Table 7. Correspondence of font metric parameters between OpenType and TEX affecting some general In OpenType math fonts, we once again find a aspects of math typesetting. number of additional parameters for each purpose, as shown in table 6. General parameters While there is a correspondence for all of the pa- The final group of OpenType parameters combines a rameters built into TEX’s typesetting algorithms, it is mixed bag of parameters for various purposes. Some interesting to note that OpenType math has also intro- of them have a straight-forward correspondence in TEX duced some additional parameters related√ to the place- (such as the math axis position), while others do not ment of the degree of an n th root ( n x ), which is have any correspondence at all. As shown in table 7, usually handled at the macro level in TEX’s format files there are some very noteworthy parameters in this plain.tex or latex.ltx: group, which deserve some further explanations in the \newbox\rootbox following paragraphs. \def\root#1\of{% \setbox\rootbox (Script)ScriptPercentScaleDown. \hbox{$\m@th\scriptscriptstyle{#1}$}% These OpenType parameters represent the font sizes \mathpalette\r@@t} of the first and second level script fonts relative to the \def\r@@t#1#2{% base font. In TEX math fonts, these parameters do not \setbox\z@\hbox{$\m@th#1\sqrtsign{#2}$}% have a correspondence in the font metrics. Instead \dimen@=\ht\z@ \advance\dimen@-\dp\z@ they are usually specified at the macro level when a \mkern5mu\raise.6\dimen@\copy\rootbox family of math fonts is loaded. \mkern-10mu\box\z@} If a font family provides multiple design sizes (as in As shown in the listing, the definition of the \root Computer Modern), font loading of math fonts in TEX macro contains a number of hard-coded parameters, might look like the following, using different design such as a positive kern before the box containing the sizes, each at their natural size: degree and negative kern thereafter, expressed in mul- \newfam\symbols tiples of the font-specific math unit. In addition, there \textfont\symbols=cmsy10 is also a raise factor expressed relative to the size of \scriptfont\symbols=cmsy7 the box containing the radical sign. \scriptscriptfont\symbols=cmsy5 Obviously, the extra OpenType parameters related If a font family does not provide multiple design to the degree of radicals correspond directly to the pa- sizes (as in MathTime), font loading of math fonts will rameters used internally in the \root macro, making it use scaled-down versions of the base font: possible to supply a set of font-specific values instead \newfam\symbols of using hard-coded values expressed in multiples of \textfont\symbols=mtsy10 at 10pt font-specific units. \scriptfont\symbols=mtsy10 at 7.6pt In LuaT X, this approach has been taken one step E \scriptscriptfont\symbols=mtsy10 at 6pt further, introducing a new \Uroot primitive as an ex- tension of the \Uradical primitive, making it possible The appropriate scaling factors depend on the font to replace the processing at the macro level by process- design, but are usually defined in macro packages or in format files using higher-level macros such as ing at the algorithmic level in LuaTEX’s extended math typesetting engine [11]. \DeclareMathSizes in LaTEX. 18 MAPS 38 Ulrik Vieth
In OpenType math fonts, it will be possible to pack- set up in the format file. age optical design variants for script sizes into a single As a result, we will typically get 10 pt or 12 pt de- font by using OpenType feature selectors to address the limiters in text style and 18 pt or 24 pt delimiters in design variants and using scaling factors as specified in display style. For typical settings, the delimiters only the MATH table. (As discussed in [12], there are many have to cover 90 % of the required size and they may issues to consider regarding the development of Open- fall short by at most 5 pt. Type math fonts besides setting up the font parameters. If a generalized fraction with delimiters is coded like One such issue is the question of font organization re- the following garding the inclusion of optical design variants into the $ {n \atopwithdelims() k} $ base font.) the contents will be treated as a delimited fraction, and The corresponding code for font loading of full- in this case the size of delimiters will depend on the featured OpenType math fonts in new T X engines E \fontdimen parameters σ and σ applicable in ei- might look like the following: 20 21 ther display style or text style. \newfam\symbols As a result, regardless of the contents, we will al- \textfont\symbols="CambriaMath" ways get 10 pt delimiters in text style and 24 pt delim- \scriptfont\symbols="CambriaMath:+ssty0" iters in display style, even if 18 pt delimiters would be scaled
So far, we have discussed only one aspect of the in- In OpenType math, these concepts have been ex- formation contained in the OpenType MATH table, fo- tended, so it would be possible to have multiple sizes cusing on the global parameters which correspond to of display style operators as well as extensible versions TEX’s \fontdimen parameters or to built-in rules of of operators, if desired. TEX’s math typesetting algorithms. While LuaTEX has already implemented most of the Besides those global parameters, there are other new features of OpenType math, it has not yet ad- data structures in the OpenType MATH table, which are dressed additional sizes of big operators, and it is not also important to consider, as we will discuss in the fol- clear how that would be done. lowing sections. Most likely, this would require some changes to the semantics of math markup at the user level, so that op- Instructions for vertical and horizontal erators would be defined to apply to a scope of a sub- variants and constructions formula, which could then be measured to determine the required size of operators. The concepts of vertical and horizontal variants and In addition, such a change might also require constructions in OpenType math are obviously very adding new parameters to decide when an opera- similar to TEX’s concepts of charlists and extensible tor is big enough, similar to the role of the parame- recipes. However, there are some subtle differences ters \delimiterfactor and \delimitershortfall regarding when and how these concepts are applied in the case of big delimiters. in the math typesetting algorithms. In TEX, charlists and extensible recipes are only used Horizontal variants and constructions in certain situations when typesetting elements such Wide accents. When typesetting wide math accents as big operators, big delimiters, big radicals or wide TEX uses charlists to switch to the next-larger horizon- accents. In OpenType math fonts, these concepts have tal variants, but it doesn’t support extensible recipes been extended and generalized, allowing them to be for horizontal constructions. used also for other stretchable elements such as long As a result, math accents in traditional TEX fonts arrows or over- and underbraces. cannot grow beyond a certain maximum size, and stretchable horizontal elements of arbitrary size have Vertical variants and constructions to be implemented using other mechanisms, such as Big delimiters. When typesetting big delimiters or alignments at the macro level. radicals TEX uses charlists to switch to the next-larger In OpenType math, these concepts have been ex- vertical variants, optionally followed by extensible tended, making it possible to introduce extensible ver- recipes for vertical constructions. In OpenType math, sions of wide math accents (or similar elements), if these concepts apply in the same way. desired. In addition, new mechanisms for bottom ac- It is customary to provide at least four fixed-size cents have also been added, complementing the exist- variants, using a progression of sizes such as 12 pt, ing mechanisms for top accents. 18 pt, 24 pt, 30 pt, before switching to an extensible version, but there is no requirement for that other than Over- and underbraces. When typesetting some compatibility and user expectations. (At the macro stretchable elements such as over- and underbraces, level these sizes can be accessed by using \big (12 pt), TEX uses an alignment construction at the macro level \Big (18 pt), \bigg (24 pt), \Bigg (30 pt).) to get an extensible brace of the required size, which Font designers are free to provide any number of ad- is then typeset as a math operator with upper or lower ditional or intermediate sizes, but in TEX they used to limits attached. be limited by constraints such as 256 glyphs per 8-bit While it would be possible to define extensible over- font table and no more than 16 different heights and and underbraces in OpenType math fonts as extensible depths in TFM files. In OpenType math fonts, they are versions of math accents, the semantics of math ac- no longer subject to such restrictions, and in the ex- cents aren’t well suited to handle upper or lower limits ample of Cambria Math big delimiters are indeed pro- attached to those elements. vided in seven sizes. In LuaTEX, new primitives \Uoverdelimiter and \Uunderdelimiter have been added as a new con- Big operators. When typesetting big operators TEX cept to represent stretchable horizontal elements uses the charlist mechanism to switch from text style which may have upper or lower limits attached. The to display style operators, but only once. There is no placement of these limits is handled similar to limits support for multiple sizes of display operators, nor are on big operators in terms of so-called ‘stretch stacks’ there extensible versions. as discussed earlier in section . 20 MAPS 38 Ulrik Vieth
Long arrows. In TEX math fonts, long horizontal ar- slightly more complicated, as it also requires some ad- rows are constructed at the macro level by overlapping ditional information which pieces are of fixed size and the glyphs of short arrows and suitable extension mod- which are extensible, such as ules (such as − or =). Similarly, arrows with hooks or integral : integralbt:0 uni23AE:1 integraltp:0 tails are constructed by overlapping the glyphs of reg- or ular arrows and suitable glyphs for the hooks or tails. In OpenType math fonts, all such constructions can arrowboth : be defined at the font level in terms of horizontal con- arrowleft.left:0 uni23AF:1 arrowright.right:0 structions rather than relying on the macro level. How- It is interesting to note that some of the building ever, in most cases such constructions will also contain blocks (such as uni23AE or uni23AF) have Unicode an extensible part, making the resulting long arrows slots by themselves, while others have to placed in the strechable as well. private use area, using private glyph names such as In LuaTEX, stretchable long arrows can also be de- glyph.left, glyph.mid, or glyph.right. fined using the new primitives \Uoverdelimiter as Moreover, vertical or horizontal constructions may discussed in the case of over- and underbraces. The also contain multiple extensible parts, such as in the placement of limits on such elements more or less cor- example of over- and underbraces, where the left, mid- responds to using macros such as \stackrel to stack dle, and right parts are of fixed size while the extensi- text on top of a relation symbol. ble part appears twice on either side.
Encoding of variants and constructions Additional glyph metric information In traditional TEX math fonts, glyphs are addressed by Besides the global parameters and the instructions a slot number in a font-specific output encoding. Each for vertical and horizontal variants and constructions, variant glyph in a charlist and each building block in there is yet another kind of information stored in the an extensible recipe needs to have a slot of its own in OpenType MATH table, containing additions to the font the font table. However, only the entry points to the metrics of individual glyphs. charlists need to be encoded at the macro level and In traditional TEX math fonts, the file format of TFM these entry points in a font-specific input encoding do fonts only provides a limited number of fields to store not even have to coincide with the slot numbers in the font metric information. As a workaround, certain output encoding. fields which are needed in math mode only are stored In OpenType math fonts, the situation is somewhat in rather non-intuitive way by overloading fields for different. The underlying input encoding is assumed to other purposes [13]. consist of Unicode characters. However these Unicode For example, the nominal width of a glyph is used to codes are internally mapped to font programs using store the subscript position, while the italic correction glyph names, which can be either symbolic (such as is used to indicate the horizontal offset between the summation or integral) or purely technical (such as subscript and superscript position. uni2345 or glyph3456). As a result, the nominal width doesn’t represent the With few exceptions, most of the variant glyphs and actual width of the glyph and the accent position may building blocks cannot be allocated in standard Uni- turn out incorrect. As a secondary correction, fake code slots, so these glyphs have to be mapped to the kern pairs with a so-called skewchar are used to store private use area with font-specific glyph names. In an offset to the accent position. Cambria Math, variant glyphs use suffix names (such In OpenType math fonts, all such non-intuitive ways as glyph.vsize
Summary and conclusions http://fontforge.sourceforge.net/math. html In this paper, we have tried to help improve the un- [ 5 ] Apostolos Syropoulos: Asana Math. derstanding of the internals of OpenType math fonts. www.ctan.org/tex-archive/fonts/ We have done this in order to contribute to the much- Asana-Math/ needed development of math support for Latin Modern [ 6 ] Bogusław Jackowski: Appendix G Illuminated. and T X Gyre fonts. E Proceedings of the 16th EuroT X Conference In the previous sections, we have discussed the pa- E 2006, Debrecen, Hungary. rameters of the OpenType MATH table in great detail, http://www.gust.org.pl/projects/ illustrating the similarities and differences between e-foundry/math-support/tb87jackowski.pdf traditional T X math fonts and OpenType math fonts. E [ 7 ] Ulrik Vieth: Understanding the æsthetics of However, we have covered other aspects of OpenType math typesetting. Biuletyn GUST, 5–12, 2008. math fonts only superficially, as it is impossible to cover Proceedings of the 16th BachoT X Conference everything in one paper. E 2008, Bachotek, Poland. For a more extensive overview of the features and http://www.gust.org.pl/projects/ functionality of OpenType math fonts as well as a dis- e-foundry/math-support/vieth2008.pdf cussion of the resulting challenges to font developers, [ 8 ] Microsoft Typography: OpenType specification. readers are also referred to [12]. Version 1.5, May 2008. In view of the conference motto, it is interesting http://www.microsoft.com/typography/ to note that recent versions of LuaT X have started to E otspec/ provide a full-featured implementation of OpenType [ 9 ] Yannis Haralambous: Fonts and Encodings. math support in LuaT X and Context [14, 15], which E O’Reilly Media, 2007. ISBN 0-596-10242-9 differs significantly from the implementation of Open- http://oreilly.com/catalog/9780596102425/ Type math support in X T X [10]. In this paper, we E E [ 10 ] Will Robertson: The unicode-math package. have pointed out some of these differences, but fur- Version 0.3b, August 2008. ther discussions of this topic are beyond of the scope http://github.com/wspr/unicode-math/tree/ of this paper. master [ 11 ] Taco Hoekwater: LuaTEX Reference Manual. Acknowledgments Version 0.37, 31 March 2009. http://www.luatex.org/svn/trunk/manual/ The author once again wishes to thank Bogusław Jack- luatexref-t.pdf owski for permission to reproduce and adapt the fig- [ 12 ] Ulrik Vieth: Do we need a ‘Cork’ math font ures from his paper Appendix G Illuminated [6]. In encoding? TUGboat, 29(3), 426–434, 2008. addition, the author also wishes to acknowledge feed- Proceedings of the TUG 2008 Annual Meeting, back and suggestions from Taco Hoekwater and Hans Cork, Ireland. Reprinted in this MAPS, Hagen regarding the state of OpenType math support 3–11. in LuaT X. E https://www.tug.org/members/TUGboat/ tb29-3/tb93vieth.pdf References [ 13 ] Ulrik Vieth: Math Typesetting: The Good, The Bad, The Ugly. MAPS, 26, 207–216, 2001. [ 1 ] Murray Sargent III: Math in Office Blog. Proceedings of the 12th EuroT X Conference http://blogs.msdn.com/murrays/default. E 2001, Kerkrade, Netherlands. aspx http://www.ntg.nl/maps/26/27.pdf [ 2 ] Murray Sargent III: High-quality editing and [ 14 ] Taco Hoekwater: Math extensions in LuaT X. display of mathematical text in Office 2007. E Published elsewhere in this MAPS issue. http://blogs.msdn.com/murrays/archive/ [ 15 ] Hans Hagen: Unicode math in Context Mk IV. 2006/09/13/752206.aspx Published elsewhere in this MAPS issue. [ 3 ] John Hudson, Ross Mills: Mathematical Type- setting: Mathematical and scientific typesetting solutions from Microsoft. Promotional Booklet, Ulrik Vieth Microsoft, 2006. Vaihinger Straße 69 http://www.tiro.com/projects/ 70567 Stuttgart [ 4 ] George Williams: FontForge. Math typesetting Germany information. ulrik dot vieth (at) arcor dot de