1.Introduction

In this chapter we describe the algorithms used by TeXmacs in order to typeset mathematical formulas. This is a difficult subject, because esthetics and effectiveness do not always go hand in hand. Until now, TeX is widely accepted for having achieved an optimal compromise in this respect. Nevertheless, we thought that several improvements could still be made, which have now been implemented in TeXmacs. We will shortly describe the motivations behind them.

In order to obtain esthetic formulas, what criteria should we use? It is often stressed that good typesetting allows the reader to concentrate on what he reads, without being distracted by ugly typesetting details. Such distracting details arise when distinct, though similar parts of text are typesetted in a non uniform way:

Different base lines

The eye expects text of a similar nature to be typesetted with respect to a same base line. For instance, in x + y + z, the bottoms of the x and z should be at the same height as the bottom of the u-part in the y. This should again be the case in 2^x + 2^y + 2^z.

Unequal spacing

Different components of text with approximately the same function should be separated by equal amounts of space. For instance, in a² + f², the typesetter should notice the hangover of the f. This should again be the case in e^a + e^f + e^x. Similarly, the distance between the baselines of the a and the i in a_i should not be disproportially large with respect to the height of an x.

Additional difficulties may arise when considering automatically generated formulas, in which case line breaking has to be dealt with in a satisfactory way.

Unfortunately, the different esthetic criteria may enter into conflict with each other. For instance, consider the formula x_p + x

2

p

. On the one hand, the baselines of the scripts should be the same, but the other hand, the first subscript should not be “disproportionally low” with respect to the x. Unfortunately, this dilemma can not been solved in a completely satisfactory way without the help of a human for the simple reason that the computer has no way to know whether the x_p and x

i

p

are “related”. Indeed, if the x_p and x

i

p

are close (like in x_p + x

i

p

), then it is natural to opt for a common base line. However, if they are further away from each other (like in x_p + ∑

∞

i = 0

c_ix

i

p

), then we might want to opt for different base lines and locally optimize the rendering of the first x_p.

Consequently, TeXmacs should offer a reasonable compromise for the most frequent cases, while offering methods for the user to make finer adjustments in the remaining ones. Currently, we just provided the Insérer→Spécial→Move and Insérer→Spécial→Resize constructs to move and resize boxes in order to perform such adjustments. For instance, if the brackets around the two sums

ϕ(∑_ia_ixⁱ) = ψ(∑_jb_jy^j)

have different sizes, then one may resize the bottom of the subscript j of the second sum to 0fn. Alternatively, one may resize the bottoms of both the i and j subscripts to (say) -0.3fn.

Notice that one should adjust by preference in a structural and not visual way. For instance, one should prefer -0.3fn to -2mm in the above example, because the second option disallows you to switch to another font size for your document. Similarly, you should try not change the semantics of the formula. For instance, in the above example, you might have added a “dummy subscript” to the i subscript of the sum. However, this would alter the meaning of the formula (whence make it non suitable as input to a computer algebra system) In the future, we plan to provide additional constructs in order to facilitate structural adjusting. For instance, in the case of a formula like

1 + x₁ + x

2

1

+ ⋯ + x₂ + x₁x₂ + x

2

1

x₂ + ⋯x

2

2

+ x₁x

2

2

+ x

2

1

x

2

2

+ ⋯,

one might think of a construct to enclose the entire formula into an area, where all scripts are forced to be double (using dummy superscripts whereever necessary).

2.The font parameters

Several font parameters are crucial for the correct positioning of the different components. The following are often needed:

quad

The main font reference space 1fn, which can be taken as the distance between successive lines of text.

y1 and y2

The bottom and top level for the font (we have y2-y1=quad).

sep

The reference minimal space between distinct components, like the minimal distance between a subscript and a superscript. In fact, sep=quad/10.

wline

The width of several types of lines, like the fraction and square root bars, wide accents, etc.

yfrac

The height of the fraction bar, which is needed for the positioning of fractions and big delimiters. Usually, yfrac is almost equal to yx/2 below.

The following parameters are mainly needed in order to deal with scripts:

yx

The height of the x character, which is needed for the positioning of scripts. All the remaining parameters are actually computed as a function of yx.

ysub lo base

Logical base line for subscripts.

ysub hi lim

Subscripts may never physically exceed this top height.

ysup lo base

Logical base line for superscripts.

ysup lo lim

Superscripts may never physically exceed this bottom height.

ysup hi lim

Suggestion for a physical top line for superscripts.

yshift

Possible shift of the base lines when we are inside fractions or scripts.

The individual strings in a font also have several important positioning properties. First of all, they always admit left and right slopes. Furthermore, they admit left and right italic corrections, which are needed for the positioning of scripts or when passing from text in upright to text in italics (or vice versa).

3.Some major mathematical constructs

3.1.Fractions

The following heuristics are used:

• The horizontal middles of the numerator and the denominator are taken to be the same.
• The vertical spaces between the numerator resp. denominator and the fraction bar is at least sep.
• The depth (resp. height) of the numerator (resp. denominator) is descended (resp. increased) to y1 (resp. y2) if necessary. This forces the base lines of not too large numerators resp. denominators to be the same in presence of multiple fractions.
• The fraction bar has a overhang of sep/2 to both sides and the logical limits of the fraction are another sep/2 further. The logical left limit is zero.

The italic corrections are not taken into account during the positioning algorithms, because this may create the impression that the numerator and denominator are not correctly centered with respect to each other. Nevertheless, the italic corrections are taken into account in order to compute the logical bounding box of the fraction (whose has italic slopes vanish at both sides).

3.2.Roots

The following heuristics are used:

• The vertical space between the main argument and the upper bar is at least sep.
• The root itself is typesetted like a large delimiter. The positioning of a potential script works only is very dependent on the usage of TeX fonts.
• The upper bar has a overhang of sep/2 at the right and the logical right limit of the root is situated another sep/2 further to the right.

We take the logical right border plus the italic correction of the main argument in order to determine the right hand limit of the upper bar. The left italic correction is not needed.

3.3.Negations

The following heuristics are used:

• The negation bar passes through the logical center of the argument.
• The italic corrections of the argument are only taken into account during the computation of the logical limits of the negation box (which has zero left and right slopes).

3.4.Wide boxes

The following heuristics are used:

• We use TeX fonts for small accents and an ad hoc algorithm for the wider ones.
• The distance between the main argument and the accent is at least sep (or a distance which depends on the TeX font for small accents).
• The accent is positioned horizintally according to the right slope of the main argument.
• The slopes for the accented box are inherited from those of the main argument and the italic corrections are adjusted accordingly.
• All script height parameters of the accented box are inherited from the main argument. The only exception is ysup_hi_lim, which may be increased by the height of the accent, or determined in the generic way, whichever leads to the least value. It is indeed better to keep superscripts positioned reasonably low, whenever possible.

4.Subscripts and superscripts

The positioning of subscripts and superscripts is a complicated affair, due to the conflict between locally and globally optimal esthetics mentioned above. The base line for a subscript is determined as follows:

1. Always pretend that the subscript has height at least y2-yshift in the script font (actually we should use the height of an M instead).
2. Try to position the script at the base line given by the main argument.
3. If the top limit (given by the main argument) is physically exceeded by the subscript, then the base line is moved further down accordingly.

The base line for a superscript is determined as follows:

1. Try to physically position the superscript beneath the suggested top line. Usually, this will place the superscript to far down.
2. Move the superscript up to the logical base line if necessary. This will usually occur: most of the time, the logical base line is the just the height of an x-script below the suggested top line.
3. If the superscript physically descends below the physical under limit given by the main box, then we move the superscript further upwards.

If both a subscript and a superscript were present, then we still have to adjust the base lines: if the top of the subscript and the bottom of the superscript are not physically separated by sep, then we both move the subscript and the superscript by the same amount away from each other. Because of step 1 in the positioning of the subscript, the base lines of double scripts will usually be the same in formulas with several of them.

The right slope and italic correction of a script box may be non trivial. In order to compute them, we first determine the script (or main argument), whose right limit (taking into account its italic correction) is furthest to the right (this may be the main box, in the case of a big integral with a tiny subscript). Then the right slope of the main box is inherited by the right slope of this script (or main argument). As to the italic correction, it is precisely the difference between the right offset of the script plus its italic correction minus the logical right coordinate of the entire box. The italic correction should be at least zero though. The left slope and italic correction are computed in a similar way.

5.Big delimiters

The automatic positioning and computation of sizes of big delimiters is again complicated because of potential conflicts between locally and globally optimal esthetics.

First of all, TeX fonts come only with a discrete set of possible sizes for large delimiters. This is an advantage from the point of view that it favorites delimiters around slightly different expressions to have the same baselines. However, it has the disadvantage that delimiters are easily made “one size to large”. For this reason, we actually diminish the height and the depth of the delimited expression by the small amount sep, before computing the sizes of the delimiters.

Secondly, it is best when the vertical middles of big delimiters occur at the height of fraction bars. However, in a formula like

f (

1

1 + 1 1 + 1 1 + 1 x

),

it may be worth it to descend the delimiters a bit. On the other hand, slight vertical shifts in the middles of the delimiters potentially have a bad effect on base lines, like in

f(∑

b

i = 1

X_i) + g(∑

a

j = 1

Y_j).

In TeXmacs, we use the following compromise: we start with the middle of the delimited expression as a first approximation to the middle of the delimiters. The real middle is obtained by shifting this middle towards the height of fraction bars by an amount which cannot exceed sep.

From a horizontal point of view, we finally have to notice that we adapted the metrics of the big delimiters in a way that potential scripts are positioned in a better way. For instance, according to the TeX tfm file, in a formula like

(A + (∑

10

i = 1

B_i)²),

the square rather seems to be a left superscript of the second closing bracket than a right superscript of the first one. This is particularly annoying in the case of automatically generated formulas, where this situation occurs quite often.