Are the widths of 𝐿, 𝑌, and 𝑍 in TeX Gyre Termes Math really equal?

Question

It was a surprise to me when feeding lualatex with

\documentclass{standalone}
\usepackage{unicode-math}
\setmathfont{TeX Gyre Termes Math}% or \setmathfont{texgyretermes-math.otf}
\begin{document}
Width of $L$: \setbox0\hbox{$L$}\the\wd0. \text{Width of}\ $Y$:\ \setbox0\hbox{$Y$}\the\wd0. \text{Width of}\ $Z$:\ \setbox0\hbox{$Z$}\the\wd0.
\end{document}

yielded

Are the widths of the three symbols REALLY equal? I find it hard to believe because these numbers look way too nice, occupying only 3 decimal digits each. I might imagine that really different lengths (given in sp or some other unit) were rounded up to the same number while converting to pt. Is it really so? If so, is it possible to distinguish between the widths of math-mode 𝐿, 𝑌, and 𝑍 from inside LuaLaTeX? If so, how? (After all, running xelatex on the same input provides us with three different numbers. Also, running pdflatex (with \usepacakge{amsmath} instead of \usepackage{unicode-math}+\setmathfont{…}) yields three different numbers.)

Answer 1

character lengths are not measured properties that depend on some accuracy of measurement, they are properties set by the font designer.

Modifying your example to

\documentclass{article}
\usepackage{unicode-math}
\setmathfont{TeX Gyre Termes Math}% or \setmathfont{texgyretermes-math.otf}
\begin{document}
Width of $L$: \setbox0\hbox{$L$}\the\wd0. \text{Width of}\ $Y$:\ \setbox0\hbox{$Y$}\the\wd0. \text{Width of}\ $Z$:\ \setbox0\hbox{$Z$}\the\wd0.


L 
\the\fontcharwd\textfont0 119871,
\the\fontcharic\textfont0 119871,

Y 
\the\fontcharwd\textfont0 119884,
\the\fontcharic\textfont0 119884,

Z 
\the\fontcharwd\textfont0 119885,
\the\fontcharic\textfont0 119885,

\end{document}

You get

with luatex, and

With xetex.

So the three characters have the same width but different italic correction.

luatex, in contrast to xetex, does not add italic correction at the end of the math which explains why you get different box widths with xetex.

Answer 2

Yes and no. The three characters have the same advance width 556. If we only look at that, the orange boxes below, we see (here the characters are squeezed together to make it more clear that the orange boxes have the same width):

We notice that the L and Z stick out to the left while Y and Z stick out to the right (in fact L also sticks out to the right, but very very little, see below). If we look at the font data, we see the following:

  [119871]={ -- L
   ["boundingbox"]={ -8, 0, 559, 653 },
   ["index"]=454,
   ["math"]={
    ["accent"]=267,
    ["italic"]=31,
   },
   ["unicode"]=119871,
   ["width"]=556,
  [119884]={ -- Y
   ["boundingbox"]={ 78, 0, 633, 653 },
   ["index"]=467,
   ["math"]={
    ["accent"]=362,
    ["italic"]=105,
   },
   ["unicode"]=119884,
   ["width"]=556,
  },
  [119885]={ -- Z
   ["boundingbox"]={ -6, 0, 606, 653 },
   ["index"]=468,
   ["math"]={
    ["accent"]=362,
    ["italic"]=78,
   },

• The (advance) widths are indeed equal.
• The bounding boxes shown are {xbottomleft, ybottomleft, xtopright, ytopright}. The negative values in the first slot for L and Z indicate that they stick out to the left. The values in the third slot indicates that they do in fact all stick out to the right (the L very little).
• We notice that the characters have different amounts of italic correction.

Here I would consider the answer to be complete, but since the other answers wrote about luatex and xetex doing things differently, let me add something about the italic correction in math. In luatex you have the possibility to add italic correction as well (by setting \mathitalicsmode to 1 or 2, I think). I think that the reason it is not added by default is that if one reads the vague specifications it seems that it should not always be added. The situation today is that there are some math fonts, like cambria, garamond-math and lucida (and probably some more) that somehow follow the spec and do not rely on italic corrections (but on kerns), and then there are the rest (like Termes) that relies on the old TeX way with italic correction.

In the last few weeks I have discussed these issues with Hans Hagen a lot, in the context (pun intended) of the newer luametatex engine. Our aim has been to have a spacing model that works both with the old TeX-like fonts and with the newer ones. If we use italic corrections, a formula can look like this (italic corrections marked in blue):

As can be seen, the italic correction is always added, and in case of a subscript, removed. Since italic correction in opentype math is not really italic correction, we thought that maybe we could get rid of it alltogether, and replace it by corner kerns. Then it can look like this:

The output is (hopefully) the same. Notice that the width of the orange box with the Z is changed, and that there are two corner kerns added (one bottom right for the subscript, and one top left for the prescript).

The reason that the orange box is enlarged to the left in order to have a method that also works for difficult characters, like lower case italic f that in some fonts sticks out a lot to the left and right from the orange box. Compare the following to examples, and you will understand why.

Answer 3

You're not measuring the width of the characters, but of some math formulas. And math formulas are treated differently by LuaLaTeX and XeLaTeX, as the former doesn't add a trailing italic correction.

If you compile the following with XeLaTeX, you get that, indeed, the characters have the same width, namely 5.56pt.

\documentclass{article}
\usepackage{unicode-math}

\setmathfont{TeX Gyre Termes Math}

\frenchspacing

\begin{document}

$x$

\the\fontcharwd\textfont0 `𝐿

\the\fontcharwd\textfont0 `𝑌

\the\fontcharwd\textfont0 `𝑍

Width of $L$: \sbox0{$L$}\the\wd0.

Width of $Y$: \sbox0{$Y$}\the\wd0.

Width of $Z$: \sbox0{$Z$}\the\wd0.

\end{document}