Nice looking square roots with LuaTeX?

Question

Very often there are questions like: "Why are the square roots so ugly? Why is there a difference in height?"

$\sqrt{x} \, \sqrt{y} \, \sqrt{X}$

Then the answers all sound like: "Use \phantom, \mathstrut or \smash[b]." Not really satisfying as all the suggestions are adjustments by hand.

Now we have had LuaTeX since more than 10 years. Why isn't there any solution with lua callbacks?

A good \sqrt sign should...

• have a minimum height (example: by inserting \phantom{0}).
• not grow into depth when there are only descenders of letters or subscripts (\smash[b]). Of couse the bottom should not be smashed when there is a big fraction or another big constuction.
• ignore \underbraces.

If I had more knowledge about LuaTeX I would try to create an MWE. In the LuaTeX manual we find radical nodes and so on. Maybe someome has an idea.

EDIT (as an answer to projetmbc's comment):

The first of my suggestions would lead to the same square root height for \sqrt x, \sqrt X and \sqrt 2.

The second one would lead to equal heigt for $\sqrt g$, \sqrt x and \sqrt{x_1}.

The third one would prevent the square root sign from growing for an underbrace: $\sqrt{\underbrace{something}_{something}}$

I cannot imagine a solution without LuaTeX.

Answer 1

I'm not convinced that it's an improvement to force (most) \sqrt expressions to have surds with the same height and depth. But, as always, if you really understand what you're doing, you'll probably be ok using the one-size-fits-almost-all approach.

The following, LuaLaTeX-based solution features (a) a utility LaTeX macro called \lsqrt and (b) two Lua functions: lsqrt does almost all of the real work, and checkroot (which is called by lsqrt) contains a list of conditions; if one (or more) of these conditions is met, \lsqrt does nothing special, i.e., it behaves like the basic \sqrt macro.

The default height of the horizontal bar of the square root symbol generated by \lsqrt is set so that something such as k^2 will just fit. Note that this is a bit taller than what the OP required; however, as the difference in heights is just 0.6pt (at least for Computer Modern with 10pt as the main document font size), I think it's justifiable to go for this ever so slightly greater height. Of course, if you prefer a height that accomodates k but not k^2, just change both instances of \\vphantom{k^2} to \\vphantom{k}. (Aside: the \\ double-backslash symbol occurs because for Lua, \ has a special meaning too; however, the Lua-special and TeX-special meanings of \ are completely different. To output a single backslash character in Lua, it's necessary to input \\.)

% !TEX TS-program = lualatex
\documentclass{article} 
\usepackage{amsmath}
\usepackage{luacode} % for 'luacode*' env. and '\luastringN' macro

%% Lua-side code: 2 Lua functions: checkoptargroot, lsqrt
\begin{luacode*}

function checkroot ( s )
  if       s:find ( "\\[dt]?frac" ) -- \frac term
        or s:find ( "\\int" )       -- integral symbol
        or s:find ( "\\sum" )       -- summation symbol
        or s:find ( "%^%s?%b{}" )   -- exponent term encased in matching curly braces
     then
       return true
  else
       return false
  end
end

function lsqrt ( b , s ) -- this function does most of the work
  if b=="" then -- no optional argument was provided to "\lsqrt"
    if checkroot ( s ) then
      tex.sprint ( "\\sqrt{".. s .."}" ) -- nothing special to do
    else
      tex.sprint ( "\\sqrt{\\smash{".. s ..
                   "}\\vphantom{k^2}}\\vphantom{"..s.."}" )
    end
  else -- need to handle non-empty optional argument
    if checkroot ( s ) then
      tex.sprint ( "\\sqrt["..b.."]{".. s .."}" ) -- nothing special to do
    else
      tex.sprint ( "\\sqrt["..b.."]{\\smash{".. s ..
                   "}\\vphantom{k^2}}\\vphantom{"..s.."}" )
    end    
  end
end

\end{luacode*}

%% LaTeX-side code: utility macro to access the 'lsqrt' function
\newcommand\lsqrt[2][]{\directlua{lsqrt(\luastringN{#1},\luastringN{#2})}}

\begin{document}

$\sqrt{y}$  $\sqrt{x}$  $\sqrt{t}$  $\sqrt{k}$  $\sqrt{b_q^2}$ vs.\
$\lsqrt{y}$ $\lsqrt{x}$ $\lsqrt{t}$ $\lsqrt{k}$ $\lsqrt{b_q^2}$

\medskip
$\sqrt[3]{y}$  $\sqrt[4]{x}$  $\sqrt[5]{t}$  $\sqrt[6]{k}$  $\sqrt[7]{b_q^2}$ vs.\
$\lsqrt[3]{y}$ $\lsqrt[4]{x}$ $\lsqrt[5]{t}$ $\lsqrt[6]{k}$ $\lsqrt[7]{b_q^2}$

\medskip
$\sqrt{\frac{1}{2}}$     $\lsqrt{\frac{1}{2}}$,
$\sqrt[3]{\dfrac{1}{5}}$ $\lsqrt[3]{\dfrac{1}{5}}$,
$\sqrt{a_q^{-x^2/2}}$    $\lsqrt{a_q^{-x^2/2}}$

\end{document} 

Answer 2

I think that you will not find one version that will fit you perfectly for all cases, and, as mentioned in a comment, I think this is something for the macro package. Since you have gotten a satisfying LaTeX answer, I provide a ConTeXt one, hopefully to help somebody.

In ConTeXt you have the possibilities to do setups and define your own instances of mathradicals. By default, the \sqrt has a strut that has height but no depth, but that can be changed. Also, as we will see, it is possible to set the height and depth of content. Let us show some examples

\usemodule[visual] % for \fakewords only

\starttext

\fakewords{25}{30}

\startformula
  \sqrt{y}+\sqrt{x} + \sqrt{b_q^2} + \sqrt{\frac{1}{5}} + \root[n=3]{x+y}
  \breakhere
  \sqrt[depth=1sp]{y} + \sqrt[strut=no]{x} + \sqrt[depth=1sp]{b_q^2} + \sqrt[depth=6pt]{\frac{1}{5}} + \sqrt[depth=1sp][3]{x+y}
\stopformula

\fakewords{25}{30}

\definemathradical
  [mysqrt]
  [depth=1sp]

\startformula
  \mysqrt{y} + \mysqrt{x} + \mysqrt{b_q^2} + \mysqrt{\frac{1}{5}} + \mysqrt[depth=6pt]{\frac{1}{5}} + \mysqrt[3]{x+y}
\stopformula

\fakewords{25}{30}

\startformula
  \sqrt{\int_a^c 2x\dd x} 
  = 
  \sqrt{\underbrace{c^2-a^2}_{=b^2}}
  = 
  \sqrt[depth=4pt]{\underbrace{c^2-a^2}_{=b^2}} 
  =
  \mysqrt{\underbrace{c^2-a^2}_{=b^2}} 
\stopformula

\fakewords{25}{30}

\stoptext

Compiling this file with context gives the result:

Some comments:

• The first line in the first displayed formula shows the default \sqrt{}
• The second line in the first displayed formula shows some variants, as we can get them by inserting options to the \sqrt macro.
• For the third displayed formula, we have introduced our own instance of a math radical, \mysqrt, with the help of \definemathradical. To set the depth explicitly we need to set it to some non-zero value (this is by old construction, and the value zero will give just the original version back). We set it to something very small, 1sp. Note in the fraction 1/5 that it looks better with a slightly modified depth (that we can add as an optional argument to \mysqrt).
• In the last displayed formula we show also some formula with an underbrace. Here you see three versions, and you can choose yourself which one you prefer.

A curiosity: After discussions with Hans, we agreed that using the name sqrt in \sqrt[3]{x} is a bit strange, so from now on the base math radical in ConTeXt will be \root instead of \sqrt (but with \sqrt as a copy of it, not to break compatibility). There will also be support for input like \root[n=3]{x} instead of \root[3]{x}.