% language=us

\startcomponent about-speed

\environment about-environment

\startchapter[title=Math Styles]

\startsection[title=Introduction]

Because \CONTEXT\ is often considered somewhat less math savvy than for instance
\LATEX\ we have more freedom to experiment with new insights and move forward. Of
course \CONTEXT\ always could deal with math, and even provides rather advanced
support when it comes to combining fonts, which at some point was needed for a
magazine that used two completely different sets of fonts in one issue. Also,
many of the mechanisms had ways to influence the rendering, but often by means of
constants and flags.

Already in an early stage of \LUATEX\ we went \UNICODE\ and after that the low
level code has been cleaned up stepwise. In fact, we probably have less code now
than before because we need less hacks. Well, this might not be that true, if we
consider that we also introduced code at the \LUA\ end which wasn't there before,
but which makes makes support easier.

Because we don't need to support all kind of third party math extensions that
themselves might depend on overloading low level implementations, we can
rigourously replace mechanisms. In the process we also make things easier to
configure, easier to define and we promote some previously low level tuning
options at the user level.

Or course, by introducing new features and more options, there is a price to pay
in terms of speed, but in practice users will seldom use the more complex
constructs thousands of times in one document. Elsewhere arrows and alike were
discussed, here I will spend some words on math styles and will use fences and
fractions as an example as these mechanisms were used to experiment.

\stopsection

\startsection[title=Math styles]

In \TEX\ a formula can used three different sizes of a font: text, script and
scriptscript. In addition a formula can be typeset using rules for display math
or rules for inline math. This means that we have the following so called math
styles:

\starttabulate[||||]
% \FL
\NC \bf keyword         \NC \bf meaning               \NC \bf command                \NC \NR
% \FL
\NC \type{display}      \NC used for display math     \NC \type {\displaystyle}      \NC \NR
\NC \type{text}         \NC used for inline math      \NC \type {\textstyle}         \NC \NR
\NC \type{script}       \NC smaller than text style   \NC \type {\scriptstyle}       \NC \NR
\NC \type{scriptscript} \NC smaller than script style \NC \type {\scriptscriptstyle} \NC \NR
% \LL
\stoptabulate

Each of these commands will force a style but in practice you seldom need to do
that because \TEX\ does it automatically. In addition there is are cramped styles
with corresponding commands.

\starttabulate
  \NC \ruledhbox{$\displaystyle             x^2 + \sqrt{x^2+2x} + \sqrt{\displaystyle             x^2+2x}$} \NC \type{\displaystyle            } \NC \NR
  \NC \ruledhbox{$\crampeddisplaystyle      x^2 + \sqrt{x^2+2x} + \sqrt{\crampeddisplaystyle      x^2+2x}$} \NC \type{\crampeddisplaystyle     } \NC \NR
  \NC \ruledhbox{$\textstyle                x^2 + \sqrt{x^2+2x} + \sqrt{\textstyle                x^2+2x}$} \NC \type{\textstyle               } \NC \NR
  \NC \ruledhbox{$\crampedtextstyle         x^2 + \sqrt{x^2+2x} + \sqrt{\crampedtextstyle         x^2+2x}$} \NC \type{\crampedtextstyle        } \NC \NR
  \NC \ruledhbox{$\scriptstyle              x^2 + \sqrt{x^2+2x} + \sqrt{\scriptstyle              x^2+2x}$} \NC \type{\scriptstyle             } \NC \NR
  \NC \ruledhbox{$\crampedscriptstyle       x^2 + \sqrt{x^2+2x} + \sqrt{\crampedscriptstyle       x^2+2x}$} \NC \type{\crampedscriptstyle      } \NC \NR
  \NC \ruledhbox{$\scriptscriptstyle        x^2 + \sqrt{x^2+2x} + \sqrt{\scriptscriptstyle        x^2+2x}$} \NC \type{\scriptscriptstyle       } \NC \NR
  \NC \ruledhbox{$\crampedscriptscriptstyle x^2 + \sqrt{x^2+2x} + \sqrt{\crampedscriptscriptstyle x^2+2x}$} \NC \type{\crampedscriptscriptstyle} \NC \NR
\stoptabulate

Here we applied the styles as follows:

\startbuffer
$\textstyle x^2 + \sqrt{x^2+2x} + \sqrt{\textstyle x^2+2x}$
\stopbuffer

\typebuffer

The differences are subtle: the superscripts in the square root are positioned a
bit lower than normal: the radical forces them to be cramped.

\startlinecorrection
\scale[width=\hsize]{\maincolor \getbuffer}
\stoplinecorrection

Although the average user will not bother about styles, a math power user might
get excited about the possibility to control the size of fonts being used, of
course wit the danger of creating a visually inconsistent document. And, as in
\CONTEXT\ we try to avoid such low level commands \footnote {Although \unknown\
it's pretty hard to convince users to stay away from \type {\vskip} and friends.}
it will be no surprise that we have ways to set them beforehand.

\startbuffer
\definemathstyle[mystyle][scriptscript]

$ 2x + \startmathstyle [mystyle] 4y^2 \stopmathstyle = 10 $
\stopbuffer

\typebuffer

So, if you want it this ugly, you can get it:

\blank \start \getbuffer \stop \blank

A style can be a combination of keywords. Of course we have \type {display},
\type {text}, \type {script} and \type {scriptscript}. Then there are \type
{uncramped} and \type {cramped} along with their synonyms \type {normal} and
\type {packed}. In some cases you can also use \type {small} and \type {big}
which will promote the size up or down, relative to what we have currently.

A style definition can be combination of such keywords:

\starttyping
\definemathstyle[mystyle][scriptscript,cramped]
\stoptyping

Gradually we will introduce the \type {mathstyle} keyword in math related
setups commands.

In most cases a user will limit the scope of some setting by using braces, like
this:

\startbuffer
$x{\setupmathstyle[script]x}x$
\stopbuffer

This gives {\maincolor \ignorespaces \getbuffer \removeunwantedspaces}: a smaller
symbol between two with text size. Equally valid is this:

\startbuffer
$x\startmathstyle[script]x\stopmathstyle x$
\stopbuffer

\typebuffer

Again we get {\maincolor \ignorespaces \getbuffer \removeunwantedspaces}, but at
the cost of more verbose coding.

The use of \type {{}} (either or not hidden in commands) has a few side effects.
In text mode, when we use this at the start of a paragraph, the paragraph will
start inside the group and when we end the group, specific settings that were
done at that time get lost. So, in practice you will force a paragraph outside
the group using \type {\dontleavehmode}, \type {\strut}, or one of the
indentation commands. \stopitem

In math mode a new math group is created which limits local style settings to
this group. But as such groups also can trigger special kinds of spacing you
sometimes don't want that. One pitfall is then to do this:

\startbuffer
$x\begingroup\setupmathstyle[script]x\endgroup x$
\stopbuffer

\typebuffer

Alas, now we get {\maincolor \ignorespaces \getbuffer \removeunwantedspaces}. A
\type {\begingroup} limits the scope of many things but it will not create a math
group! This kind of subtle issues is the reason why we have pre|-|built solutions
that take care of style switching, grouping, spacing and positioning.

\stopsection

\startsection[title=Fences]

Fences are symbols at the left and right of an expression: braces, brackets,
curly braces, and bars are the most well known. Often they are supposed to adapt
their size to the content that they wrap. Here you see some in action:

\starttabulate[||c||]
\NC \type {$|x|$}                            \NC $|x|$                             \NC okay  \NC \NR
\NC \type {$||x||$}                          \NC $||x||$                           \NC okay  \NC \NR
\NC \type {$a\left | \frac{1}{b}\right | c$} \NC $a\left |  \frac{1}{b}\right | c$ \NC okay  \NC \NR
\NC \type {$a\left ||\frac{1}{b}\right ||c$} \NC $a\left || \frac{1}{b}\right ||c$ \NC wrong \NC \NR
\NC \type {$a\left ‖ \frac{1}{b}\right ‖ c$} \NC $a\left ‖  \frac{1}{b}\right ‖ c$ \NC okay  \NC \NR
\stoptabulate

Often authors like to code their math with minimal structure and if you use
\UNICODE\ characters that is actually quite doable. Just look at the double bar
in the example above: if we input \type {||} we don't get what we want, but with
\type {‖} the result is okay. This is because the \type {\left} and \type
{\right} commands expect one character. But, even then, coding a bit more
verbose sometimes makes sense.

In stock \CONTEXT\ we have a couple of predefined fences:

\starttyping
\definemathfence [parenthesis] [left=0x0028,right=0x0029]
\definemathfence [bracket]     [left=0x005B,right=0x005D]
\definemathfence [braces]      [left=0x007B,right=0x007D]
\definemathfence [bar]         [left=0x007C,right=0x007C]
\definemathfence [doublebar]   [left=0x2016,right=0x2016]
\definemathfence [angle]       [left=0x003C,right=0x003E]
\stoptyping

\startbuffer
test $a \fenced[bar]      {\frac{1}{b}} c$ test
test $a \fenced[doublebar]{\frac{1}{b}} c$ test
test $a \fenced[bracket]  {\frac{1}{b}} c$ test
\stopbuffer

You use these by name:

\typebuffer

and get

\startlines \getbuffer \stoplines

\startbuffer
\definemathfence [nooffence] [left=0x005B]
\stopbuffer

You can stick to only one fence:

\typebuffer \getbuffer

\startbuffer
on $a \fenced[nooffence]{\frac{1}{b}} c$ off
\stopbuffer

Here \CONTEXT\ will take care of the dummy fence that \TEX\ expects instead.

\startlines \getbuffer \stoplines

You can define new fences and clone existing ones. You can also assign some
properties:

\startbuffer
\definemathfence
  [fancybracket]
  [bracket]
  [command=yes,
   color=blue]
\stopbuffer

\typebuffer \getbuffer

\startbuffer
test $a\fancybracket{\frac{1}{b}}c$ test
test \color[red]{$a\fancybracket{\frac{1}{b}}c$} test
\stopbuffer

\typebuffer

The color is only applied to the fence. This makes sense as the formula can
follow the main color but influencing the fences is technically somewhat more
complex.

\getbuffer

Here are some more examples:

\startbuffer
\definemathfence
  [normalbracket]
  [bracket]
  [command=yes,
   color=blue]

\definemathfence
  [scriptbracket]
  [normalbracket]
  [mathstyle=script]

\definemathfence
  [smallbracket]
  [normalbracket]
  [mathstyle=small]
\stopbuffer

\typebuffer \getbuffer

\starttabulate
\NC \type{$a                \frac{1}{b} c$}  \NC $a                \frac{1}{b}  c$ \NC \NR
\TB
\NC \type{$a \normalbracket{\frac{1}{b} c$}} \NC $a \normalbracket{\frac{1}{b}} c$ \NC \NR
\TB
\NC \type{$a \scriptbracket{\frac{1}{b} c$}} \NC $a \scriptbracket{\frac{1}{b}} c$ \NC \NR
\TB
\NC \type{$a \smallbracket {\frac{1}{b} c$}} \NC $a \smallbracket {\frac{1}{b}} c$ \NC \NR
\stoptabulate

As with most commands, the fences inherit from the parents so we can say:

\starttyping
\setupmathfences [color=red]
\stoptyping

and get all our fences colored red. The \type {command} option results in a
command being defined, which saves you some keying.

\stopsection

\startsection[title=Fractions]

In \TEX\ the mechanism to put something on top of something else, separated by a
horizontal rule, is driven by the \type {\over} primitive. That one has a
(compared to other commands) somewhat different specification, in the sense that
one of its arguments sits in front:

\starttyping
$ {{2x}\over{x^1}} $
\stoptyping

Although to some extend this is considered to be more readable, macro packages
often provide a \type {\frac} commands that goes like this:

\starttyping
$ \frac{2x}{x^1} $
\stoptyping

There we have less braces and the arguments come after the command. As with the
fences in the previous section, you can define your own fractions:

\startbuffer
\definemathfraction
  [innerfrac]
  [frac]
  [alternative=inner,
   mathstyle=script,
   color=red]

\definemathfraction
  [outerfrac]
  [frac]
  [alternative=outer,
   mathstyle=script,
   color=blue]
\stopbuffer

\typebuffer \getbuffer

The mathstyle and color are already discussed but the \type {alternative} is
specific for these fractions. It determines if the style is applied to the whole
fraction or to its components.

\startbuffer
\startformula
\outerfrac{2a}{3b} = \innerfrac{2a}{3b} = \frac{2a}{3b}
\stopformula
\stopbuffer

\typebuffer

As with fences, the color is only applied to the horizontal bar as there is no
other easy way to color that otherwise.

\getbuffer

As \TEX\ has a couple of low level stackers, we provide an interface to that as
well, but we hide the dirty details. For instance you can define left and right
fences and influence the rule

\startbuffer
\definemathfraction[fraca][rule=no,left=0x005B,right=0x007C]
\definemathfraction[fracb][rule=yes,left=0x007B,right=0x007D]
\definemathfraction[fracc][rule=auto,left=0x007C]
\definemathfraction[fracd][rule=yes,rulethickness=2pt,left=0x007C]
\stopbuffer

\typebuffer \getbuffer

When \type {rule} is set to \type {auto}, we use \TEX's values (derived from font
metrics) for the thickness of rules, while \type {yes} triggers usage of the
specified \type {rulethickness}.

\startbuffer
\startformula
\fraca{a}{b} + \fracb{a}{b} + \fracc{a}{b} + \fracd{a}{b}
\stopformula
\stopbuffer

\typebuffer

Gives:

\getbuffer

\startbuffer
\definemathfraction
  [frace]
  [rule=yes,
   color=blue,
   rulethickness=1pt,
   left=0x005B,
   right=0x007C]
\stopbuffer

\typebuffer \getbuffer

This fraction looks as follows (scaled up):

\startlinecorrection
\midaligned{\scale[height=5ex]{$\displaystyle\frace{a}{b}$}}
\stoplinecorrection

So, the color is applied to the (optional) fences as well as to the (optional)
rule. And when you color the whole formula as part of the context, you get

\startlinecorrection
\midaligned{\scale[height=5ex]{\color[maincolor]{$\displaystyle\frace{a}{b}$}}}
\stoplinecorrection

There is a (maybe not so) subtle difference between fences that come with
fractions and regular fences, Take these definitions:

\startbuffer
\definemathfence    [parenta] [left=0x28,right=0x29,command=yes]
\definemathfraction [parentb] [left=0x28,right=0x29,rule=auto]
\stopbuffer

\typebuffer \getbuffer

Of course the \type {b} variant takes less code:

\startbuffer
\startformula
\parenta{\frac{a}{b}} + \parentb{a}{b}
\stopformula
\stopbuffer

\typebuffer

But watch how the parentheses are also larger. At some point \CONTEXT\ will
provide a bit more control over this,

\getbuffer

You can also influence the width of the rule, but that is not related to the
style.

\startbuffer
\definemathfraction
  [wfrac]
  [margin=.25em]

\definemathfraction
  [wwfrac]
  [margin=.50em]

\startformula
  \frac   {  a } { \frac {  b } {  c } } +
  \wfrac  {  a } { \frac {  b } {  c } } =
  \wwfrac { 2a } { \frac { 2b } { 2c } }
\stopformula
\stopbuffer

\typebuffer

Both the nominator and denominator are widened by the margin:

\getbuffer

\stopsection

\stopcomponent
