% Preliminary writeup about the rules for "gray fonts".

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\def\MF{{\logo META}\-{\logo FONT}}
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\line{Gray fonts for \MF\ proofs.\hfil
	(Preliminary draft: \today)}
\medskip
The \.{GFtoDVI} program converts a \.{GF} file into a \.{DVI} file that,
when printed, gives a hardcopy proof of the characters. The proof diagrams
can be regarded as an array of rectangles, where each rectangle is either
blank or filled with a special symbol that we shall call $x$. A blank
rectangle represents a white pixel, while $x$ represents a black pixel.
Additional labels and reference lines are often superimposed on this
array of rectangles; hence it is usually best to choose a symbol $x$ that
has a somewhat gray appearance, although any symbol can actually be used.

In order to construct such proofs, \.{GFtoDVI} needs to work with
a special type of font known as a ``gray font''; it's possible to
obtain a wide variety of different sorts of proofs by using different
sorts of gray fonts. The purpose of this
memo is to explain exactly what gray fonts are supposed to contain.

The simplest gray font contains only two characters, namely $x$
and a another symbol that is used for dots that identify key points.
If proofs with relatively large pixels are desired, a two-character
gray font is all that's needed. However, if the pixel size is to be
relatively small, practical considerations make a two-character
font too inefficient, since it requires the typesetting of tens
of thousands of tiny little characters; printing device drivers
rarely work very well when they are presented with data that is
so different from ordinary text. Therefore a gray font with small
pixels usually has a number of characters that replicate $x$ in
such a way that comparatively few characters actually need to be
typeset.

Since many printing devices are not able to cope with
arbitrarily large or complex characters, it is not possible for a
single gray font to work well on all machines. In fact,
$x$ must have a width that is an even multiple of the printing
device's unit of horizontal position, since rounding the positions of grey
characters would otherwise produce unsightly streaks on proof output.
Thus, there is no way to make the gray font as device independent as
the rest of the system, in the sense that we would expect approximately
identical output on machines with different resolution. Fortunately,
proof sheets are rarely considered to be final documents; hence
\.{GFtoDVI} is set up to provide results that adapt suitably to
local conditions.

This understood, we can now take a look at what \.{GFtoDVI} expects to
see in a gray font. The character~$x$ always appears in position~1. It
must have positive height~$h$ and positive width~$w$; its depth
and italic correction are ignored.

Positions 2--120 of a gray font are reserved for special combinations
of $x$'s and blanks, stacked on top of each other. None of these character
codes need be present in the font; but if they are, the slots should be
occupied by characters of width~$w$ that have certain configurations of
$x$'s and blanks, prescribed for each character position. For example,
position~3 of the font should either contain no character at all,
or it should contain a character consisting of two $x$'s one above
the other; one of these $x$'s should appear immediately above the
baseline, and the other should appear immediately below.

It will be convenient to use a horizontal notation like `\.{XOXXO}'
to stand for a vertical stack of $x$'s and blanks. The convention
will be that the stack is built from bottom to top, and the topmost
rectangle should sit on the baseline. Thus, `\.{XOXXO}' stands
actually for a character of depth~$4h$ that looks like this:
$$\vcenter{\halign{\hfil#\hfil\cr
blank\cr
$x$\rlap{\qquad\raise8pt\hbox{\smash{\hbox{$\longleftarrow$ baseline}}}}\cr
$x$\cr
blank\cr
$x$\cr
}}$$
We use a horizontal notation instead of a vertical one because column
vectors take too much space, and because the horizontal notation corresponds
to binary numbers in a convenient way.

Positions 1--63 of a gray font are reserved for the patterns \.X, \.{XO},
\.{XX}, \.{XOO}, \.{XOX}, \dots, \.{XXXXXX}, just as in the normal
binary notation of the numbers 1--63. Positions 64--70 are reserved for
the special patterns \.{XOOOOOO}, \.{XXOOOOO}, \dots, \.{XXXXXXO},
\.{XXXXXXX} of length seven; positions 71--78 are, similarly, reserved for
the length-eight patterns \.{XOOOOOOO} through \.{XXXXXXXX}. The
length-nine patterns \.{XOOOOOOOO} through \.{XXXXXXXXX} are assigned
to positions 79--87, the length-ten patterns to positions 88--97,
the length-eleven patterns to positions 98--108, and the length-twelve
patterns to positions 109--120.

Position 0 of a gray font is reserved for the ``dot'' character, which
should have positive height~$h'$ and positive width~$w'$. When \.{GFtoDVI}
wants to put a dot at some place $(x,y)$ on the figure, it positions
the dot character so that its reference point is at $(x,y)$. The
dot will be considered to occupy a rectangle $(x+\delta,y+\epsilon)$
for $-w'\leq\delta\leq w'$ and $-h'\leq\epsilon\leq h'$; the rectangular
box for a label will butt up against the rectangle enclosing the dot.

All other character positions of a gray font (namely, positions 121--255)
are unreserved, in the sense that they have no predefined meaning.
But \.{GFtoDVI} may access them via the ``character list'' feature of
\.{TFM} files, starting with any of the characters in positions
1--120. In such a case each succeeding character in a list should be
equivalent to two of its predecessors, horizontally adjacent to each other.
For example, in a character list like
$$53,\;121,\;122,\;123$$
character 121 will stand for two 53's, character 122 for two 121's (i.e.,
four 53's), and character 123 for two 122's (i.e., eight 53's). Since
position~53 contains the pattern \.{XXOXOX}, character~123 in this example
would have height~$h$, depth~$5h$, and width~$8w$, and it would stand for
the pattern
$$\vcenter{\halign{&$\hfil#\hfil$\cr
x&x&x&x&x&x&x&x\cr
&&&&&&&\cr
x&x&x&x&x&x&x&x\cr
&&&&&&&\cr
x&x&x&x&x&x&x&x\cr
x&x&x&x&x&x&x&x\cr}}$$
Such a pattern is, of course, rather unlikely to occur in a \.{GF} file,
but \.{GFtoDVI} would be able to use it if it were present. Designers
of gray fonts should provide characters only for patterns that they think
will occur often enough to make the doubling worthwhile. For example,
the character in position 120 (\.{XXXXXXXXXXXX}), or whatever is the
tallest stack of $x$'s present in the font, is a natural candidate for
repeated doubling.

Here's how \.{GFtoDVI} decides what characters of the gray font will be used,
given a configuration of black and white pixels: If there are no black
pixels, stop. Otherwise look at the top row that contains at least one
black pixel, and the eleven rows that follow. For each such column,
find the largest~$k$ such that $1\leq k\leq120$ and the gray font contains
character~$k$ and the pattern assigned to position~$k$ appears in the
given column. Typeset character $k$ (unless no such character exists)
and erase the corresponding black pixels; use doubled characters,
if they are present in the gray font, if two or more consecutive equal
characters need to be typeset. Repeat the same process on the remaining
configuration, until all the black pixels have been erased.

If all characters in positions 1--120 are present, this process is guaranteed to
take care of at least six rows each time; and it usually takes care of
twelve, since all patterns that contain at most one ``run'' of $x$'s
are present.

Fonts have optional parameters, as described in Appendix~F of {\sl The
\TeX book}, and some of these are important in gray fonts. The
slant parameter~$s$, if nonzero, will cause \.{GFtoDVI} to skew its
output; in this case the character $x$ will presumably be a parallelogram
with a corresponding slant, rather than the usual rectangle. \MF's
coordinate $(x,y)$ will appear in physical position $(xw+yhs,yh)$
on the proofsheets.

Parameter number~8 of a gray font specifies the thickness of rules
that go on the proofs. If this parameter is zero, \TeX's default
rule thickness (0.4\thinspace pt) will be used.

The other parameters of a gray font are ignored by \.{GFtoDVI}, but
it is conventional to set the space parameter to~$w$ and the
xheight parameter to~$h$.

For best results the designer of a gray font should choose $h$ and~$w$
so that the user's \.{DVI}-to-hardcopy software will not make any
rounding errors. Furthermore, the dot should be an even number~$2m$ of
pixels in diameter, and the rule thickness should work out to an
even number~$2n$ of pixels; then the dots and rules will be centered on
the correct positions, in case of integer coordinates. Gray fonts
are almost always intended for particular output devices, even though
`\.{DVI}' stands for `device independent'; we use \.{DVI} files for \MF\
proofs chiefly because software to print \.{DVI} files is already in place.

\bye