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<h2 class="title"><a id="chap.fileext"
name="chap.fileext"></a>Appendix A. Filename
Extension Summary</h2>
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<p class="releaseinfo">$Revision: 1.1 $</p>
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<p class="pubdate">$Date: 2002/08/23 14:31:13 $</p>
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<p>This chapter summarizes many common filename extensions.
The extensions are listed in alphabetical order. All
extensions can be shortened to three letters for consistency
with operating systems that do not allow longer file
extensions. On other file systems, they may be slightly
different. For example, <tt>EPS</tt> files are sometimes
called <tt>EPSF</tt> files on unix systems, which allow
longer filenames.</p>
<div class="variablelist">
<dl>
<dt><span class="term">abf</span></dt>
<dd>
<p>An Adobe binary screen font file contains a binary
encoding of a BDF (bitmap distribution format) file.
Binary encoding makes the files smaller, but it also
makes them less portable and unintelligible to humans.
The binary format is described in Adobe's ABF Format
Specification [<a
href="bi01.html#abffiles">abffiles</a>]. BDF files are
described below.</p>
</dd>
<dt><span class="term">afm</span></dt>
<dd>
<p>Adobe font metrics files are ASCII files distributed
with \ps Type 1 fonts. Type 1 fonts are the
linearly scalable fonts that \ps printer users are most
familiar with. Bounding boxes, an encoding vector (what
characters go where), kerning, and ligature information
are among the things described in this file. The AFM
file format is described completely in Adobe's AFM
Format Specification [<a
href="bi01.html#afmfiles">afmfiles</a>].</p>
<p>\ps fonts (available through commercial vendors or
from the Internet) are supplied with AFM files.
Generally, the only occasion that you would have to
modify an AFM file would be to change the encoding
vector.</p>
</dd>
<dt><span class="term">aux</span></dt>
<dd>
<p>Auxiliary files are built by LaTeX each time it
formats a document. LaTeX writes information about
cross references, citations, etc., to the auxiliary
file for post-processing by other tools, or for TeX
processing the next time this document is
formatted.</p>
</dd>
<dt><span class="term">bbl</span></dt>
<dd>
<p>Bibliography files are created by BibTeX from the
citations in your document, the bibliography databases
(BIB) that you specify, and the bibliography style
(BST) you use. BibTeX writes the resulting bibliography
to the BBL file, which is automatically included in
your LaTeX document at the place where you define the
bibliography.</p>
</dd>
<dt><span class="term">bdf</span></dt>
<dd>
<p>Bitmap distribution format files are ASCII files
that describe a bitmap font. They are frequently used
to distribute bitmap versions of scalable fonts in
screen resolution at common sizes. They are resolution
specific, but they are portable from one architecture
to another. The BDF file format is described completely
in Adobe's BDF Format Specification [<a
href="bi01.html#bdffiles">bdffiles</a>].</p>
<p>Some fonts packages are distributed with BDF files.
Other BDF files are created as part of the conversion
process from native format to X11 format. It is
unlikely that you would ever create one purely by
hand.</p>
</dd>
<dt><span class="term">bib</span></dt>
<dd>
<p>Bibliography databases contain bibliographic
information. These are generally handwritten and may
contain bibliographic information for all of the
sources that you are (ever) likely to cite. The BibTeX
program reads information about each work that you
FIXME: from the BIB file. Consult the documentation for
BibTeX for more information about the format of BIB
files.</p>
</dd>
<dt><span class="term">blg</span></dt>
<dd>
<p>BibTeX log files record the status of the last run
of BibTeX.</p>
</dd>
<dt><span class="term">bst</span></dt>
<dd>
<p>Bibliography style files are used by BibTeX to
define the layout of the citations. BibTeX produces
LaTeX commands in the BBL file that define the
citations in the format specified by the BST file.</p>
<p>You may eventually write or modify a bibliography
style file, but it is less common than modifying LaTeX
style files because bibliographies have a more rigidly
defined format. Consult the documentation for BibTeX
for more information about the format of BST files.</p>
</dd>
<dt><span class="term">bzr</span></dt>
<dd>
<p>The GNU fontutils define the BZR format to hold
generic scalable font data. The file actually contains
the specification for a series of bezier curves. The
BZR file format is defined in the TeXinfo pages that
accompany the GNU fontutils. The GNU fontutils create
BZR files.</p>
</dd>
<dt><span class="term">dvi</span></dt>
<dd>
<p>TeX produces device-independent output in the DVI
file. This file describes the TeXed document in a
simple stack language that can be rendered on any
device. The format of DVI files is described in the
\web documentation for <b>DVItype</b>, or in <span
class="emphasis"><em>The DVI Drivers
Standard</em></span> [<a
href="bi01.html#dvi:standard">dvi:standard</a>].</p>
<p>TeX (and some <b>MFware</b> utilities) produces DVI
files.</p>
<p>\hyphenation{encap-sulated}</p>
</dd>
<dt><span class="term">epsf</span></dt>
<dd>
<p>Encapsulated \ps files contain scalable \ps images
and extra information (such as the size of the image's
bounding box) that is necessary to scale the image
appropriately for printing, unlike generic \ps. Using
encapsulated \ps images in your TeX document requires a
DVI driver that understands \ps \specials. How to
include pictures and figures via encapsulated
PostScript is described in detail in Chapter <a
href="ch06.html"
title="Chapter 6. Pictures and Figures">Chapter 6</a>,
<span class="emphasis"><em><a href="ch06.html"
title="Chapter 6. Pictures and Figures">Chapter 6</a></em></span>.</p>
<p>You are unlikely to create encapsulated \ps files by
hand, but many drawing and drafting programs can create
them for you.</p>
</dd>
<dt><span class="term">fig</span></dt>
<dd>
<p>FIG files are created by the <b>XFig</b> program
(and possibly other programs). The scalable
representation of a collection of graphics objects is
stored in ASCII form in FIG files. The <b>transfig</b>
program can translate FIG files into a number of other
formats including EPSF, HPGL, and a variety of LaTeX
environments.</p>
</dd>
<dt><span class="term">fli</span></dt>
<dd>
<p>Font libraries are distributed with emTeX. They
contain a collection of PK files. Font libraries have
several advantages over a directory full of PK files:
they are easier to maintain (because you don't have to
deal with hundreds of files); they are faster to search
(because they are indexed more efficiently than a
directory); they are smaller (because <span
class="emphasis"><em>each</em></span> PK file wastes an
average of half a cluster of disk space); and the name
of each font is not limited to eight characters as it
is under MS-DOS file naming conventions.</p>
<p>Note: <b>dvips</b> can also use emTeX FLI files.</p>
</dd>
<dt><span class="term">gf</span></dt>
<dd>
<p>Generic font files contain bitmap data for the
characters of a font. The GF format is very simple, and
many TeX related programs that create fonts produce GF
files. The disadvantage of GF files is that they are
very large (because no compression is performed). The
format of GF files is described in the \web
documentation for <b>GFtoPK</b> (or any of the
GF-related <b>MFware</b> programs).</p>
<p>MetaFont is the primary source for GF files. Some
other programs (some of the GNU fontutils, for example)
also produce GF files.</p>
</dd>
<dt><span class="term">gif</span></dt>
<dd>
<p>Graphics interchange format is a CompuServe bitmap
graphics standard. GIF files are very popular, and a
number of converters (e.g., <b>BM2FONT</b>) can
translate GIF files into a format usable by TeX.</p>
</dd>
<dt><span class="term">glo</span></dt>
<dd>
<p>Glossary files are produced by the LaTeX \glossary
command. They are analogous to the IDX files produced
by the \index commands. The glossary is inserted in
your document wherever the \makeglossary command
occurs.</p>
</dd>
<dt><span class="term">gsf</span></dt>
<dd>
<p>Ghostscript fonts are scalable fonts very similar to
\ps Type 1 fonts. Theoretically,
<b>Ghostscript</b> can use \ps Type 1 fonts
directly, although I have never tried. Several GSF
fonts are distributed with <b>Ghostscript</b>.</p>
</dd>
<dt><span class="term">hpgl</span></dt>
<dd>
<p>Hewlett-Packard GL is a plotter language. Many
programs can produce vector graphics in HPGL
format.</p>
</dd>
<dt><span class="term">hptfm</span></dt>
<dd>
<p>Hewlett-Packard tagged font metric files are a lot
like TeX TFM files. It is unfortunate that both files
have the extension TFM because they are completely
incompatible. You can generate TeX TFM files from
HPTFMs with the <b>HPTFM2PL</b> program.</p>
</dd>
<dt><span class="term">idx</span></dt>
<dd>
<p>Index files are produced automatically when you use
the \index commands in LaTeX. The IDX file contains raw
indexing data that will be used by the <b>MakeIndex</b>
program to build an index for your document. You must
include the <tt>makeidx</tt> style in your
<tt>documentstyle</tt> command, and you must turn on
indexing with \makeindex in the preamble of your
document if you wish to (re)build the index. See the
entry for IND files below for more information.</p>
</dd>
<dt><span class="term">ilg</span></dt>
<dd>
<p><b>MakeIndex</b> log files record the status of the
last run of <b>MakeIndex</b>.</p>
</dd>
<dt><span class="term">img</span></dt>
<dd>
<p>The IMG format is a particular bitmapped image
format used by the GEM Window System (a PC-based
windowed desktop interface product). The GNU fontutils
read IMG files as their default format. The
<b>PBMplus</b> utilities<sup>[<a id="id2942243"
name="id2942243" href="#ftn.id2942243">129</a>]</sup>
can convert between many graphics file formats,
including IMG.</p>
<p>Some scanning software produces IMG files directly.
Other IMG files are distributed by the Free Software
Foundation as part of an ongoing project to produce
high-quality, free typefaces.</p>
</dd>
<dt><span class="term">ind</span></dt>
<dd>
<p>Index files are produced by the <b>MakeIndex</b> and
automatically get included into your LaTeX document
wherever you put the \printindex command. The \index
commands in your LaTeX document write raw indexing data
to the IDX file. <b>MakeIndex</b> reads the IDX file,
sorts and formats the index according to the IST file,
and produces an IND file for your document.</p>
</dd>
<dt><span class="term">ist</span></dt>
<dd>
<p>Index specification files are used by
<b>MakeIndex</b> to format the index file. Consult the
documentation for <b>MakeIndex</b> for more
information.</p>
</dd>
<dt><span class="term">jpeg</span></dt>
<dd>
<p>JPEG files are compressed bitmap images. Because
JPEG files use a “lossy” compression
algorithm, they are frequently much smaller than other
formats.</p>
</dd>
<dt><span class="term">lof</span></dt>
<dd>
<p>List of figures files are produced by the
\listoffigures command in LaTeX. After seeing
\listoffigures, LaTeX writes figure captions to the LOF
file. The next time the document is formatted, LaTeX
will insert the LOF file at the point where you issue
the \listoffigures command.</p>
</dd>
<dt><span class="term">log</span></dt>
<dd>
<p>Log files are always produced by TeX and MetaFont.
The LOG file is generally uninteresting. Status and
warning messages deemed too trivial (or too detailed)
for the display are written to the log file (all
messages written to the display are also written to the
log).</p>
</dd>
<dt><span class="term">lot</span></dt>
<dd>
<p>List of tables files are exactly analogous to LOF
files.</p>
</dd>
<dt><span class="term">mf</span></dt>
<dd>
<p>Just as TeX reads TEX files, which are plain ASCII
descriptions of a typeset document, MetaFont reads MF
files, which are plain ASCII descriptions of a
typeface. MetaFont and MF files are the topic of
Knuth's \MFbook [<a
href="bi01.html#kn:mfbook">kn:mfbook</a>]. Unlike \ps
fonts, MetaFont fonts are not linearly scaled.<sup>[<a
id="id2942471" name="id2942471"
href="#ftn.id2942471">130</a>]</sup></p>
<p>The standard TeX distribution contains the MF files
for the Computer Modern fonts. Knuth has produced
several more MF files to demonstrate MetaFont. The
American Mathematical Society has extended Computer
Modern with several more. The <b>MFpic</b> macro
package produces MF files from a picture-like
environment in TeX. \TheMFbook describes how to create
your own fonts with MetaFont.</p>
<p>The <span class="emphasis"><em>List of
MetaFonts</em></span> [<a
href="bi01.html#lreq:metafonts">lreq:metafonts</a>] is
posted occasionally to the newsgroups
\path|comp.text.tex| and \path|comp.fonts|.</p>
<p>Chapter <a href="ch11.html"
title="Chapter 11. Introducing MetaFont">Chapter 11</a>,
<span class="emphasis"><em><a href="ch11.html"
title="Chapter 11. Introducing MetaFont">Chapter 11</a></em></span>,
describes MetaFont in more detail. The TeX fonts
available in MetaFont format are listed in
Chapter <a href="ch05.html"
title="Chapter 5. Fonts">Chapter 5</a>,
<span class="emphasis"><em><a href="ch05.html"
title="Chapter 5. Fonts">Chapter 5</a></em></span>.</p>
</dd>
<dt><span class="term">mfj</span></dt>
<dd>
<p>MFjob files are plain ASCII files that contain
instructions for <b>MFjob</b>, an emTeX program that
builds groups of MetaFont fonts. MFJ files can be
created by hand to automate the process of building a
set of fonts. They are also created by the emTeX DVI
drivers if automatic font generation is being used.</p>
</dd>
<dt><span class="term">msp</span></dt>
<dd>
<p>Microsoft Paint files contain bitmapped graphic
images. They can be included in a TeX document with
\special commands recognized by the emTeX DVI
drivers.</p>
</dd>
<dt><span class="term">pbm</span></dt>
<dd>
<p>The portable bitmap format is a flexible bitmap
representation introduced by the <b>PBMplus</b>
package. The <b>PBMplus</b> utilities allow for the
conversion of PBM format files to and from almost
anything else. The PBM format (and all the utilities)
are described in the manpages that accompany the
<b>PBMplus</b> toolkit distribution.</p>
<p>The PBM toolkit and many other X11 graphics
utilities can read and write PBM files (e.g. XV).</p>
</dd>
<dt><span class="term">pcf</span></dt>
<dd>
<p>The PCF format is one of several X11 bitmap font
formats. Architecture-specific versions of X11 use PCF
files. Other architectures use one of a number of other
architecture-specific formats (e.g., SNF). PCF files
are used by at least the DEC versions of the X11
server. The X11 distribution for your architecture
includes a program that will convert BDF files to the
standard adopted for your architecture.</p>
<p>PCF files are almost invariably created from some
other source. It is unlikely that you will ever create
one by hand.</p>
</dd>
<dt><span class="term">pcl</span></dt>
<dd>
<p>PCL files contain printer commands for HP LaserJet
printers. DVI drivers for HP LaserJet printers create
PCL files. It is possible to get information out of
some PCL files with <b>pcltomsp</b>.</p>
</dd>
<dt><span class="term">pcx</span></dt>
<dd>
<p>PCX files contain bitmapped graphic images. They can
be included in a TeX document with \special commands
recognized by the emTeX DVI drivers.</p>
</dd>
<dt><span class="term">pfa</span></dt>
<dd>
<p>Printer font ASCII files contain scalable outline
data that describes each character in a Type 1
font. A large portion of this file is encrypted, so it
is an ASCII file only in the sense that the binary
portion is represented as a string of hexadecimal ASCII
digits. This is traditional \ps because it is pure
ASCII. See PFB below.</p>
<p>Type 1 outline fonts are created by special
font editing programs or conversion tools (e.g. the GNU
fontutils).</p>
</dd>
<dt><span class="term">pfb</span></dt>
<dd>
<p>Printer font binary files, like PFA files, contain
the outline data for \ps Type 1 fonts. The binary
format was adopted to save space (they are generally
about half the size of their PFA counterparts).<sup>[<a
id="id2942788" name="id2942788"
href="#ftn.id2942788">131</a>]</sup> Because they are
binary files, it is more difficult to transfer them
from one architecture to another (endian-ness, binary
transmission, etc.). \ps purists are apt to disparage
them.</p>
</dd>
<dt><span class="term">pfm</span></dt>
<dd>
<p>Printer font metric files are a Microsoft Windows
standard. They are encountered frequently in archives
that contain Type 1 fonts. Unfortunately, these
archives occasionally fail to include AFM files, which
are more standard outside of the Windows community.
Even more unfortunately, PFM files do not contain all
of the information that is in an AFM file. However, the
<b>PFM2AFM</b> program can construct a partial AFM
file. I believe that the PFM file format is described
in a Microsoft technical note; however, I have never
seen it.</p>
<p>Unless you use Microsoft Windows, PFM files are
likely to be useless. If you need PFM files, the MS-DOS
program <b>Refont</b> can create them from AFM
files.</p>
</dd>
<dt><span class="term">pk</span></dt>
<dd>
<p>Most TeX DVI conversion programs read packed bitmap
font files. The PK font format defines a clever scheme
that allows bitmap fonts to be compressed
significantly. The format of PK files is described in
the \web documentation for <b>PKtype</b> (or any of the
PK-related <b>MFware</b> programs).</p>
<p>You are unlikely to create PK files by hand, per se,
but there are a number of utility programs that
ultimately create PK files (e.g., <b>GFtoPK</b>,
MetaFont, <b>MFpic</b>, <b>PS2PK</b>).</p>
</dd>
<dt><span class="term">pl</span></dt>
<dd>
<p>A property list file contains an ASCII
representation of a binary file. The property list
format was created during TeX development to allow
binary files (specifically TFM files) to be hand-coded.
Most users have no reason to create PL files; however,
some programs create PL files that must be converted
into TFM files with the TeXware program <b>PLtoTF</b>.
The PL format is described in the \web documentation
for <b>PLtoTF</b>.</p>
<p>If you need to edit TeX font metric information for
a particular font, you will almost certainly do so by
editing the PL file. You can create a PL file from a
TFM file with the <b>TFtoPL</b> utility.</p>
</dd>
<dt><span class="term">ps</span></dt>
<dd>
<p>\ps is a page description language. The \ps language
is described in a series of volumes from Adobe Systems.
PS is a common extension for \ps files.</p>
<p>Unless you are inclined to enter the Obfuscated \ps
Contest, you are unlikely to create \ps files by hand.
\ps files are created by many common tools.</p>
</dd>
<dt><span class="term">pxl</span></dt>
<dd>
<p>This format is obsolete. It has been completely
superseded by the PK format. If you still have PXL
files, you can convert them to PK format with the
<b>PXtoPK</b> program. If you are still using a DVI
driver that needs PXL files, you need an upgrade.</p>
</dd>
<dt><span class="term">sfl</span></dt>
<dd>
<p>These files contain HP LaserJet softfonts in
landscape orientation. LaserJet softfonts are device
specific bitmap representations of a typeface. The
bitmap versions are described thoroughly in the <span
class="emphasis"><em>LaserJet Technical Reference
Manual</em></span> [<a
href="bi01.html#pcl5:techref">pcl5:techref</a>] for
each of the HP LaserJet printers. Newer laser printers
can perform automatic rotation of fonts (in 90 degree
increments, at least), so the distinction between
landscape and portrait font files is disappearing.</p>
</dd>
<dt><span class="term">sfp</span></dt>
<dd>
<p>These files contain HP LaserJet softfonts in
portrait orientation. See the entry for SFL files,
above.</p>
</dd>
<dt><span class="term">sfs</span></dt>
<dd>
<p>Scalable softfonts are HP LaserJet softfonts for the
new (HPLJ III and higher) LaserJet printers. These are
really in AGFA IntelliFont Scalable format [<a
href="bi01.html#intellifont">intellifont</a>].</p>
</dd>
<dt><span class="term">snf</span></dt>
<dd>
<p>Server native format fonts are another version of
X11 bitmap font. See the entry for PCF files, above,
for more information.</p>
</dd>
<dt><span class="term">sty</span></dt>
<dd>
<p>Style files are used by LaTeX to define the layout
of a LaTeX document (by redefining the meaning of
commands like \section, for example). They are also
used commonly to extend LaTeX. See the LaTeX
manual [<a
href="bi01.html#ll:latexbook">ll:latexbook</a>] for
more information.</p>
<p>Style files are really just TeX files that perform
specific tasks. You will eventually write or modify a
style file, but it isn't something you are likely to do
every day.</p>
</dd>
<dt><span class="term">tex</span></dt>
<dd>
<p>TEX files describe the layout of a typeset document
in the TeX programming language,<sup>[<a id="id2943178"
name="id2943178" href="#ftn.id2943178">132</a>]</sup>
as defined by <span class="emphasis"><em>The
TeXbook</em></span> [<a
href="bi01.html#kn:texbook">kn:texbook</a>]. Most
people use some form of macro package on top of TeX to
make the language easier to swallow. If a TeX file
begins with \documentstyle{} or has \begin{document}
somewhere near the top, it is probably a LaTeX
document. Otherwise, look for the \input commands to
see what macro packages are being included.</p>
<p>Documents that do not appear to be LaTeX documents
and do not appear to \input special macro packages may
be using a special <span
class="emphasis"><em>format</em></span>. Formats are
fast-loading precompiled macro packages. If you know
the name of the format file, you can tell TeX to use it
by typing &<span
class="emphasis"><em>format-name</em></span> as a
parameter to TeX.</p>
</dd>
<dt><span class="term">tiff</span></dt>
<dd>
<p>TIFF files contain bitmapped or vector graphic
images in a very flexible form. The “T” in
TIFF stands for “tagged.” All of the
different kinds of information (regarding number of
colors, compression, etc.) that might appear in a TIFF
file are given unique tags that allow a TIFF file
reader to skip over information that it does not
understand.</p>
</dd>
<dt><span class="term">tfm</span></dt>
<dd>
<p>TeX font metric files contain information about
fonts. TeX doesn't know anything about the intrinsic
shape of the characters that it lays down on the page.
TeX deals entirely with boxes. Every character is
described by the rectangular box that (usually)
surrounds it. The TFM file for a font describes the
size of each character's box, as well as ligature and
kerning information for the font. A human-readable
version of a TFM file can be produced with the
<b>TFtoPL</b> program. The format of TFM files is
described thoroughly in the \web documentation for
<b>TFtoPL</b>.</p>
<p>If you have reason to modify a TFM file, you will
almost certainly do so by converting it to PL format
first. You can convert it back into a TFM file with the
<b>PLtoTF</b> utility.</p>
<p>See also HPTFM files.</p>
</dd>
<dt><span class="term">toc</span></dt>
<dd>
<p>Table of contents files are produced by the
\tableofcontents command in LaTeX. After seeing
\tableofcontents, LaTeX writes chapter, section,
subsection, etc., names to the TOC file. The next time
the document is formatted, LaTeX will insert the TOC
file at the point where you issue the \tableofcontents
command.</p>
</dd>
<dt><span class="term">txt</span></dt>
<dd>
<p>Generic ASCII text.</p>
</dd>
<dt><span class="term">vf</span></dt>
<dd>
<p>Virtual font files. They are described in more
detail in Chapter <a href="ch05.html"
title="Chapter 5. Fonts">Chapter 5</a>,
<span class="emphasis"><em><a href="ch05.html"
title="Chapter 5. Fonts">Chapter 5</a></em></span>.
In short, a virtual font maps a character to an
arbitrary sequence of <tt>DVI</tt> file commands. This
may be another character in a different font, a
different character in the same font, or something else
entirely.</p>
</dd>
<dt><span class="term">vpl</span></dt>
<dd>
<p>The virtual property list is a property list file
for virtual fonts (as opposed to being some sort of
property list file that was itself virtual ;-). VPL
files serve the same purpose for VF files that PL files
serve for TFM files. The VPL format is defined in the
\web documentation for <b>VPtoVF</b>.</p>
</dd>
<dt><span class="term">xbm</span></dt>
<dd>
<p>X11 bitmap files contain a bitmapped image. X11
icons are frequently stored in XBM files. They also
occur in <tt>.icon</tt> files and files without
extensions (e.g., in
<tt>/usr/include/X11/bitmaps</tt>). I mention them here
only because I like to use icons on my X11 desktop, and
I have used <b>PKtoBM</b> to create several nice ones
from TeX PK files.</p>
<p>X11 bitmap files are used for all bitmap displays in
the X11 server (not just icons). Because they are ASCII
and not binary, they are architecture independent,
which makes them very portable.\par</p>
</dd>
</dl>
</div>
<div class="footnotes">
<br />
<hr width="100" align="left" />
<div class="footnote">
<p><sup>[<a id="ftn.id2942243" name="ftn.id2942243"
href="#id2942243">129</a>]</sup> {The <b>PBMplus</b>
utilities are a collection of programs that allow
conversion between different graphic formats by using the
PBM format as a transition step.}</p>
</div>
<div class="footnote">
<p><sup>[<a id="ftn.id2942471" name="ftn.id2942471"
href="#id2942471">130</a>]</sup> {Linear versus
non-linear scaling is a typographic issue better
discussed elsewhere. I mention it here just for
completeness.}</p>
</div>
<div class="footnote">
<p><sup>[<a id="ftn.id2942788" name="ftn.id2942788"
href="#id2942788">131</a>]</sup> {The proof is left as an
exercise to the reader (I always wanted to say
that).}</p>
</div>
<div class="footnote">
<p><sup>[<a id="ftn.id2943178" name="ftn.id2943178"
href="#id2943178">132</a>]</sup> You already knew this,
didn't you?</p>
</div>
</div>
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