texdimens
=========
## Copyright and License
Copyright (c) 2021 Jean-François Burnol
This file is part of the texdimens package distributed under the
LPPL 1.3c. See file LICENSE.md.
Development: https://github.com/jfbu/texdimens
Release: `0.9 2021/07/21`
## Aim of this package
Utilities and documentation related to TeX dimensional units, usable
both with Plain (`\input texdimens`) and with LaTeX (`\usepackage{texdimens}`).
The aim of this package is to address the issue of expressing dimensions
(or dimension expressions evaluated by `\dimexpr`) in the various TeX
units, to the extent possible.
The descriptions that follow do not explain the details of the exact
internal TeX procedures of parsing dimensional input, they only describe
in a faithful manner the exact outcome of those internal procedures.
The reader is supposed to be familiar with TeX basics.
This project requires the e-TeX extensions `\dimexpr` and `\numexpr`.
The notation `` in the macro descriptions refers to a
*dimensional expression* as accepted by `\dimexpr`. The syntax has some
peculiarities: among them the fact that `-(...)` (for example `-(3pt)`)
is illegal, one must use alternatives such as `0pt-(...)` or a
sub-expression `-\dimexpr...\relax` for example.
Notice that this is WIP and inaccuracies may exist even relative to
descriptions of TeX handlings due to limited time available for the
project.
## TeX points and scaled points
TeX dimensions are represented internally by a signed integer which is
in absolute value at most `0x3FFFFFFF`, i.e. `1073741823`. The
corresponding unit is called the "scaled point", i.e. `1sp` is `1/65536`
of one TeX point `1pt`, or rather `1pt` is represented internally as `65536`.
If `\foo` is a dimen register:
- `\number\foo` produces the integer `N` such as `\foo` is the same as `Nsp`,
- inside `\numexpr`, `\foo` is replaced by `N`,
- `\the\foo` produces a decimal `D` (with at most five places) followed
with `pt` (catcode 12 tokens) and this output `Dpt` can serve as input
in a dimen assignment to produce the same dimension as `\foo`. One can
also use the catcode 11 characters `pt` for this. Digits and decimal
mark must have their standard catcode 12.
When TeX encounters a dimen denotation of the type `Dpt` it will
compute `N` in a way equivalent to `N = round(65536 D)` where ties are
rounded away from zero. Only 17 decimal places of `D` are
kept as it can be shown that going beyond can not change the result.
When `\foo` has been assigned as `Dpt`, `\the\foo` will produce some
`Ept` where `E` is not necessarily the same as `D`. But it is guaranteed
that `Ept` defines the same dimension as `Dpt̀`.
## Further units known to TeX on input
TeX understands on input further units: `bp`, `cm`, `mm`, `in`, `pc`,
`cc`, `nc`, `dd` and `nd`. It also understands font-dependent units `ex` and
`em`, and PDFTeX adds the `px` dimension unit. Japanese engines also
add specific units.
`ex`, `em`, `px` and other engine-dependent units, are all currently
excluded from consideration in this project and the rest of this
documentation is only for the `bp`, `cm`, `mm`, `in`, `pc`, `cc`, `nc`, `dd`
and `nd` units. When we say "unit" we mean one of those or the `pt` (the
`sp` is a special case and will be included or not tacitly in the "unit"
denomination depending on the case).
TeX associates to each such unit `uu` a fraction `phi` which is a conversion
factor. It is always `>1`:
uu phi reduced real approximation 1uu in sp= x=[65536phi]/65536 \the<1uu>
(Python output) [65536phi] (real approximation)
-- ---------- ------- ------------------ --------- -------------------- ----------
bp 7227/7200 803/800 1.00375 65781 1.0037384033203125 1.00374pt
nd 685/642 same 1.0669781931464175 69925 1.0669708251953125 1.06697pt
dd 1238/1157 same 1.070008643042351 70124 1.07000732421875 1.07pt
mm 7227/2540 same 2.8452755905511813 186467 2.8452606201171875 2.84526pt
pc 12/1 12 12.0 786432 12.0 12.0pt
nc 1370/107 same 12.80373831775701 839105 12.803726196289062 12.80373pt
cc 14856/1157 same 12.84010371650821 841489 12.840103149414062 12.8401pt
cm 7227/254 same 28.45275590551181 1864679 28.452743530273438 28.45274pt
in 7227/100 same 72.27 4736286 72.26998901367188 72.26999pt
When TeX parses an assignment `U uu` with a decimal `U` and a unit `uu`,
among those listed above, it first handles `U` as with the `pt` unit.
This means that it computes `N = round(65536*U)`. It then multiplies
this `N` by the conversion factor `phi` and truncates towards zero the
mathematically exact result to obtain an integer `T`:
`T=trunc(N*phi)`. The assignment `Uuu` is concluded by defining the
value of the dimension to be `Tsp`.
Attention that although the mnemotic is `phi=1uu/1pt`, this formula
definitely does not apply with numerator and denominator interpreted as
TeX dimensions. See the above table. Also, the last column looks like
`round(.,5)` is applied to the previous one, but `\the\dimexpr1dd\relax`
is an exception (in the table `[x]` is the integer part, aka for
non-negative values, the `\trunc()` function). Notice also that
`1.00375` is the exact value of the `phi` factor for the `bp` unit but
`1.00375pt>1bp` (`65782>65781`).
As `phi>1` (and is not exceedingly close to `1`), the increasing
sequence `0<=trunc(phi)<=trunc(2phi)<=...` is *strictly increasing* and
**it has jumps**: not all TeX dimensions can be obtained from an
assignment not using the `pt` unit.
The "worst" unit is the largest i.e. the `in` whose conversion factor is
`72.27`. The simplest unit to understand is the `pc` as it corresponds
to an integer ratio `12`: only dimensions which in scaled points are
multiple of `12` are exactly representable in the `pc` unit.
This also means that some dimensions expressible in one unit may not be
available with another unit. For example, and perhaps surprisingly,
there is no decimal `D` which would achieve `1in==Dcm`: the "step"
between attainable dimensions is `72--73sp` for the `in` and `28--29sp`
for the `cm`, and as `1in` differs internally from `2.54cm` by only
`12sp` (see below the `xintsession` verbatim) it is impossible to adjust
either the `in` side or the `cm` side to obtain equality. See in the
[Extras?] section the closest dimension attainable both via `in` and via
`cm`.
In particular `1in==2.54cm` is **false** in TeX, but it is true that
`100in==254cm`... It is also false that `10in==25.4cm` but it is true that
`10in==254mm`... It is false though that `1in==25.4mm`!
>>> (\dimexpr1in, \dimexpr2.54cm);
@_1 4736286, 4736274
>>> (\dimexpr10in, \dimexpr25.4cm);
@_2 47362867, 47362855
>>> (\dimexpr100in, \dimexpr254cm);
@_3 473628672, 473628672
>>> (\dimexpr1in, \dimexpr25.4mm);
@_4 4736286, 4736285
>>> (\dimexpr10in, \dimexpr254mm);
@_5 47362867, 47362867
`\maxdimen` can be expressed only with `pt`, `bp`, and `nd`. For the
other units the maximal attainable dimensions in `sp` unit are given in
the middle column of the next table.
maximal allowed the corresponding minimal TeX dimen denotation
(with 5 places) maximal attainable dim. causing "Dimension too large"
--------------- ------------------------- --------------------------
16383.99999pt 1073741823sp (=\maxdimen) 16383.99999237060546875pt
16322.78954bp 1073741823sp (=\maxdimen) 16322.78954315185546875bp
15355.51532nd 1073741823sp (=\maxdimen) 15355.51532745361328125nd
15312.02584dd 1073741822sp 15312.02584075927734375dd
5758.31742mm 1073741822sp 5758.31742095947265625mm
1365.33333pc 1073741820sp 1365.33333587646484375pc
1279.62627nc 1073741814sp 1279.62627410888671875nc
1276.00215cc 1073741821sp 1276.00215911865234375cc
575.83174cm 1073741822sp 575.83174896240234375cm
226.70540in 1073741768sp 226.70540618896484375in
Perhaps for these various peculiarities with dimensional units, TeX does
not provide an output facility for them similar to what `\the` achieves for
the `pt`.
## Macros of this package
The macros defined by the package are expandable, and will expand
completely in an `\edef`, or in a `\dimexpr...\relax` construct.
As they parse their inputs via `\dimexpr` they can be nested (with
the suitable dimension unit added as postfix to nested macro).
Apart from the `\texdimenup` in case of a negative input, they will
even expand completely under f-expansion.
Negative dimensions behave as if replaced by their absolute value, then
at last step the sign (if result is not zero) is applied (so "down" means
"towards zero", and "up" means "away from zero").
1. For input `X` equal to (or sufficiently close to) `\maxdimen` and
those units `uu` for which `\maxdimen` is not exactly representable
(i.e. all units except `pt`, `bp` and `nd`), the output `D` of the
"up" macros `\texdimenup{X}`, if used as `Duu` in a dimension
assignment or expression, will (naturally) trigger a "Dimension too
large" error.
2. For `dd`, `nc` and `in`, and input `X` equal to (or sufficiently
close to) `\maxdimen` it turns out that `\texdimen{X}` produces
an output `D` such that `Duu` is the first "virtually attainable" TeX
dimension *beyond* `\maxdimen`. Hence `Duu` will trigger on use
"Dimension too large error".
3. Again for the `dd`, `nc` and `in` units, both the "down" and "up" macros
will trigger "Dimension too large" during their execution if used
with an input equal to (or sufficiently close to) `\maxdimen`.
`\texdimenpt{}`
> Does `\the\dimexpr \relax` then removes the `pt`.
`\texdimenbp{}`
> Produces a decimal (with up to five decimal places) `D` such that `Dbp`
> represents the dimension exactly if possible. If not possible it
> will differ by `1sp` from the original dimension, but it is not
> known in advance if it will be above or below.
> `\maxdimen` on input produces `16322.78954` and indeed is realized as `16322.78954bp`.
`\texdimenbpdown{}`
> Produces a decimal (with up to five decimal places) `D` such that `Dbp`
> represents the dimension exactly if possible. If not possible it
> will be smaller by `1sp` from the original dimension.
`\texdimenbpup{}`
> Produces a decimal (with up to five decimal places) `D` such that `Dbp`
> represents the dimension exactly if possible. If not possible it
> will be larger by `1sp` from the original dimension.
`\texdimennd{}`
> Produces a decimal (with up to five decimal places) `D` such that `Dnd`
> represents the dimension exactly if possible. If not possible it
> will differ by `1sp` from the original dimension, but it is not
> known in advance if it will be above or below.
> `\maxdimen` on input produces `15355.51532` and indeed is realized as `15355.51532nd`.
`\texdimennddown{}`
> Produces a decimal (with up to five decimal places) `D` such that `Dnd`
> represents the dimension exactly if possible. If not possible it
> will be smaller by `1sp` from the original dimension.
`\texdimenndup{}`
> Produces a decimal (with up to five decimal places) `D` such that `Dnd`
> represents the dimension exactly if possible. If not possible it
> will be larger by `1sp` from the original dimension.
`\texdimendd{}`
> Produces a decimal (with up to five decimal places) `D` such that `Ddd`
> represents the dimension exactly if possible. If not possible it
> will differ by `1sp` from the original dimension, but it is not
> known in advance if it will be above or below.
> Warning: the output for `\maxdimen` is `15312.02585` but `15312.02585dd`
> will trigger "Dimension too large" error.
> `\maxdimen-1sp` is attainable via `15312.02584dd`.
`\texdimendddown{}`
> Produces a decimal (with up to five decimal places) `D` such that `Ddd`
> represents the dimension exactly if possible. If not possible it
> will be smaller by `1sp` from the original dimension.
`\texdimenddup{}`
> Produces a decimal (with up to five decimal places) `D` such that `Ddd`
> represents the dimension exactly if possible. If not possible it
> will be larger by `1sp` from the original dimension.
`\texdimenmm{}`
> Produces a decimal (with up to five decimal places) `D` such that `Dmm`
> represents the dimension exactly if possible. If not possible it
> will either be the closest from below or from above, but it is not
> known in advance which one (and it is not known if the other choice
> would have been closer).
> `\maxdimen` as input produces on output `5758.31741` and indeed the
> maximal attainable dimension is `5758.31741mm` (`1073741822sp`).
`\texdimenmmdown{}`
> Produces a decimal (with up to five decimal places) `D` such that `Dmm`
> represents the dimension exactly if possible. If not possible it
> will be largest representable dimension smaller than the original one.
`\texdimenmmup{}`
> Produces a decimal (with up to five decimal places) `D` such that `Dmm`
> represents the dimension exactly if possible. If not possible it
> will be smallest representable dimension larger than the original one.
`\texdimenpc{}`
> Produces a decimal (with up to five decimal places) `D` such that `Dpc`
> represents the dimension exactly if possible. If not possible it
> will be the closest representable one (in case of tie, the approximant
> from above is chosen).
> `\maxdimen` as input produces on output `1365.33333` and indeed the
> maximal attainable dimension is `1365.33333pc` (`1073741820sp`).
`\texdimenpcdown{}`
> Produces a decimal (with up to five decimal places) `D` such that `Dpc`
> represents the dimension exactly if possible. If not possible it
> will be largest representable dimension smaller than the original one.
`\texdimenpcup{}`
> Produces a decimal (with up to five decimal places) `D` such that `Dpc`
> represents the dimension exactly if possible. If not possible it
> will be smallest representable dimension larger than the original one.
`\texdimennc{}`
> Produces a decimal (with up to five decimal places) `D` such that `Dnc`
> represents the dimension exactly if possible. If not possible it
> will either be the closest from below or from above, but it is not
> known in advance which one (and it is not known if the other choice
> would have been closer).
> Warning: the output for `\maxdimen` is `1279.62628` but `1279.62628nc`
> will trigger "Dimension too large" error.
> `\maxdimen-9sp` is attainable via `1279.62627nc`.
`\texdimenncdown{}`
> Produces a decimal (with up to five decimal places) `D` such that `Dnc`
> represents the dimension exactly if possible. If not possible it
> will be largest representable dimension smaller than the original one.
`\texdimenncup{}`
> Produces a decimal (with up to five decimal places) `D` such that `Dnc`
> represents the dimension exactly if possible. If not possible it
> will be smallest representable dimension larger than the original one.
`\texdimencc{}`
> Produces a decimal (with up to five decimal places) `D` such that `Dcc`
> represents the dimension exactly if possible. If not possible it
> will either be the closest from below or from above, but it is not
> known in advance which one (and it is not known if the other choice
> would have been closer).
> `\maxdimen` as input produces on output `1276.00215` and indeed the
> maximal attainable dimension is `1276.00215cc` (`1073741821sp`).
`\texdimenccdown{}`
> Produces a decimal (with up to five decimal places) `D` such that `Dcc`
> represents the dimension exactly if possible. If not possible it
> will be largest representable dimension smaller than the original one.
`\texdimenccup{}`
> Produces a decimal (with up to five decimal places) `D` such that `Dcc`
> represents the dimension exactly if possible. If not possible it
> will be smallest representable dimension larger than the original one.
`\texdimencm{}`
> Produces a decimal (with up to five decimal places) `D` such that `Dcm`
> represents the dimension exactly if possible. If not possible it
> will either be the closest from below or from above, but it is not
> known in advance which one (and it is not known if the other choice
> would have been closer).
> `\maxdimen` as input produces on output `575.83174` and indeed the
> maximal attainable dimension is `575.83174cm` (`1073741822sp`).
`\texdimencmdown{}`
> Produces a decimal (with up to five decimal places) `D` such that `Dcm`
> represents the dimension exactly if possible. If not possible it
> will be largest representable dimension smaller than the original one.
`\texdimencmup{}`
> Produces a decimal (with up to five decimal places) `D` such that `Dcm`
> represents the dimension exactly if possible. If not possible it
> will be smallest representable dimension larger than the original one.
`\texdimenin{}`
> Produces a decimal (with up to five decimal places) `D` such that `Din`
> represents the dimension exactly if possible. If not possible it
> will either be the closest from below or from above, but it is not
> known in advance which one (and it is not known if the other choice
> would have been closer).
> Warning: the output for `\maxdimen` is `226.70541` but `226.70541in`
> will trigger "Dimension too large" error.
> `\maxdimen-55sp` is maximal attainable dimension (via `226.7054in`).
`\texdimenindown{}`
> Produces a decimal (with up to five decimal places) `D` such that `Din`
> represents the dimension exactly if possible. If not possible it
> will be largest representable dimension smaller than the original one.
`\texdimeninup{}`
> Produces a decimal (with up to five decimal places) `D` such that `Din`
> represents the dimension exactly if possible. If not possible it
> will be smallest representable dimension larger than the original one.
`\texdimenbothcmin{}` (new with `0.9dev`)
> Produces a decimal (with up to five decimal places) `D` such that `Din`
> is the largest dimension smaller than the original one and
> exactly representable both in the `in` and `cm` units.
`\texdimenbothincm{}` (new with `0.9dev`)
> Produces a decimal (with up to five decimal places) `D` such that `Dcm`
> is the largest dimension smaller than the original one and
> exactly representable both in the `in` and `cm` units. It thus represents
> the same dimension as `\texdimenbothcmin{}in`.
`\texdimenbothcminpt{}` (new with `0.9dev`)
> Produces a decimal (with up to five decimal places) `D` such that `Dpt`
> is the largest dimension smaller than the original one and
> exactly representable both in the `in` and `cm` units. It thus represents
> the same dimension as the one provided by `\texdimenbothcmin` and
> `\texdimenbothincm`.
`\texdimenbothincmpt{}` (new with `0.9dev`)
> Same as `\texdimenbothcminpt`.
`\texdimenbothcminsp{}` (new with `0.9dev`)
> Produces an integer (explicit digit tokens) `N` such that `Nsp`
> is the largest dimension smaller than the original one and
> exactly representable both in the `in` and `cm` units.
`\texdimenbothincmsp{}` (new with `0.9dev`)
> Same as `\texdimenbothcminsp`.
## Extras?
As already stated the "up" and also the "down" macros for the `dd`, `nc`
and `in` units will trigger "Dimension too large" if used with inputs
equal to or very near `\maxdimen`. "Safe" variants which are guaranteed
never to trigger this error but have some extra overhead to filter out
inputs very close to `\maxdimen` will *perhaps* be provided if there is
some demand for it.
But of course anyhow the output from the "up" macros if used
as input with the corresponding unit will be beyond `\maxdimen` if the
latter is not attainable, i.e. for all units except `bp`, and `nd`
(and `pt` but there is no "up" macro for it).
According to a reference on the web by an anonymous contributor the
dimensions representable with both `in` and `cm` units have the shape
`trunc(3613.5*k)sp` for some integer `k`. The largest one smaller
than a given dimension will thus differ from it by at most about `0.055pt`,
which is also about `0.02mm`.
For example `\texdimenbothincm{1cm}` expands to `0.99994cm` which maps
internally to `1864566sp` which differs from TeX's `1cm` by only
`-113sp`. It can be obtained from `0.39368in` or `28.45102pt`.
And `\texdimenbothcmin{1in}` expands to `0.99945in`, maps internally to
`4733685sp` which differs from TeX's `1in` by `-2601sp`. It can be obtained
as `2.5386cm` or `72.2303pt`.
Currently the package does not provide analogous approximations from above.
For the `1in` for example it would be `4737298sp`, i.e. `1.00021in` which
differs from TeX's `1in` by `+1012sp` and is obtained also as `2.54054cm`
and `72.28543pt`.