[UEB Maths] new draft of sections 13-15 for comment

[Janet Reynolds]

I am pasting below the new draft of Sections 13 on arrows, 14 on shapes
and 15 on matrices. I will send it separately as a Word attachment.=20

I have made little change to Section 13, arrows were very well defined
by Committee 2 and the groupings I did last time still seem to work
well. The same applies to Section 14 on shapes.
However I did have a bit of trouble with the grade 1 indicators. Suppose
you are in grade 2 mode and you have an arrow between two words.
Initially I gave it a grade 1 word symbol but then I wondered whether
the arrow indicator would set grade 1 mode for the rest of the arrow
construct, in which case you would only need a single grade 1 symbol
indicator for the initial arrow indicator. The same applies for shapes.
I reread those sections in the Committee 2 report and although it talks
about setting "arrow mode" or "shape mode" it wasn't clear whether grade
1 mode was also implicit. In the end I did the examples as if it was
implicit but I will add it to the agenda for our teleconference.

Section 15 on matrices is much as it was before. When it came to wide
matrices we included two options, one from the US and one from the UK, I
think it's good to have both.

I've not used passage indicators around the matrices, just added grade 1
indicators for any isolated letters. I'm assuming we don't need one for
a negative letter like -b, because of the dot 5 you couldn't read it as
hyphen but.

I must admit that those 2 cell minus signs do feel unwieldy in a matrix,
and that reminded me that we have talked about having some rules for
situations when you could miss the dot 5's out before operation signs
and brackets. My main concern here is for diagrams, where space is at a
premium, and back translation is unlikely to be used. Anyway I'll add
that to the agenda for our conference call.=20
Janet

13 Arrows
13.1 Simple arrows
.=3D\		arrow indicator
.=3D^\	bold arrow indicator
.=3D\o	simple right pointing arrow (east)
.=3D\[	simple left pointing arrow (west)
.=3D\+	simple up pointing arrow (north)
.=3D\%	simple down pointing arrow (south)
.=3D\s	simple up and right pointing arrow (northeast)
.=3D\<	simple down and right pointing arrow (southeast)
.=3D\:	simple up and left pointing arrow (northwest)
.=3D\>	simple down and left pointing arrow (southwest)

A simple arrow has a standard barbed tip at one end (like a v on its
side, pointing away from the shaft). The shaft is straight and its
length and thickness are not significant. These arrows are represented
by an opening arrow indicator and the appropriate closing arrow
indicator. Notice that all these terminating symbols have three dots,
arranged in a consistent pattern that best describes the direction.

Note that unless you are already in grade 1 mode, the arrow indicator
will need a grade 1 symbol indicator.  The bold arrow indicator will not
need one, as it already has a prefix. Both arrow indicators set grade 1
mode until the next space, in the same way that the numeric indicator
does.

n right arrow 0 ( n tends to zero)

;n ;|o #j
or
;;;n |o #j;'

input "right arrow" process "bold right arrow" output

9put ;|o process ^|o |tput
or
;;;input |o process ^|o output;'


Arrows are signs of comparison so should usually be spaced. An exception
is when they are written below the limit function (see Section 9.6).

Do not use arrow indicators when a simple right pointing arrow is the
only modifier above or below an item. See "arrow over previous item" and
"arrow below previous item" in Section 12.
13.2 Arrows with unusual shafts and a standard barbed tip
Shaft symbols:
.=3D3		short single straight line
.=3D33	medium single straight line
.=3D333	long single straight line
.=3D7		double, short
.=3D111	dotted, long
.=3D9		curved or bent to the left (anticlockwise in line of
direction)
.=3D5		curved or bent to the right (clockwise in line of
direction)
.=3D4		sharp turn to the right (in line of direction)
.=3D0		sharp turn to the left (in line of direction)

All shaft symbols can be elongated by repetition. The shaft symbols are
placed between the opening and closing arrow indicators. These examples
still have standard barbed tips.=20


Double shafted medium length right pointing arrow
;\77o

Dotted long arrow pointing northwest
;\111:

Medium length right pointing arrow bending clockwise
;\eeo

Short down pointing arrow
;\3%
13.3 Arrows with unusual tips
Barb symbols:=20
.=3Dr		regular barb, full, in line of direction
.=3Dw		regular barb, full, counter to line of direction
.=3D@w	regular barb, upper half, counter
.=3D,w	regular barb, lower half, counter
.=3D@r	regular barb, upper half, in line
.=3D,r	regular barb, lower half, in line
.=3D&		curved, full, counter
.=3Dy		curved, full, in line
.=3D@&	curved, upper half, counter
.=3D,&	curved, lower half, counter
.=3D@y	curved, upper half, in line
.=3D,y	curved, lower half, in line
.=3D\		straight, full, (directionless)
.=3D@\	straight, upper half, (directionless)
.=3D,\	straight, lower half, (directionless)

If an arrow has unusual tips, decide which is the head before you choose
the direction of your closing indicator. The complete rules for deciding
arrow direction are:

(1) If there are directional tips, and all lead in the same direction,
the head is the end that lies in that direction.
(2) If there are no directional tips, but one end has a tip and the
other does not, the end with the tip is the head.
(3) In all other cases, the head of the arrow is deemed to be the end at
the right, or in the case of strictly vertical arrows, at the top.

The tip(s) and shaft segment(s) are transcribed between the opening and
closing indicators. These items are expressed in logical order, that is
starting with the arrow tail and progressing towards the head, even if
that runs counter to the physical order (as in the case of a left
pointing arrow).  Certain elements are omitted, corresponding to these
reader rules:

(1) If no tip is transcribed, it is understood that an ordinary full
barbed tip occurs at the arrow head, and there is no other tip.
(2) If no shaft is transcribed, it is understood that the shaft is a
straight line of medium length.  In this case, if no tip is transcribed,
rule (1) also applies; if one tip is transcribed, it is at the head; if
two tips are transcribed, the first is at the tail and the second at the
head.
=20
otherwise ordinary right arrow, with curved head
;\yo =20

common horizontal bidirectional arrow
;\wro =20

bidirectional horizontal double shaft arrow, with curved heads
;\&77yo =20

arrow like the previous, but tilted from lower left to upper right
;\&77ys =20

otherwise ordinary right arrow, with tail and head tips
;\rro =20

otherwise ordinary right arrow, with two head tips
;\33rro =20

otherwise ordinary left arrow, with tail and head tips
;\rr[ =20

otherwise ordinary left arrow, with two head tips
;\33rr[ =20

right arrow with three tips separated by 2 short shaft segments
;\r3r3ro =20

bold arrow up, followed by ordinary arrow down
^\+\% =20

common right arrow over common left arrow
(see section 14.3c regarding vertical juxtaposition)
;\o]\[ =20

Half barbed left arrow over half barbed right arrow (reversible chemical
reaction)
;\@r[]\,ro

Other arrow symbols (such as equilibrium arrows) that occur in Chemistry
can be found in the UEBC symbols list.
14 Shape Symbols and Composite Symbols
Listing of shape indicators:
.=3D$ 	shape indicator
.=3D_$	filled (solid) shape indicator
.=3D.$	shaded shape indicator
.=3D@$	transcriber assigned shape indicator
.=3D@_$  transcriber assigned filled (solid) shape indicator
.=3D@.$  transcriber assigned shaded shape indicator
.=3D:		shape terminator

Listing of specific shapes:
.=3D$#c	regular (equilateral) triangle
.=3D$#d	square
.=3D$#e	regular pentagon
.=3D$#f	regular hexagon
.=3D$#g	regular heptagon
.=3D$#h	regular octagon
.=3D$#i	regular nonagon
.=3D$#aj	 regular decagon
.=3D$#aa	 regular undecagon
.=3D$#ab	 regular dodecagon (etc. for all regular polygons)
.=3D$=3D	 circle
.=3D$@#d	 parallelogram

Composite Symbols:
.=3D&  superposition indicator
.=3D=3D  horizontal juxtaposition indicator=20
.=3D]  vertical juxtaposition indicator
.=3D[  physical enclosure indicator

14.1 Use of the shape termination indicator

14.1.1 If a shape is followed by a space then no termination symbol is
needed

triangle symbol space ABC
;$#c ,,abc

14.1.2 If the shape symbol is followed by punctuation, or unspaced from
a following symbol, then the shape terminator must be used.

;$#c:,,abc

,:at is ! next %ape3
_<;$#d:1 .$=3D:1 _$#c:1 .$#d 444_>

Note that unless you are already in grade 1 mode, a grade 1 symbol
indicator will be needed before the shape indicator. This does not
however apply to the shaded and filled shape indicators because they
already have a prefix.  All the initial shape indicators set grade 1
mode until the next space, in the same way that the numeric indicator
and arrow indicators do.
14.2 Transcriber defined shapes=20
The description within transcriber defined shapes should be a short
series of initials or a single grade 1 word. They should not be used if
the print symbol is already covered elsewhere in the code. The
definitions of all shape symbols should be available to the reader in
either a transcriber's note or on a special symbols page.

For example, a smiling face used as an icon throughout a book could be
defined as=20
@$sf or @$smile
rather than=20
@$<smiling face>
which is too long and requires braille grouping symbols to stop the
space terminating the shape.

14.3 Combined shapes=20
If two print symbols have been combined to form a new previously
undefined symbol, then it must be decided whether the second symbol is
enclosed, superimposed, combined on the right or combined below. Each of
the four composite symbol indicators signals a combining of the item
just prior with the item immediately following it, where "item" is as
defined in Section 7. =20

Each composite symbol indicator will need a grade 1 symbol indicator
unless the whole expression is already in grade 1 mode.

14.3.1 Physical Enclosure, dots 246:

circle enclosing a plus sign
;$=3D["6

In the example below, the circle enclosing a plus sign is being used as
an operation sign. In the first version the operation sign is unspaced
so a termination sign is needed but no grade indicators are needed
because the number signs initiate numeric mode. In the second version
the operation sign is spaced for clarity so no termination sign is
needed but the shape symbols do need grade 1 indicators.=20

2 "circled plus" 3 =3D 3 "circled plus" 2
#b$=3D["6:#c "7 #c$=3D["6:#b
#b ;$=3D["6 #c "7 #c ;$=3D["6 #b

14.3.2 Superposition, dots 12346

Note that this structure should not be used for negation. See "line
through previous item" in Section 12.

R with superimposed x (prescription symbol)
,r;&x

Integral sign with a small square superimposed half way up
(the termination could be omitted if there was a following space)
;;!&$#d:

Integral sign with a small circle superimposed half way up=20
(closed integral defined in Section 11.1)
@!=20

14.3.3 Vertical Juxtaposition, dots 12456

The upper symbol should be given first, followed by the vertical
juxtaposition indicator, then the lower symbol.
This structure should not be used for bars, arrows, dots, tildes or hats
over or under other symbols (see Section 12). Neither should it be used
for superscripts or subscripts written directly over or under (see
Section 7).

A hollow dot with an equal sign underneath.=20
.4;]"7

binomial coefficient (refer Section 11.2)
"<n;]r">

14.3.4 Horizontal Juxtaposition, dots 123456

"Horizontal juxtaposition" is to be invoked only when two symbols are
written in close proximity and it is clear from the usage that a new
single symbol, distinct from the elementary symbols considered in
sequence, is intended.  Otherwise, symbols written one after the other
should simply be brailled accordingly.
=20
15 Matrices and vectors
15.1 Enlarged grouping symbols
When enlarged brackets are used in print for vectors, matrices, systems
of equations, function definitions etc., the appropriate enlarged
grouping symbols should be used in braille. These are the usual grouping
symbols preceded by a dot 6. See the full list in Section 5. These
should be placed directly under each other. Blank lines before and after
such arrangements may be needed for clarity. =20
15.2 Matrices
The columns should be left adjusted except for minus signs which should
be brailled to stand out. One column of blank cells should be left
between columns. Material outside the matrix, such as punctuation or
signs of operation and comparison, should be placed on the top line,
even if they are centered in print.
=20
,i "7 ,"<#a #j,">4
      ,"<#j #a,">

,.<#a #b #c,.>,.<  #a   #b,.>
,.<#d #e #f,.>,.<"-#c   #d,.>
              ,.<  #e "-#f,.>    =20

,"<  a "-b,">
,"<"-c  ;d,">
15.3 Determinants
These have the same structure as matrices but are enclosed in print with
enlarged vertical bars.

;;;_|,p_| "7 ,_|a b_| "7 ad"-bc;'
             ,_|c d_|
15.4  Omission dots
The placement of dots used to indicate the omission of one or more rows
or columns can follow the print.

,_|a5#aa   a5#ab   444 a5<#an>,_|
,_|a5#ba   a5#bb   444 a5<#bn>,_|
,_|  4       4     444    4   ,_|
,_|a5<m#a> a5<m#b> 444 a5<mn> ,_|
15.5 Dealing with wide matrices
If a matrix or determinant is too wide for the braille page, runovers
within entries may be necessary. If there is not room to indent these
runovers, they can be blocked and a blank line left between rows.=20

,"<a5#ax5#A   a5#ay5#A   a5#az5#A  ,">
,"<"6b5#ax5#b "6b5#ay5#b "6b5#az5#b,">
,"<"6c5#ax5#c "6c5#ay5#c "6c5#az5#c,">
,"<                                ,">
,"<a5#bx5#A   a5#by5#A   a5#bz5#A  ,">
,"<"6b5#bx5#b "6b5#by5#b "6b5#bz5#b,">
,"<"6c5#bx5#c "6c5#by5#c "6c5#bz5#c,">
,"<                                ,">
,"<a5#cx5#A   a5#cy5#A   a5#cz5#A  ,">
,"<"6b5#cx5#b "6b5#cy5#b "6b5#cz5#b,">
,"<"6c5#cx5#c "6c5#cy5#c "6c5#cz5#c,">

Another approach is to complete the first column without overruns and
then to place the next column below this, indented two cells.=20

,"<a5#ax5#A"6b5#ax5#b"6c5#ax5#c
,"<a5#bx5#A"6b5#bx5#b"6c5#bx5#c
,"<a5#cx5#A"6b5#cx5#b"6c5#cx5#c
  a5#ay5#A"6b5#ay5#b"6c5#ay5#c =20
  a5#by5#A"6b5#by5#b"6c5#by5#c =20
  a5#cy5#A"6b5#cy5#b"6c5#cy5#c =20
    a5#az5#A"6b5#az5#b"6c5#az5#c,">
    a5#bz5#A"6b5#bz5#b"6c5#bz5#c,">
    a5#cz5#A"6b5#cz5#b"6c5#cz5#c,">

Notice that in the first example the structure of the matrix is clearer
but in the second example the individual entries are easier to read.
Notice also the different placement of the enlarged grouping signs in
the two examples.
A transcriber's note should be included in either case to explain the
placement of each entry.
15.6 Vectors
Vectors are a special type of matrix with only one column, usually
enclosed by enlarged round parentheses.=20

,"<  #b,">
,"<"-#a,">

Names of vectors are usually written in bold in print and have arrows or
bars above or below. Boldface only needs to be shown if it is the only
method used.=20
For arrows and bars see Section 12.
If the vector above was called p and went from point A to point B here
are some of the most likely forms:

p with a bar under
p,:

p with a bar over
p:=20

bold p
^2p

AB with an arrow over
<,,ab>^:

AB with a bar under
<,,ab>,:
15.7 Grouping of equations=20
Opening enlarged curly braces are often used to group equations. Print
spacing should be followed where possible.

;;;,solve3
  ,_<  x"6#by "7 #g
  ,_<#bx"-  y "7 "-#d

f"<x"> "7 ,_<#j if x @< #j
          ,_<x9#b if x @> #j;'=20