USER'S MANUAL FOR DIAGRAM MACROS - J. C. Reynolds - December 1987
The file diagmac.tex contains TEX macros for producing various kinds of
diagrams. It consists of two parts: a collection of general macros for
producing a wide variety of diagrams, and a second collection of macros
(which call upon the first) that are specifically oriented to category-theory
diagrams.
A second file diagmactest.tex is an input file for LATEX that tests the
macros in diagmac.tex.
USE OF THE LATEX PICTURE FACILITY
These macros use the LATEX picture facility to draw lines, arrows, and
circles. Thus all lines and arrowheads are subject to the limitations of
this facility. In particular, the slope of any solid or dashed line must be
a pair of integers whose magnitudes, after division by their greatest common
divisor, are no more than six. When an arrowhead is placed on a line or a
circular arc, the slope of the line (or the tangent to the arc)
must be a pair of integers whose magnitudes, after division by their
greatest common divisor, are no more than four. Also, lines (or dashes in
dashed lines) that are not either horizontal or vertical will not appear
unless they are longer than a minimum length, which is about 10 to 15 points.
Since circles and circular arcs are also drawn via the LATEX picture
facility, they are limited to a fixed variety of diameters. A list of these
diameters is the meaning of the control symbol \diameterlist, which should
be changed if a different circle font is used. (Note that it is a list
of dimensions in increasing order that ends with a comma.)
The latex declarations \thinlines and \thicklines may be used to vary
the thickness of lines, arrowheads, and circles.
PROGRAMS AND STATES
Certain parameters to these macros are ``programs''. A program is a
TEX text that does not directly produce any output but causes state changes
by calling macros. For example, in LATEX, the text read in picture mode,
i.e. the text between \begin{picture} and \end{picture} commands, is a
program that causes state changes by calling the macro \put. (Internally,
such macros cause state changes by assigning to hidden registers and
redefining hidden control symbols. As a consequence, a program cannot
call state-changing macros within a group.)
The diagram-producing macros use two kinds of program, called diagram
programs and expression programs. The state manipulated by a diagram
program, called a diagram state, is a plane containing symbols, lines,
and circles. Locations on this plane are specified by an x,y-coordinate
system, in which x specifies horizontal distance, with increasing values
to the right, and y specifies vertical distance, with increasing values
upwards. The diagram state also contains a ``vertex list'', which is a
list of points (i.e. x,y-coordinate pairs) paired with polygonal regions
called ``shadows''.
The diagram state may also contain a ``current edge'', which is a
(perhaps invisible) directed line segment. When the current edge is defined,
it is determined by four dimension registers:
\xstart: the x-coordinate of the start point
\ystart: the y-coordinate of the start point
\xend: the x-coordinate of the end point
\yend: the y-coordinate of the end point,
and two number registers:
\xslope: the x-component of the slope
\yslope: the y-component of the slope
giving the slope of the edge, reduced to lowest terms. A diagram program may
refer to any of these quantities, and may also alter the dimension registers
explicitly (as well as by calling diagram macros), providing this alteration
preserves the slope of the edge.
The state manipulated by an expression program, called an expression state,
is also a plane, containing an expression and other symbols, etc.,
upon which is imposed an x,y-coordinate system. This state contains an
invisible ``current rectangle'', determined by the four dimension registers:
\lexpr: the x-coordinate of the left side
\rexpr: the x-coordinate of the right side
\texpr: the y-coordinate of the top
\bexpr: the y-coordinate of the bottom,
and a ``center point'', determined by the two dimension registers:
\xcenter: the x-coordinate of the center point
\ycenter: the y-coordinate of the center point.
An expression program may refer to or alter these six dimension registers
explicitly (as well as by calling various macros).
The expression state may also contain a (perhaps invisible) polygon called
the ``current shadow'', and a (perhaps invisible) circle called the
``current circle''. When the current circle is defined, it is determined
by three dimension registers:
\dcircle: the diameter
\xcircle: the x-coordinate of the center
\ycircle: the y-coordinate of the center.
An expression program may refer to or alter these three dimension registers
explicitly (as well as by calling various macros).
The qualification ``perhaps invisible'' is meant to indicate that the
position, shape, and size of edges, shadows, and circles are established
by one group of macros (e.g. \setedge, \rect, \octagon, \setcircle), but
that these entities are actually drawn, i.e. made to appear on the plane
of the diagram or expression state, by another group of macros (e.g.
\drawsolidedge, \outline, \drawcircle).
In calls of the diagram macros, a coordinate is sometimes specified by a
dimension, but often it is specified by a number (i.e. integer) that gives
the coordinate as a multiple of the dimension that is the meaning of the
control symbol \diagramunit. This control symbol is defined to be 1pt,
but the user may redefine it to be some other dimension, either in his main
program or at the beginning of a diagram program.
In addition to the control symbols discussed in this description, this
collection of macros defines a large number of control symbols that are
normally no concern of the user. To avoid the accidental redefinition
of these symbols by the user, they are all given names beginning with \zz.
THE GENERAL MACROS FOR DIAGRAMS
We now describe the general macros for drawing diagrams. The main level
macro is
\diagram{}
It executes the diagram program that is its only parameter, and then issues
the final state produced by this program as a horizontal box whose height,
width, and depth are just enough to enclose all of the symbols and lines
in this state, plus the origin (0,0) of the coordinate system. The height
(depth) will be the distance from the horizontal line y=0 to the highest
(lowest) extent of any symbol or line.
Within a diagram program, one can call the following macros:
\vertex,:
{}{}
\vertex sets the in math mode, with text style,
and creates an expression state containing the resulting expression, with
the current rectangle just enclosing the expression. The center point is
placed midway between the left and right sides of the current rectangle,
at a height above the baseline of the expression given by the control
symbol \centerheight, which is defined to be 3pt. (The effect is to place
the center point on the axis of the expression. However, the user may need
to change the definition of \centerheight if he is using unusual fonts or
script style.) The reference point of the expression will lie at the
origin of the coordinate system.
Next, \vertex executes the to modify the expression
state. Then the material in the expression state is placed in the
current diagram state, at a position so that the center point lies at
the point ,. Finally, if the expression
state contains a current shadow, the point ,
is paired with the shadow and placed on the vertex list.
\place,:
{}{}
\placed{}{}
{}{}
\place behaves the same way as \vertex, except that nothing is placed on the
vertex list. \placed behaves the same way as \place, except that the
coordinates at which the center point is placed are expressed by dimensions
rather than numbers.
\setedge,,
,:
\setedge makes the current edge a directed line segment from the point
``start'' given by its first two parameters to the point ``end'' given
by its last two parameters. This line segment is invisible (until it
is drawn by one of the macros discussed below).
\setedge also examines the vertex list to obtain any shadows that have been
associated with the start or end points by prior executions of \vertex.
\shiftedge{}
\shiftedge displaces the current edge by a vector whose length is determined
by the parameter, and whose direction is obtained by rotating
the current edge 90 degrees counterclockwise.
\shadeedge
\shadeedge changes the extent of the current edge, without displacing
or rotating it, to exclude the portions of the edge lying within shadows
associated with its start and end points. If the execution of \setedge that
established the current edge found a shadow associated with the start point,
then \shadedge will shorten (or conceivably lengthen) the current edge
so that its start point lies on the boundary of the shadow. (If this is
not possible, the start point will be adjusted to be as close as possible
to the shadow.) The end point is adjusted similarly.
\drawsolidedge
\drawsolidedge draws the current edge as a solid line. It is subject to
the constraints of the LATEX picture facility.
\drawdashedge{}{}{}{}
\drawdashedge draws the current edge as a dashed line. It is subject to
the constraints of the LATEX picture facility (particularly regarding the
minimum length of printable dashes for lines that are not horizontal or
vertical). The dashed line will always begin and end with a dash. The
number of dashes will be as large as possible subject to the constraint
that, if one or more blanks occur, the dashes will be at least as long
as the first parameter and the blanks will be at least as long as the
second parameter. If one or more blanks occur, the excess length of the
dashes and of the blanks will be proportional to the third and fourth
parameters respectively. The first two parameters must be positive
dimensions, and the last two parameters must be nonnegative numbers whose
sum is positive.
\drawdotedge{}{<1 or 0>}
\drawdotedge draws the current edge as a dotted line. The number of dots
will be the largest number such that the distance between dots is at least
as large as the first parameter, which must be a positive dimension.
A dot will always appear at the start point, and will appear at the end
point if the second parameter is 1. If the second parameter is 0 then
the final dot will be omitted.
\drawedgehead{}{<1 or 0>}{<1 or 0>}
\drawedgehead draws an arrowhead on the current edge at a distance from
the start point of p times the length of the edge, where p is the first
parameter divided by 100. The arrowhead will point to the end point if
the second parameter is 1, or to the start point if the second parameter
is 0. If the third parameter is 1, the arrowhead will be advanced towards
its tip by the value of the control symbol \edgeheaddisp, which is defined
to be 4pt, but may be redefined by the user.
\abutleft:
{}{}
\abutright:
{}{}
\abutbelow:
{}{}
\abutabove:
{}{}
Each of these macros uses the to initialize
an expression state (in the same way as \vertex) and then executes
the , which must establish a shadow. The material in
the expression state is then placed in the diagram state, at a location
such that the shadow touches the current edge (or its extension as an
infinite line), and lies to the left (or to the right, below, or above,
as determined by the macro name). For \abutleft and \abutright, which
must not be used when the current edge is horizontal, the first parameter
gives the y-coordinate of the point at which the center point is to be
located. For \abutbelow and \abutabove, which must not be used when the
current edge is vertical, the first parameter gives the x-ccordinate.
\abutleftd{}
{}{}
\abutrightd{}
{}{}
\abutbelowd{}
{}{}
\abutaboved{}
{}{}
Each of these macros behaves the same way as its cousin, described above,
except that the first parameter is a dimension instead of a number.
Within an expression program, one can call the following macros:
\leftghost{}
\rightghost{}
These macros change \xcenter (the x-coordinate of the center point).
The is set in an hbox, using math mode,
text style, which is ignored except for its width. \leftghost sets
\xcenter to the left of the current rectangle plus half the width of
the hbox. \rightghost sets \xcenter to the right of the current rectangle
minus half the width of the hbox. The effect is to place the ``ghost
expression'' (invisibly) within the current rectangle at the left or
right side, and to move the center point horizontally to the midpoint of
the ghost expression.
\border{}{}
\borderto{}{}
\symmetrize
These macros enlarge the current rectangle. \border moves the left and
right sides outwards by its first parameter, and raises the top and lowers
the bottom by its second parameter. (If either parameter is negative,
the rectangle will contract.) \borderto enlarges the current rectangle
so that its width is at least the first parameter and its height (including
depth) is at least the second parameter. (Equal amounts will be added at
the left and right, and at the top and bottom.) \symmetrize raises the top
or lowers the bottom so that they are equally distant from the center point.
\place,:
{}{}
\placed{}{}
{}{}
These macros can be called from expression programs as well as diagram
programs. They have no effect on the current rectangle or center point.
\rect
\rect defines the current shadow to be the current rectangle.
\hexagon
\hexagon defines the current shadow to be a hexagon with two horizontal
sides identical with the top and bottom of the current rectangle, and
four sides of slope (+ or - 1), (+ or - 2).
\octagon{}
\octagon defines the current shadow to be an octagon inscribed in the
current rectangle. The horizontal sides and vertical sides are shorter than
those of the current rectangle by twice the parameter, and the remaining
sides have slope (+ or - 1), (+ or - 1).
\diamond
\diamond defines the current shadow to be a square, just large enough to
enclose the current rectangle, whose sides have slope (+ or - 1), (+ or - 1).
\rorect{}{<1 or 0>}{<1 or 0>}
\rorect defines the current shadow to be a rectangle with rounded (i.e.
quarter-circle) corners. The diameter of the corners is determined as
follows:
(1) Take the maximum of:
(a) The first parameter,
(b) If the second parameter is 1, then the width of the current
rectangle, else 0,
(c) If the third parameter is 1, then the height of the current
rectangle, else 0.
(2) Take the diameter of the smallest printable circle larger or equal
to (1), or if no such printable circle exists, take the diameter
of the largest printable circle.
The shadow is then the smallest rounded rectangle with corners of this
diameter such that the corresponding true (unrounded) rectangle encloses
the current rectangle.
The effect (if there is a sufficiently large printable circle) is to produce:
A rounded rectangle 00
A vertical oblong if the second and third parameters are 10
A horizontal oblong 01
A circle 11
If the shadow is drawn (using \outline, as described below) its shape will
be the rounded rectangle just described. However, if the shadow is used
to shade an edge or to abut an expression to an edge or circle, then a
slight fudge occurs: the shadow is taken to be the smallest octagon
(with the same shape as that produced by \octagon) enclosing the specified
rounded rectangle.
\outline
\outline draws the current shadow.
\setcircle{}{}{}
\setcircle defines the current circle to have a diameter given by the first
parameter and a center defined by the second and third parameter.
\shiftcircle{}{}
\shiftcircle displaces the current circle by the vector described by its
parameters.
\drawcircle<1 or 0:upper right quadrant><1 or 0:lower right quadrant>
<1 or 0:lower left quadrant><1 or 0:upper left quadrant>
\drawcircle draws the current circle. More precisely, it draws those
quadrants of the current circle for which the corresponding parameter is 1.
\drawcirclehead{}{}{<1 or 0>}
\drawcirclehead draws an arrowhead on the current circle, at the
intersection with a directed line segment starting at the center with a
slope determined by the first two parameters. If the third parameter
is 1 (0) the arrowhead will point in a clockwise (counterclockwise)
direction. The arrowhead will be advanced towards its tip by the distance
\circleheaddisp. This control symbol is defined to be 2pt, but may be
redefined by the user.
\abutcircleleft{}
{}{}
\abutcircleright{}
{}{}
\abutcirclebelow{}
{}{}
\abutcircleabove{}
{}{}
Each of these macros uses the to initialize
an expression state (in the same way as \vertex) and then executes the
, which must establish a shadow. The material in the
final expression state produced by this program is then placed in the
expression state of the expression program containing the call of
\abutcircle... , at a location such that shadow touches the current circle
on the outside of this circle. For \abutcircleleft and \abutcircleright
the first parameter gives the y-coordinate of the point at which the center
is to be located. For \abutcirclebelow and \abutcircleabove the first
parameter gives the x-coordinate.
Actually, the abutment is approximate. For \abutcircleabove, the shadow
is abutted against three tangents to the current circle, that touch at the
top of the circle and at the two points 45 degrees to the left and right
of the top, and is then given the lowest of the three positions obtained
by these abutments. The other three macros behave similarly.
AN EXAMPLE
As a simple example, consider
$$\diagram{
\vertex 0,100:{A}{\border{3pt}{4pt}\rect}
\vertex 150,100:{B}{\border{3pt}{4pt}\rect}
\vertex 0,0:{A'}{\border{3pt}{4pt}\rect}
\vertex 150,0:{B'}{\border{3pt}{4pt}\rect}
\setedge 0,100,150,100:
\shadeedge
\drawsolidedge
\drawedgehead{100}10
\abutabove 75:{\textstyle c}{\border{2pt}{2pt}\octagon{3pt}}
\setedge 0,0,150,0:
\shadeedge
\drawsolidedge
\drawedgehead{100}10
\abutbelow 75:{\textstyle c'}{\border{2pt}{2pt}\octagon{3pt}}
\setedge 0,100,0,0:
\shadeedge
\drawsolidedge
\drawedgehead{100}10
\abutleft 50:{\textstyle a}{\border{2pt}{2pt}\octagon{3pt}}
\setedge 150,100,150,0:
\shadeedge
\drawsolidedge
\drawedgehead{100}10
\abutright 50:{\textstyle b}{\border{2pt}{2pt}\octagon{3pt}}
}$$
This call of \diagram contains a diagram program in which the four calls
of \vertex place the expressions A, B, A', and B' at the four corners of
a 100pt by 150pt rectangle. Then come four groups of five calls that
draw edges along the sides of this rectangle and abut expressions to
the middles of these edges.
In each group, \setedge determines the position of the edge, \shadeedge
adjusts the end points to exclude the shadows of the expressions that
have been placed at these points by \vertex, \drawsolidedge draws the
edge as a solid line, and \drawedgehead places an arrowhead at the end
of the edge. Then \abut... places an expression above, below, to the
left, or to the right of the midpoint of the edge, so that its shadow
touches the edge.
In the calls of \vertex, {\border{3pt}{4pt}\rect} is an expression program
that enlarges the current rectangle by 3pt at the left and right and by 4pt
at the top and bottom, and then establishes this expanded rectangle as the
shadow. In the calls of \abut... , {\border{2pt}{2pt}\octagon{3pt}} is an
expression program that enlarges the current rectangle by 2pt on each side
and then defines the shadow to be an octagon inscribed in this expanded
rectangle, with slanted edges of length 4.24pt.
The result is a display that looks approximately like:
c
A --------------------> B
| |
| |
| |
a| |b
| |
| |
V V
A'--------------------> B'
c'
(except, of course that the arrows are solid).
THE MACROS FOR CATEGORY-THEORY DIAGRAMS
Now we describe the additional macros oriented towards category-theory
diagrams. The main level program is
\ctdiagram{}
\ctdiagram is similar to \diagram, except that it executes \ctsolid,
\cthead, and \ctoutermid (described below) before the ,
so that the category-theory macros for drawing edges will draw solid edges
with arrowheads and will calculate midpoints of edges before shading or
displacement.
Within a diagram program, one can call the following macros (in addition
to the general macros described previously):
\ctvg,:
{}{}
\ctv,:{}
\ctvg is similar to \vertex, except that:
(1) The is set in \ctvertexstyle.
The control symbol \ctvertexstyle is defined to be \displaystyle,
but may be redefined by the user.
(2) The execution of the is followed by a
``standard expression program'' that enlarges the current rectangle
by \ctvertexborderlr on the left and right and by \ctvertexbordertb
on the top and bottom, and then creates a rectangular shadow of the
same size. The control symbols \ctvertexborderlr and \ctvertexbordertb
are defined to be 3pt and 4pt respectively, but may be redefined
by the user.
\ctv is similar to \ctvg except that only the standard expression program
is executed.
\ctsolid
\ctdash
\ctdot
These macros cause subsequent executions of the edge-drawing macros described
below to draw solid, dashed, or dotted edges respectively. Horizontal and
vertical dashed edges are drawn by \drawdashedge{7pt}{7pt}11, but other
dashed edges are drawn by \drawdashedge{15pt}{7pt}01. Dotted edges are
drawn by \drawdotedge{8pt}1. (These conventions can be alter by redefining
the macros \zzctdrawdashedge and \zzctdrawdotedge.)
\cthead
\ctnohead
\cthead (\ctnohead) causes subsequent executions of the edge-drawing macros
described below to draw (not to draw) arrowheads.
\cten,,,
:
\cten draws an edge from x-start to x-end, after shading the start and end
points with any shadows associated with these points on the vertex list.
The edge will be solid, dashed, or dotted depending upon whether \ctsolid,
\ctdash, or \ctdot was called last. An arrowhead will or will not be placed
at the end point depending upon whether \cthead or \ctnohead was called last.
\ctetg,,,
;:{}
\ctebg,,,
;:{}
\ctelg,,,
;:{}
\cterg,,,
;:{}
Each of these macros draws an edge in the same way as \cten, and then abuts
the to the
top \ctetg
bottom for \ctebg
left \ctelg
right \cterg
of the edge, with its center placed at the x-coordinate (for \ctetg or
\ctebg) or y-coordinate (for \ctelg or \cterg) specified by the fifth
parameter. The abutted expression is set in \ctabutstyle, with an octagonal
shadow (of the shape produced by \octagon). This octagon will be inscribed
in a rectangle obtained by bordering the expression by \ctabutborderlr
on the left and right, and by \ctabutbordertb on the top and bottom;
the length of the slanted sides of the octagon will be \ctabutborderinset
times the square root of 2.
The relevant control symbols are defined to be:
\ctabutstyle \textstyle
\ctabutborderlr 2pt
\ctabutbordertb 2pt
\ctabutborderinset 3pt
These symbols may be redefined by the user, but \ctabutborderinsetdouble
must also be redefined so that its value is twice \ctabutborderinset.
\ctetg and \ctebg should not be used to draw a vertical edge; \ctelg and
\cterg should not be used to draw a horizontal edge.
\ctetbg,,,
;,:
{<1 or 0>}{<1 or 0>}
{}{}
\ctetbg draws a pair of edges in the same manner as \cten and then abuts
the first above the pair, in the same manner
as \ctetg, with its center placed at the x-coordinate specified by the
fifth parameter, and abuts the second below
the pair, in the same manner as \ctebg, with its center placed at the
x-coordinate specified by the sixth parameter. If the seventh parameter
is 1 (and \cthead has been called most recently), the arrowhead on the upper
edge will occur at the end point; otherwise it will occur (pointing
backwards) at the start point. The eighth parameter controls the arrowhead
on the lower edge similarly. The distance between the edges will be twice
the control symbol \ctdoubleedgedisp, which is defined to be 2pt, but may
be redefined by the user.
\ctetbg should not be used to draw a vertical edge.
\ctelrg,,,
;,:
{<1 or 0>}{<1 or 0>}
{}{}
\ctelrg draws a pair of edges in the same manner as \cten and then abuts
the first to the left, in the same manner
as \ctetg, with its center placed at the y-coordinate specified by the
fifth parameter, and abuts the second to
the right, in the same manner as \ctebg, with its center placed at the
y-coordinate specified by the sixth parameter. If the seventh parameter
is 1 (and \cthead has been called most recently), the arrowhead on the left
edge will occur at the end point; otherwise it will occur (pointing
backwards) at the start point. The eighth parameter controls the arrowhead
on the right edge similarly. The distance between the edges will be twice
the control symbol \ctdoubleedgedisp, which is defined to be 2pt, but may
be redefined by the user.
\ctelrg should not be used to draw a horizontal edge.
\ctet,,,
:{}
\cteb,,,
:{}
\ctel,,,
:{}
\cter,,,
:{}
\ctetb,,,
:{<1 or 0>}{<1 or 0>}
{}{}
\ctelr,,,
:{<1 or 0>}{<1 or 0>}
{}{}
These macros behave similarly to their cousins described above, except that
the fifth parameter (and also the sixth parameter in the case of \ctetb and
\ctelr) is omitted. In its place, these macros use the x- or y-coordinate
of the midpoint between the start and end points of the edge. If \ctoutermid
(described below) has been called most recently, then the midpoint will
be calculated from the start and end coordinates given as parameters to the
macros. If \ctinnermid (described below) has been called most recently,
then the midpoint will be computed after displacement and shading, so that
it will be the midpoint of the actual line segment that is printed.
(In the case of \ctetb and \ctelr, this midpoint will be calculated
separately for the two edges that are printed.)
\ctoutermid
\ctinnermid
These macros control the calculation of edge midpoints as described above.
Within a expression program, one can call the following macros (in addition
to the general macros described previously):
\ctgl{}
\ctgr{}
These macros are similar to \leftghost and \rightghost except that the
is set in \ctvertexstyle.
\ctlptl{}
\ctlptr{}
\ctlpbr{}
\ctlpbl{}
These macros print a loop (three quarters of a circle) of diameter
\ctloopdiameter on the exterior of the current rectangle, with its center
at the
top left \ctlptl
top right for \ctlptr
bottom right \ctlpbr
bottom left \ctlpbl
corner of the current rectangle, and with a clockwise arrowhead at the
clockwise end of the loop. Then the is
abutted to the
left \ctlptl
right for \ctlptr
right \ctlpbr
left \ctlpbl
of the loop, with its center
above \ctlptl
above for \ctlptr
below \ctlpbr
below \ctlpbl
the center of the loop by the distance \ctabutcircledisp.
The control symbols \ctloopdiameter and \ctabutcircledisp are defined to be
20pt and 5pt respectively, but may be redefined by the user.
The current rectangle is expanded by \ctvertexborderlr at the left and right
and and \ctvertexbordertb at the top and bottom before the loop center
is determined, and is contracted to its original size afterwards. Thus
the loop center will lie at a corner of the shadow that will be produced
by the ``standard expression program'' executed by \ctvg. (Actually, the
loop center is displaced by \circleheaddisp, so that the tip of the
arrowhead will just touch the shadow.) The arrowhead is always printed,
regardless of the use of \cthead and \ctnohead.
The is set in \ctabutstyle, and is given an
octagonal shadow in the same manner as by \ctetg. The abutment to the loop
is similar to that performed by \abutcircleleft or \abutcircleright.
\ctlptlcc{}
\ctlptrcc{}
\ctlpbrcc{}
\ctlpblcc{}
These macros are similar to their cousins described above, except that a
counterclockwise arrowhead is placed at the counterclockwise end of the loop.
AN EXAMPLE
For example, the following produces the same display as the previous
example:
$$\ctdiagram{
\ctv 0,100:{A}
\ctv 150,100:{B}
\ctv 0,0:{A'}
\ctv 150,0:{B'}
\ctet 0,100,150,100:{c}
\cteb 0,0,150,0:{c'}
\ctel 0,100,0,0:{a}
\cter 150,100,150,0:{b}
}$$
Less trivial examples of the usage of these macros are found in the file
diagmactest.tex.