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author | Norbert Preining <norbert@preining.info> | 2019-09-02 13:46:59 +0900 |
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committer | Norbert Preining <norbert@preining.info> | 2019-09-02 13:46:59 +0900 |
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diff --git a/macros/latex/contrib/diagmac2/doc/diagmac2.pdf b/macros/latex/contrib/diagmac2/doc/diagmac2.pdf Binary files differnew file mode 100644 index 0000000000..a9c041255e --- /dev/null +++ b/macros/latex/contrib/diagmac2/doc/diagmac2.pdf diff --git a/macros/latex/contrib/diagmac2/doc/diagmac2.tex b/macros/latex/contrib/diagmac2/doc/diagmac2.tex new file mode 100644 index 0000000000..42d4e439fc --- /dev/null +++ b/macros/latex/contrib/diagmac2/doc/diagmac2.tex @@ -0,0 +1,1241 @@ +\documentclass[11pt]{article} +\title{User's Manual for Diagram Macros} +\date{December, 1987\\[2ex] Revised for \texttt{diagmac2.sty}\\by Bob Tennent\\ Version 2.1 May, 2009} +\author{J. C. Reynolds} +\usepackage{fullpage} +\usepackage{url} +\usepackage{diagmac2} +\RequirePackage{amssymb} +\RequirePackage{fancyvrb} +\DefineVerbatimEnvironment{myverb}{Verbatim}{baselinestretch=0.95} +\DefineVerbatimEnvironment{smallverb}{Verbatim}{fontsize=\footnotesize,baselinestretch=0.95} +\RequirePackage{doc} +\MakeShortVerb{\"} % "..." is equivalent to \verb"..." +\PassOptionsToPackage{urlcolor=black,linkcolor=blue,colorlinks}{hyperref} +\RequirePackage{hyperref} +\newcommand{\myurl}[1]{\textcolor{blue}{\underline{\textcolor{black}{\url{#1}}}}} + + +\begin{document} +\sloppy +\maketitle +\thispagestyle{empty} +\tableofcontents + +\section{Introduction} +\begin{list}{}{\setlength{\leftmargin}{0.6in}} +\item +\footnotesize +\begin{flushright} + These macros are in the public domain, and have not changed in many + years. Acknowledgement of their usage is not necessary. However, + neither I nor CMU accept any responsibility for the consequences of + errors in these macros or their documentation. This is more than the + usual disclaimer; \TeX\ is a beastly language for programming anything + complex, and I am not an expert in its use, so that there are probably + errors lurking in the macros. +\\ [1ex] +John Reynolds\footnote{\myurl{ftp://ftp.cs.cmu.edu/user/jcr/README}} +\end{flushright} +\end{list} + + + +\enlargethispage*{4ex} +\par\noindent +The file \texttt{diagmac2.sty} 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. + +\texttt{diagmac2.sty} is fully compatible with Reynolds's original \texttt{diagmac}, but +takes advantage of the \texttt{pict2e} implementation of the \texttt{picture} environment +to allow arbitrary slopes for edges and diameters for circles. In +this manual, the sections of the original manual that refered to the +limitations have been deleted. Sections~\ref{extensions} and \ref{examples} are new +in Version~2.1. + +The \LaTeX\ declarations "\thinlines" and "\thicklines" or the "pict2e" declaration +"\linethickness" may be used to vary +the thickness of lines, arrowheads, and circles. + +\cleardoublepage +\section{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: +\begin{itemize} + +\item "\xstart": the $x$-coordinate of the start point +\item "\ystart": the $y$-coordinate of the start point +\item "\xend": the $x$-coordinate of the end point +\item "\yend": the $y$-coordinate of the end point, +\end{itemize} +and two number registers: +\begin{itemize} +\item "\xslope": the $x$-component of the slope +\item "\yslope": the $y$-component of the slope +\end{itemize} +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: +\begin{itemize} +\item "\lexpr": the $x$-coordinate of the left side +\item "\rexpr": the $x$-coordinate of the right side +\item "\texpr": the $y$-coordinate of the top +\item "\bexpr": the $y$-coordinate of the bottom, +\end{itemize} +and a ``center point,'' determined by the two dimension registers: +\begin{itemize} +\item "\xcenter": the $x$-coordinate of the center point +\item "\ycenter": the $y$-coordinate of the center point. +\end{itemize} +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: +\begin{itemize} +\item "\dcircle": the diameter +\item "\xcircle": the $x$-coordinate of the center +\item "\ycircle": the $y$-coordinate of the center. +\end{itemize} +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". + +\cleardoublepage +\section{The General Macros for Diagrams} + +We now describe the general macros for drawing diagrams. The main level +macro is +\begin{myverb} + \diagram{<diagram program>} +\end{myverb} +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: +\begin{myverb} + \vertex<number:x-coord>,<number:y-coord>: + {<balanced mathematical text>}{<expression program>} +\end{myverb} +"\vertex" sets the <balanced mathematical text> 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 "<expression program>" 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 "<number:x-coord>,<number:y-coord>". Finally, if the expression +state contains a current shadow, the point "<number:x-coord>,<number:y-coord>" +is paired with the shadow and placed on the vertex list. +\begin{myverb} + \place<number:x-coord>,<number:y-coord>: + {<balanced mathematical text>}{<expression program>} + + \placed{<dimen:x-coord>}{<dimen:y-coord>} + {<balanced mathematical text>}{<expression program>} +\end{myverb} +"\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. +\begin{myverb} + \setedge<number:x-start-coord>,<number:y-start-coord>, + <number:x-end-coord>,<number:y-end-coord>: +\end{myverb} +"\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". +\begin{myverb} + \shiftedge{<dimen:length>} +\end{myverb} +"\shiftedge" displaces the current edge by a vector whose length is determined +by the "<dimen:length>" parameter, and whose direction is obtained by rotating +the current edge 90 degrees counterclockwise. +\begin{myverb} + \shadeedge +\end{myverb} +"\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. +\begin{myverb} + \drawsolidedge +\end{myverb} +"\drawsolidedge" draws the current edge as a solid line. +\begin{myverb} + \drawdashedge{<dimen:length>}{<dimen:length>}{<number>}{<number>} +\end{myverb} +"\drawdashedge" draws the current edge as a dashed line. 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. +\begin{myverb} + \drawdotedge{<dimen:length>}{<1 or 0>} +\end{myverb} +"\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. +\begin{myverb} + \drawedgehead{<number:0 to 100>}{<1 or 0>}{<1 or 0>} +\end{myverb} +"\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. +\begin{myverb} + \abutleft<number:y-coord>: + {<balanced mathematical text>}{<expression program>} + + \abutright<number:y-coord>: + {<balanced mathematical text>}{<expression program>} + + \abutbelow<number:x-coord>: + {<balanced mathematical text>}{<expression program>} + + \abutabove<number:x-coord>: + {<balanced mathematical text>}{<expression program>} +\end{myverb} +Each of these macros uses the "<balanced mathematical text>" to initialize +an expression state (in the same way as "\vertex") and then executes +the "<expression program>", 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 $x$-coordinate. +\begin{myverb} + \abutleftd{<dimen:y-coord>} + {<balanced mathematical text>}{<expression program>} + + \abutrightd{<dimen:y-coord>} + {<balanced mathematical text>}{<expression program>} + + \abutbelowd{<dimen:x-coord>} + {<balanced mathematical text>}{<expression program>} + + \abutaboved{<dimen:x-coord>} + {<balanced mathematical text>}{<expression program>} +\end{myverb} +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: +\begin{myverb} + \leftghost{<balanced mathematical text>} + + \rightghost{<balanced mathematical text>} +\end{myverb} +These macros change "\xcenter" (the $x$-coordinate of the center point). +The "<balanced mathematical text>" 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. +\begin{myverb} + \border{<dimen:x-length>}{<dimen:y-length>} + + \borderto{<dimen:x-length>}{<dimen:y-length>} + + \symmetrize +\end{myverb} +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. +\begin{myverb} + \place<number:x-coord>,<number:y-coord>: + {<balanced mathematical text>}{<expression program>} + + \placed{<dimen:x-coord>}{<dimen:y-coord>} + {<balanced mathematical text>}{<expression program>} +\end{myverb} +These macros can be called from expression programs as well as diagram +programs. They have no effect on the current rectangle or center point. +\begin{myverb} + \rect +\end{myverb} +"\rect" defines the current shadow to be the current rectangle. +\begin{myverb} + \hexagon +\end{myverb} +"\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). +\begin{myverb} + \octagon{<dimen:length>} +\end{myverb} +"\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). +\begin{myverb} + \diamond +\end{myverb} +"\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). +\begin{myverb} + \rorect{<dimen:diameter>}{<1 or 0>}{<1 or 0>} +\end{myverb} +"\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. +\begin{enumerate} + \item \label{one} Take the maximum of: +\begin{enumerate} + \item the first parameter; + \item if the second parameter is 1, then the width of the current + rectangle, else 0; + \item if the third parameter is 1, then the height of the current + rectangle, else 0. +\end{enumerate} + + \item Take the diameter of the smallest printable circle larger or equal + to (\ref{one}), or if no such printable circle exists, take the diameter + of the largest printable circle. +\end{enumerate} +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: +\begin{verse} +\begin{tabular}{lcc} + & if the 2nd parameter is & and the 3rd parameter is\\ +\hline + a rounded rectangle & 0& 0 \\ + a vertical oblong & 1& 0 \\ + a horizontal oblong & 0& 1 \\ + a circle & 1& 1 + +\end{tabular} +\end{verse} +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. +\begin{myverb} + \outline +\end{myverb} +"\outline" draws the current shadow. +\begin{myverb} + \setcircle{<dimen:diameter>}{<dimen:x-coord>}{<dimen:y-coord>} +\end{myverb} +"\setcircle" defines the current circle to have a diameter given by the first +parameter and a center defined by the second and third parameter. +\begin{myverb} + \shiftcircle{<dimen:x-length>}{<dimen:y-length>} +\end{myverb} +"\shiftcircle" displaces the current circle by the vector described by its +parameters. +\begin{myverb} + \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> +\end{myverb} +"\drawcircle" draws the current circle. More precisely, it draws those +quadrants of the current circle for which the corresponding parameter is 1. +\begin{myverb} + \drawcirclehead{<number:x-slope>}{<number:y-slope>}{<1 or 0>} +\end{myverb} +"\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. +\begin{myverb} + \abutcircleleft{<dimen:y-length>} + {<balanced mathematical text>}{<expression program>} + + \abutcircleright{<dimen:y-length>} + {<balanced mathematical text>}{<expression program>} + + \abutcirclebelow{<dimen:x-length>} + {<balanced mathematical text>}{<expression program>} + + \abutcircleabove{<dimen:x-length>} + {<balanced mathematical text>}{<expression program>} +\end{myverb} +Each of these macros uses the "<balanced mathematical text>" to initialize +an expression state (in the same way as "\vertex") and then executes the +"<expression program>", 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. + +\paragraph{An Example} + +Consider the simple example in Figure~\ref{example1}. +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. + +\begin{figure}[t] +\begin{smallverb} + \[ + \diagram{ + \vertex 0,100:{A}{\border{3pt}{4pt}\rect} + \vertex 150,100:{B}{\border{3pt}{4pt}\rect} + \vertex 0,0:{A'}{\border{3pt}{4pt}\rect} + \vertex 150,0:{B'}{\border{3pt}{4pt}\rect} + + \setedge 0,100,150,100: + \shadeedge + \drawsolidedge + \drawedgehead{100}10 + \abutabove 75:{\textstyle c}{\border{2pt}{2pt}\octagon{3pt}} + + \setedge 0,0,150,0: + \shadeedge + \drawsolidedge + \drawedgehead{100}10 + \abutbelow 75:{\textstyle c'}{\border{2pt}{2pt}\octagon{3pt}} + + \setedge 0,100,0,0: + \shadeedge + \drawsolidedge + \drawedgehead{100}10 + \abutleft 50:{\textstyle a}{\border{2pt}{2pt}\octagon{3pt}} + + \setedge 150,100,150,0: + \shadeedge + \drawsolidedge + \drawedgehead{100}10 + \abutright 50:{\textstyle b}{\border{2pt}{2pt}\octagon{3pt}} + } + \] +\end{smallverb} +\caption{A Simple Example} +\label{example1} +\end{figure} +The result is as follows: +\[ +\diagram{ +\vertex 0,100:{A}{\border{3pt}{4pt}\rect} +\vertex 150,100:{B}{\border{3pt}{4pt}\rect} +\vertex 0,0:{A'}{\border{3pt}{4pt}\rect} +\vertex 150,0:{B'}{\border{3pt}{4pt}\rect} +\setedge 0,100,150,100: +\shadeedge +\drawsolidedge +\drawedgehead{100}10 +\abutabove 75:{\textstyle c}{\border{2pt}{2pt}\octagon{3pt}} +\setedge 0,0,150,0: +\shadeedge +\drawsolidedge +\drawedgehead{100}10 +\abutbelow 75:{\textstyle c'}{\border{2pt}{2pt}\octagon{3pt}} +\setedge 0,100,0,0: +\shadeedge +\drawsolidedge +\drawedgehead{100}10 +\abutleft 50:{\textstyle a}{\border{2pt}{2pt}\octagon{3pt}} +\setedge 150,100,150,0: +\shadeedge +\drawsolidedge +\drawedgehead{100}10 +\abutright 50:{\textstyle b}{\border{2pt}{2pt}\octagon{3pt}} +} +\] + +\cleardoublepage +\section{The Macros for Category-Theory Diagrams} + +Now we describe the additional macros oriented towards category-theory +diagrams. The main level program is +\begin{myverb} + \ctdiagram{<diagram program>} +\end{myverb} +"\ctdiagram" is similar to "\diagram", except that it executes "\ctsolid", +"\cthead", and "\ctoutermid" (described below) before the "<diagram program>", +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): +\begin{myverb} + \ctvg<number:x-coord>,<number:y-coord>: + {<balanced mathematical text>}{<expression program>} + + \ctv<number:x-coord>,<number:y-coord>:{<balanced mathematical text>} +\end{myverb} +"\ctvg" is similar to "\vertex", except that: +\begin{enumerate} + \item The "<balanced mathematical text>" is set in "\ctvertexstyle". + The control symbol "\ctvertexstyle" is defined to be "\displaystyle", + but may be redefined by the user. + + \item The execution of the "<expression program>" 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. +\end{enumerate} +"\ctv" is similar to "\ctvg" except that only the standard expression program +is executed. +\begin{myverb} + \ctsolid + + \ctdash + + \ctdot +\end{myverb} +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 altered by redefining +the macros "\zzctdrawdashedge" and "\zzctdrawdotedge".) +\begin{myverb} + \cthead + + \ctnohead +\end{myverb} +"\cthead" ("\ctnohead") causes subsequent executions of the edge-drawing macros +described below to draw (not to draw) arrowheads. +\begin{myverb} + \cten<number:x-start-coord>,<number:y-start-coord>,<number:x-end-coord>, + <number:y-end-coord>: +\end{myverb} +"\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. +\begin{myverb} + \ctetg<number:x-start-coord>,<number:y-start-coord>,<number:x-end-coord>, + <number:y-end-coord>;<number:x-coord>:{<balanced mathematical text>} + + \ctebg<number:x-start-coord>,<number:y-start-coord>,<number:x-end-coord>, + <number:y-end-coord>;<number:x-coord>:{<balanced mathematical text>} + + \ctelg<number:x-start-coord>,<number:y-start-coord>,<number:x-end-coord>, + <number:y-end-coord>;<number:y-coord>:{<balanced mathematical text>} + + \cterg<number:x-start-coord>,<number:y-start-coord>,<number:x-end-coord>, + <number:y-end-coord>;<number:y-coord>:{<balanced mathematical text>} +\end{myverb} +Each of these macros draws an edge in the same way as "\cten", and then abuts +the "<balanced mathematical text>" to the +\begin{verse} +\begin{tabular}{lll} + top & for & "\ctetg" \\ + bottom & for & "\ctebg" \\ + left & for & "\ctelg" \\ + right & for & "\cterg" \\ +\end{tabular} +\end{verse} +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: +\begin{verse} +\begin{tabular}{lc} + + "\ctabutstyle" & "\textstyle" \\ + "\ctabutborderlr" & 2pt \\ + "\ctabutbordertb" & 2pt \\ + "\ctabutborderinset" & 3pt \\ +\end{tabular} +\end{verse} +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. +\begin{myverb} + \ctetbg<number:x-start-coord>,<number:y-start-coord>,<number:x-end-coord>, + <number:y-end-coord>;<number:x-coord>,<number:x-coord>: + {<1 or 0>}{<1 or 0>} + {<balanced mathematical text>}{<balanced mathematical text>} +\end{myverb} +"\ctetbg" draws a pair of edges in the same manner as "\cten" and then abuts +the first "<balanced mathematical text>" 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 "<balanced mathematical text>" 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. +\begin{myverb} + \ctelrg<number:x-start-coord>,<number:y-start-coord>,<number:x-end-coord>, + <number:y-end-coord>;<number:y-coord>,<number:y-coord>: + {<1 or 0>}{<1 or 0>} + {<balanced mathematical text>}{<balanced mathematical text>} +\end{myverb} +"\ctelrg" draws a pair of edges in the same manner as "\cten" and then abuts +the first "<balanced mathematical text>" 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 "<balanced mathematical text>" 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. +\begin{myverb} + \ctet<number:x-start-coord>,<number:y-start-coord>,<number:x-end-coord>, + <number:y-end-coord>:{<balanced mathematical text>} + + \cteb<number:x-start-coord>,<number:y-start-coord>,<number:x-end-coord>, + <number:y-end-coord>:{<balanced mathematical text>} + + \ctel<number:x-start-coord>,<number:y-start-coord>,<number:x-end-coord>, + <number:y-end-coord>:{<balanced mathematical text>} + + \cter<number:x-start-coord>,<number:y-start-coord>,<number:x-end-coord>, + <number:y-end-coord>:{<balanced mathematical text>} + + \ctetb<number:x-start-coord>,<number:y-start-coord>,<number:x-end-coord>, + <number:y-end-coord>:{<1 or 0>}{<1 or 0>} + {<balanced mathematical text>}{<balanced mathematical text>} + + \ctelr<number:x-start-coord>,<number:y-start-coord>,<number:x-end-coord>, + <number:y-end-coord>:{<1 or 0>}{<1 or 0>} + {<balanced mathematical text>}{<balanced mathematical text>} +\end{myverb} +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.) +\begin{myverb} + \ctoutermid + + \ctinnermid +\end{myverb} +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): +\begin{myverb} + \ctgl{<balanced mathematical text>} + + \ctgr{<balanced mathematical text>} +\end{myverb} +These macros are similar to "\leftghost" and "\rightghost" except that the +"<balanced mathematical text>" is set in "\ctvertexstyle". +\begin{myverb} + \ctlptl{<balanced mathematical text>} + + \ctlptr{<balanced mathematical text>} + + \ctlpbr{<balanced mathematical text>} + + \ctlpbl{<balanced mathematical text>} +\end{myverb} +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 +\begin{verse} +\begin{tabular}{lll} + + top left & for & "\ctlptl" \\ + top right & for & "\ctlptr" \\ + bottom right & for & "\ctlpbr" \\ + bottom left & for & "\ctlpbl" +\end{tabular} +\end{verse} +corner of the current rectangle, and with a clockwise arrowhead at the +clockwise end of the loop. Then the "<balanced mathematical text>" is +abutted to the +\begin{verse} +\begin{tabular}{lll} + + left & for & "\ctlptl" \\ + right & for & "\ctlptr" \\ + right & for & "\ctlpbr" \\ + left & for & "\ctlpbl" +\end{tabular} +\end{verse} +of the loop, with its center +\begin{verse} +\begin{tabular}{lll} + + above & for & "\ctlptl" \\ + above & for & "\ctlptr" \\ + below & for & "\ctlpbr" \\ + below & for & "\ctlpbl" +\end{tabular} +\end{verse} +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 by "\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 "<balanced mathematical text>" 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". +\begin{myverb} + \ctlptlcc{<balanced mathematical text>} + + \ctlptrcc{<balanced mathematical text>} + + \ctlpbrcc{<balanced mathematical text>} + + \ctlpblcc{<balanced mathematical text>} +\end{myverb} +These macros are similar to their cousins described above, except that a +counterclockwise arrowhead is placed at the counterclockwise end of the loop. + +\paragraph{An Example} + +The following program produces the same display as the previous +example. +\begin{myverb} + \[ + \ctdiagram{ + \ctv 0,100:{A} + \ctv 150,100:{B} + \ctv 0,0:{A'} + \ctv 150,0:{B'} + \ctet 0,100,150,100:{c} + \cteb 0,0,150,0:{c'} + \ctel 0,100,0,0:{a} + \cter 150,100,150,0:{b} + } + \] +\end{myverb} +Less trivial examples of the usage of these macros are found in Section~\ref{examples} and +in "diagmactest.tex". + +\cleardoublepage +\section{Extensions} +\label{extensions} + +This section describes two macros that have been +added to "diagmac2.sty" +by Bob Tennent (and are not in Reynolds's original "diagmac"). + +\begin{myverb} + \drawedgebar +\end{myverb} +"\drawedgebar" draws a bar across the end of the current edge. +\begin{myverb} + \ctec<number:x-start-coord>,<number:y-start-coord>, + <number:x-end-coord>,<number:y-end-coord>, + <number:x-ctrl-coord>,<number:y-ctrl-coord>:{<balanced mathematical text>} +\end{myverb} +"\ctec" draws a bezier-curve edge from the start point to the end point, using the +third point as a control point. The "<balanced mathematical text>" is centered at +the control point. + +\section{Examples} +\label{examples} + +\begin{minipage}[t]{2.9in} + \[ +\ctdiagram{ + \def\diagramunit{0.8pt} + \ctinnermid + \ctv 0,100:{a} + \ctv 150,100:{f(a)} + \ctv 0,0:{g(a)} + \ctv 150,0:{f(g(a))=g(f(a))} + \ctet 0,100,150,100:{f} + \drawedgebar + \cteb 0,0,150,0:{f} + \drawedgebar + \ctel 0,100,0,0:{g} + \drawedgebar + \cter 150,100,150,0:{g} + \drawedgebar +} +\] +\end{minipage} +\qquad +\begin{minipage}[t]{3.0in} +\begin{smallverb} +\[ +\ctdiagram{ + \def\diagramunit{0.8pt} + \ctinnermid + \ctv 0,100:{a} + \ctv 150,100:{f(a)} + \ctv 0,0:{g(a)} + \ctv 150,0:{f(g(a))=g(f(a))} + \ctet 0,100,150,100:{f} + \drawedgebar + \cteb 0,0,150,0:{f} + \drawedgebar + \ctel 0,100,0,0:{g} + \drawedgebar + \cter 150,100,150,0:{g} + \drawedgebar +} +\] +\end{smallverb} +\end{minipage} +\\[4ex] + +\begin{minipage}[t]{3.1in} +\[ +\ctdiagram{ +\def\diagramunit{1.1pt} +\ctv 0,0: {I\otimes v} +\ctv 0,60: {(I\otimes I)\otimes v} +\ctv -80,60: {I\otimes (I\otimes v)} +\ctv 80,60: {I\otimes (I\otimes v)} +\ctelg -80,60,0,0;25:{I\otimes\lambda_v\!} +\cterg 80,60,0,0;25:{\,\lambda{_I\otimes v}} +\ctec -80,60,80,60,0,80:{\mathsf{id}} +\cteb -80,60,0,60:{\alpha^{-1}} +\cteb 0,60,80,60:{\alpha^{-1}} +\ctv 0,36:{\rho_I\otimes v = + \lambda_I\otimes v} +\ctnohead +\cten 0,60,0,36: +\cthead +\cten 0,36,0,0: +} +\] +\end{minipage} +\qquad +\begin{minipage}[t]{3.0in} +\begin{smallverb} +\[ +\ctdiagram{ + \ctv 0,0: {I\otimes v} + \ctv 0,60: {(I\otimes I)\otimes v} + \ctv -80,60: {I\otimes (I\otimes v)} + \ctv 80,60: {I\otimes (I\otimes v)} + \ctelg -80,60,0,0;25:{I\otimes\lambda_v\!} + \cterg 80,60,0,0;25:{\,\lambda{_I\otimes v}} + \ctec -80,60,80,60,0,80:{\mathsf{id}} + \cteb -80,60,0,60:{\alpha^{-1}} + \cteb 0,60,80,60:{\alpha^{-1}} + \ctv 0,36:{\rho_I\otimes v = + \lambda_I\otimes v} + \ctnohead + \cten 0,60,0,36: + \cthead + \cten 0,36,0,0: +} +\] +\end{smallverb} +\end{minipage} + +\clearpage + +\begin{minipage}[t]{3.0in} +\[ +\ctdiagram{ +\ctv -60,0:{V} +\ctv 60,0:{E} +\ctv 0,0: {I} +\cten 0,0,60,0: +\ctnohead +\cten -60,0,0,0: +\ctet -50,20,50,20:{\delta_0} +\setcircle{20pt}{\xstart}{\ystart} +\shiftcircle{0pt}{-10pt} +\drawcircle0001 +\drawcirclehead{-1}{0}{0} +\setcircle{20pt}{\xend}{\yend} +\shiftcircle{0pt}{-10pt} +\drawcircle1000 +\cteb -50,-18,50,-18:{\delta_1} +\setcircle{20pt}{\xstart}{\ystart} +\shiftcircle{0pt}{10pt} +\drawcircle0010 +\drawcirclehead{-1}{0}{1} +\setcircle{20pt}{\xend}{\yend} +\shiftcircle{0pt}{10pt} +\drawcircle0100 +} +\] +\end{minipage} +\qquad +\begin{minipage}[t]{3.0in} +\begin{smallverb} +\[ +\ctdiagram{ + \ctv -60,0:{V} + \ctv 60,0:{E} + \ctv 0,0: {I} + \cten 0,0,60,0: + \ctnohead + \cten -60,0,0,0: + \ctet -50,20,50,20:{\delta_0} + \setcircle{20pt}{\xstart}{\ystart} + \shiftcircle{0pt}{-10pt} + \drawcircle0001 + \drawcirclehead{-1}{0}{0} + \setcircle{20pt}{\xend}{\yend} + \shiftcircle{0pt}{-10pt} + \drawcircle1000 + \cteb -50,-18,50,-18:{\delta_1} + \setcircle{20pt}{\xstart}{\ystart} + \shiftcircle{0pt}{10pt} + \drawcircle0010 + \drawcirclehead{-1}{0}{1} + \setcircle{20pt}{\xend}{\yend} + \shiftcircle{0pt}{10pt} + \drawcircle0100 +} +\] +\end{smallverb} +\end{minipage} +\\[4ex] + +\begin{minipage}[t]{3.0in} +\[ +\ctdiagram{ +\ctv 0,0:{\widetilde{B}} +\ctv 60,0:{B} +\ctv 0,60:{E_b} +\ctv 60,60:{E} +\ctinnermid +\cteb 0,0,60,0:{b} +\cter 0,60,0,0:{q_b} +\cteb 0,60,60,60:{\overline{b}} +\cter 60,60,60,0:{q} +\ctv -40,100: {\widetilde{E}} +\ctec -40,100,0,0,-40,50:{\tilde{q}} +\ctec -40,100,60,60,10,100:{p} +\def\ctvertexborderlr{1pt} +\def\ctvertexbordertb{1pt} +\ctv -20,80:{\hat{p}} +\ctnohead\ctdot +\def\zzctdrawdotedge{\drawdotedge{2.5pt}1} +\cten -40,100,-20,80: +\cthead +\def\zzctdrawdotedge{\drawdotedge{2.5pt}0} +\cten -20,80,-6,66: +\ctv 5,55:{\mbox{\Large$\lrcorner$}} +} +\] +\end{minipage} +\qquad +\begin{minipage}[t]{3.0in} +\begin{smallverb} +\[ +\ctdiagram{ + \ctv 0,0:{\widetilde{B}} + \ctv 60,0:{B} + \ctv 0,60:{E_b} + \ctv 60,60:{E} + \ctinnermid + \cteb 0,0,60,0:{b} + \cter 0,60,0,0:{q_b} + \cteb 0,60,60,60:{\overline{b}} + \cter 60,60,60,0:{q} + \ctv -40,100: {\widetilde{E}} + \ctec -40,100,0,0,-40,50:{\tilde{q}} + \ctec -40,100,60,60,10,100:{p} + \def\ctvertexborderlr{1pt} + \def\ctvertexbordertb{1pt} + \ctv -20,80:{\hat{p}} + \ctnohead\ctdot + \def\zzctdrawdotedge{\drawdotedge{2.5pt}1} + \cten -40,100,-20,80: + \cthead + \def\zzctdrawdotedge{\drawdotedge{2.5pt}0} + \cten -20,80,-6,66: + \ctv 5,55:{\mbox{\Large$\lrcorner$}} +} +\] +\end{smallverb} +\end{minipage} + +\clearpage +\vspace*{-4ex} +\[ +\ctdiagram{ +\ctinnermid +\ctv 0,0: {(I\otimes I) \otimes(v\otimes w)} +\ctv 0,60: {I\otimes \bigl(I\otimes(v\otimes w)\bigr)} +\ctv 0,120: {I\otimes \bigl((I\otimes v)\otimes w\bigr)} +\ctv 80,180: {I\otimes\bigl((v\otimes I)\otimes w\bigr)} +\ctv 200,0: {I\otimes (v\otimes w)} +\ctv 200,180:{I\otimes \bigl(v\otimes(I\otimes w)\bigr)} +\ctv 280,0:{v\otimes w} +\ctv 280,60:{(I\otimes v)\otimes w} +\ctv 280,120: { (I\otimes v)\otimes(I\otimes w)} +\ctel 0,0,0,60:{\alpha^{-1}} +\ctel 0,60,0,120:{I\otimes \alpha^{-1}} +\ctet 0,0,200,0:{\rho_I\otimes(v\otimes v)} +\ctet 200,0,280,0:{\lambda_{v\otimes w}} +\ctelg 0,120,80,180;155:{I\otimes\sigma_{I,v}\otimes w} +\ctet 80,180,200,180:{I\otimes \alpha^{-1}} +\cterg 200,180,280,120;155:{\alpha^{-1}} +\cter 280,120,280,60:{(I\otimes v)\otimes \lambda_w} +\cter 280,60,280,0: {\lambda_v\otimes w} +\ctv 240,30:{\alpha^{-1}} +\ctnohead +\cten 200,0,240,30: +\cthead +\cten 240,30,280,60: +\ctv 100,30:{I\otimes\lambda_{v\otimes w}} +\ctnohead +\cten 0,60,100,30: +\cthead +\cten 100,30,200,0: +\ctv 100,60:{I\otimes(\lambda_v\otimes w)} +\ctnohead +\cten 0,120,100,60: +\cthead +\cten 100,60,200,0: +\ctv 120,120:{I\otimes (\rho_v\otimes w)} +\ctnohead +\cten 80,180,120,120: +\cthead +\cten 120,120,200,0: +\ctv 200,90:{I\otimes (v\otimes \lambda_w)} +\ctnohead +\cten 200,180,200,90: +\cthead +\cten 200,90,200,0: +} +\] +\\ +\begin{smallverb} + \[ + \ctdiagram{\ctinnermid + \ctv 0,0: {(I\otimes I) \otimes(v\otimes w)} + \ctv 0,60: {I\otimes \bigl(I\otimes(v\otimes w)\bigr)} + \ctv 0,120: {I\otimes \bigl((I\otimes v)\otimes w\bigr)} + \ctv 80,180: {I\otimes\bigl((v\otimes I)\otimes w\bigr)} + \ctv 200,0: {I\otimes (v\otimes w)} + \ctv 200,180:{I\otimes \bigl(v\otimes(I\otimes w)\bigr)} + \ctv 280,0:{v\otimes w} + \ctv 280,60:{(I\otimes v)\otimes w} + \ctv 280,120: { (I\otimes v)\otimes(I\otimes w)} + \ctel 0,0,0,60:{\alpha^{-1}} + \ctel 0,60,0,120:{I\otimes \alpha^{-1}} + \ctet 0,0,200,0:{\rho_I\otimes(v\otimes v)} + \ctet 200,0,280,0:{\lambda_{v\otimes w}} + \ctelg 0,120,80,180;155:{I\otimes\sigma_{I,v}\otimes w} + \ctet 80,180,200,180:{I\otimes \alpha^{-1}} + \cterg 200,180,280,120;155:{\alpha^{-1}} + \cter 280,120,280,60:{(I\otimes v)\otimes \lambda_w} + \cter 280,60,280,0: {\lambda_v\otimes w} + \ctv 240,30:{\alpha^{-1}} + \ctnohead\cten 200,0,240,30: + \cthead\cten 240,30,280,60: + \ctv 100,30:{I\otimes\lambda_{v\otimes w}} + \ctnohead\cten 0,60,100,30: + \cthead\cten 100,30,200,0: + \ctv 100,60:{I\otimes(\lambda_v\otimes w)} + \ctnohead\cten 0,120,100,60: + \cthead \cten 100,60,200,0: + \ctv 120,120:{I\otimes (\rho_v\otimes w)} + \ctnohead\cten 80,180,120,120: + \cthead\cten 120,120,200,0: + \ctv 200,90:{I\otimes (v\otimes \lambda_w)} + \ctnohead\cten 200,180,200,90: + \cthead\cten 200,90,200,0: + } + \] +\end{smallverb} + +\clearpage + +\newcommand{\op}{\mathsf{op}} +\newcommand{\vnat}{\Downarrow\mskip-\medmuskip} +\[ +\ctdiagram{ +\ctinnermid +\ctv 0,0:{W^\op\times W^\op} +\ctv 180,0:{S\times S} +\ctv 0,60:{RW^\op} +\ctv 180,60:{RS} +\ctel 0,60,0,0:{\mathit{rw}^\op} +\cter 180,60,180,0:{\mathit{rs}} +\def\ctdoubleedgedisp{6.5pt} +\ctetb 0,60,180,60:11{\widetilde{F}}{\widetilde{G}} +\ctv 98,60:{\vnat\widetilde{\eta}} +\ctetb 0,0,180,0:11{F_0\times F_1}{G_0\times G_1} +\ctv 112,0:{\vnat\eta_0\times\eta_1} +} +\] +\\[4ex] + +\begin{smallverb} + \newcommand{\op}{\mathsf{op}} + \newcommand{\vnat}{\Downarrow\mskip-\medmuskip} + \[ + \ctdiagram{ + \ctinnermid + \ctv 0,0:{W^\op\times W^\op} + \ctv 180,0:{S\times S} + \ctv 0,60:{RW^\op} + \ctv 180,60:{RS} + \ctel 0,60,0,0:{\mathit{rw}^\op} + \cter 180,60,180,0:{\mathit{rs}} + \def\ctdoubleedgedisp{6.5pt} + \ctetb 0,60,180,60:11{\widetilde{F}}{\widetilde{G}} + \ctv 98,60:{\vnat\widetilde{\eta}} + \ctetb 0,0,180,0:11{F_0\times F_1}{G_0\times G_1} + \ctv 112,0:{\vnat\eta_0\times\eta_1} + } + \] +\end{smallverb} + + +See also the examples in Reynolds's "diagmactest". + + + +\end{document} diff --git a/macros/latex/contrib/diagmac2/doc/diagmactest.pdf b/macros/latex/contrib/diagmac2/doc/diagmactest.pdf Binary files differnew file mode 100644 index 0000000000..ec4f0136fa --- /dev/null +++ b/macros/latex/contrib/diagmac2/doc/diagmactest.pdf diff --git a/macros/latex/contrib/diagmac2/doc/diagmactest.tex b/macros/latex/contrib/diagmac2/doc/diagmactest.tex new file mode 100644 index 0000000000..a4ee5e99f4 --- /dev/null +++ b/macros/latex/contrib/diagmac2/doc/diagmactest.tex @@ -0,0 +1,403 @@ +%TESTS OF DIAGRAM MACROS - J. C. Reynolds - December 1987 + +%This is an input file for LATEX that inputs the macros in diagmac.tex +%and tests them. A user's manual for these macros is in diagmac.doc + +\documentclass[12pt]{article} +\input diagmac +\oddsidemargin=0in +\evensidemargin=0in +\textwidth=6.5in +\begin{document} + +\thispagestyle{empty} + +\begin{centering} +{\large\bf TESTS OF DIAGRAM MACROS} \\[14 pt] +\today \\[21 pt] +\end{centering} + + +%These are the two examples given in the user's manual. + +$$\diagram{ +\vertex 0,100:{A}{\border{3pt}{4pt}\rect} +\vertex 150,100:{B}{\border{3pt}{4pt}\rect} +\vertex 0,0:{A'}{\border{3pt}{4pt}\rect} +\vertex 150,0:{B'}{\border{3pt}{4pt}\rect} +\setedge 0,100,150,100: +\shadeedge +\drawsolidedge +\drawedgehead{100}10 +\abutabove 75:{\textstyle c}{\border{2pt}{2pt}\octagon{3pt}} +\setedge 0,0,150,0: +\shadeedge +\drawsolidedge +\drawedgehead{100}10 +\abutbelow 75:{\textstyle c'}{\border{2pt}{2pt}\octagon{3pt}} +\setedge 0,100,0,0: +\shadeedge +\drawsolidedge +\drawedgehead{100}10 +\abutleft 50:{\textstyle a}{\border{2pt}{2pt}\octagon{3pt}} +\setedge 150,100,150,0: +\shadeedge +\drawsolidedge +\drawedgehead{100}10 +\abutright 50:{\textstyle b}{\border{2pt}{2pt}\octagon{3pt}} +}$$ + +$$\ctdiagram{ +\ctv 0,100:{A} +\ctv 150,100:{B} +\ctv 0,0:{A'} +\ctv 150,0:{B'} +\ctet 0,100,150,100:{c} +\cteb 0,0,150,0:{c'} +\ctel 0,100,0,0:{a} +\cter 150,100,150,0:{b} +}$$ + +\newpage + +%This gives a thorough workout to the general macros for diagrams. +%The result looks like an eye-chart for Martians. + +$$\diagram{ +\vertex -150,0:{X+Y}{\border{4pt}{3pt}\rorect{2pt}01\outline} +\vertex 0,-50:Y{\border{10pt}{10pt}\hexagon\outline} +\vertex 150,0:\sum{\border{10pt}{10pt}\octagon{10pt}\outline + \border{5pt}{5pt}\octagon{12pt}\thicklines\outline\thinlines} +\vertex -100,150:\alpha{\border{4pt}{3pt}\diamond\outline} +\vertex 100,150:\sum{\border{10pt}{10pt}\rorect{20pt}00\outline + \border{5pt}{5pt}\rorect{24pt}00\thicklines\outline\thinlines} +\vertex 0,200:{X^2+Y^2}{\border{4pt}{3pt}\rect\outline} +\place -150,-150:{X+Y^{Z^2}} + {\leftghost X\symmetrize\borderto{0pt}{0pt}\border{4pt}{3pt}\rect\outline + \setcircle{16pt}{\xcenter}{\bexpr}\drawcircle0110 + \drawcirclehead{0}{-1}1 + \abutcirclebelow{-10pt}\alpha{\border{2pt}{2pt}\rect\outline} + \abutcirclebelow{10pt}\alpha{\border{2pt}{2pt}\rect\outline}} +\placed{150pt}{-150pt}{X+Y} + {\rightghost Y\symmetrize\borderto{0pt}{26pt}\border{4pt}{0pt}\rect\outline + \placed{\lexpr}{\ycenter}{\vrule height3.2pt depth-2.8pt width10pt}{} + \place 0,-3:{\vrule height3.2pt depth-2.8pt width10pt}{\xcenter=\lexpr} + \setcircle{16pt}{\rexpr}{\texpr}\shiftcircle{8pt}{8pt}\drawcircle1101 + \drawcirclehead{0}{-1}1\drawcirclehead{-1}00 + \abutcircleabove{0pt}\alpha{\border{2pt}{2pt}\rect\outline}} +\vertex 0,-150:{}{\setcircle{40pt}{\xcenter}{\ycenter}\drawcircle1111 + \drawcirclehead231\drawcirclehead{-2}30 + \drawcirclehead6{-9}0\drawcirclehead{-4}{-6}1 + \abutcircleleft{0pt}\alpha{\border{2pt}{2pt}\rect\outline} + \abutcircleright{20pt}\alpha{\border{2pt}{2pt}\rect\outline} + \abutcircleright{0pt}\alpha{\border{2pt}{2pt}\rect\outline} + \abutcircleright{-20pt}\alpha{\border{2pt}{2pt}\rect\outline}} +\setedge 0,200,-100,150:\shadeedge\drawsolidedge\drawedgehead{100}10 + \abutleft 185:{\alpha+\beta}{\border{2pt}{2pt}\rorect{5pt}01\outline} +\setedge 0,200,100,150:\shadeedge\drawsolidedge\drawedgehead{100}10 + \abutright 185:{\alpha+\beta}{\border{2pt}{2pt}\rorect{5pt}01\outline} +\setedge -150,0,0,-50:\shadeedge\drawdashedge{11pt}{10pt}01\drawedgehead{80}01 + \abutleftd{-25pt}{\alpha\beta} + {\border{2pt}{2pt}\borderto{25pt}{0pt}\rect\outline} +\setedge -150,0,150,0:\drawedgehead{50}01\shadeedge\drawsolidedge + \abutabove -10:\rho{\border{2pt}{2pt}\diamond\outline} +\setedge -150,0,-100,150:\shadeedge\drawsolidedge\drawedgehead{100}10 + \abutleft 75:\rho{\border{10pt}{10pt}\octagon{10pt}\outline} +\setedge -150,0,100,150:\shadeedge\drawdotedge{7pt}1 + \abutaboved{-100pt}\rho{\border{10pt}{10pt}\hexagon\outline} +\setedge 0,-50,150,0:\shadeedge\drawsolidedge\drawedgehead{20}11 + \abutrightd{-25pt}\rho{\border{2pt}{2pt}\borderto{25pt}{0pt}\rect\outline} +\setedge 0,-50,-100,150:\shadeedge\drawsolidedge +\setedge 0,-50,100,150:\shadeedge\drawsolidedge +\setedge 150,0,-100,150:\shadeedge\drawsolidedge + \abutbelowd{100pt}\rho{\border{10pt}{10pt}\hexagon\outline} +\setedge 150,0,100,150:\shadeedge\drawdashedge{40pt}{40pt}11\drawedgehead000 + \abutleft 75:\rho{\border{10pt}{10pt}\hexagon\outline} + \shiftedge{-10pt}\shadeedge\drawdashedge{30pt}{30pt}10\drawedgehead000 + \shiftedge{-10pt}\shadeedge\drawdashedge{11pt}{5pt}01\drawedgehead000 + \shiftedge{-10pt}\shadeedge\drawdotedge{8pt}0\drawedgehead{100}10 + \abutright 75:\rho{\border{5pt}{5pt}\rorect{5pt}11\outline} +\setedge -100,150,100,150:\drawedgehead{50}11\shadeedge\drawsolidedge + \abutbelow 0:\rho{\border{2pt}{2pt}\rect\outline} +\setedge 0,-50,-150,-150:\thicklines\drawedgehead{50}01\thinlines + \shadeedge\drawdashedge{13pt}{3pt}01\drawedgehead{50}11 + \abutbelow -50:{X \atop Y}{\border{2pt}{2pt}\rorect{5pt}10\outline} +\setedge 0,-50,0,-150:\thicklines\shadeedge\drawsolidedge\thinlines +\setedge 0,-50,150,-150:\shadeedge\drawsolidedge + \abutabove 75:{X \atop Y}{\border{2pt}{2pt}\rorect{5pt}10\outline} +\setedge -175,-50,-175,-100:\drawdashedge{10pt}{31pt}11 +\setedge -165,-100,-165,-50:\drawdashedge{15pt}{15pt}01 +\setedge -155,-50,-155,-100:\drawdashedge{5pt}{5pt}11 +\setedge -145,-100,-145,-50:\drawdotedge{26pt}1 +\setedge -135,-50,-135,-100:\drawdotedge{25pt}1 +\setedge -125,-100,-125,-50:\drawdotedge{5pt}1 +\setedge 125,-50,175,-50:\drawdashedge{10pt}{31pt}11 +\setedge 175,-60,125,-60:\drawdashedge{15pt}{15pt}01 +\setedge 125,-70,175,-70:\drawdashedge{5pt}{5pt}11 +\setedge 175,-80,125,-80:\drawdotedge{26pt}1 +\setedge 125,-90,175,-90:\drawdotedge{25pt}1 +\setedge 175,-100,125,-100:\drawdotedge{5pt}1 +}$$ + +\newpage + +%These three diagrams test the macros for category-theory diagrams. + +$$\ctdiagram{ +\ctvg0,0:{Y'}{\ctlpbl{I_{Y'}}} +\ctvg150,0:{Z=Z_0}{\ctgl{Z}\ctlpbr{I_Z}} +\ctvg0,100:{X_0=X}{\ctgr{X}\ctlptl{I_X}} +\ctvg150,100:{Y}{\ctlptr{I_Y}} +\ctet0,100,150,100:\alpha +\cteb0,0,150,0:{\beta'} +\ctel0,100,0,0:{\alpha'} +\cter150,100,150,0:\beta +\ctetb0,100,150,0:11{\alpha;\beta}{\alpha';\beta'} +}$$ + +$$\ctdiagram{\ctdash +\ctvg0,0:{Y'}{\ctlpblcc{I_{Y'}}} +\ctvg150,0:{Z=Z_0}{\ctgl{Z}\ctlpbrcc{I_Z}} +\ctvg0,100:{X_0=X}{\ctgr{X}\ctlptlcc{I_X}} +\ctvg150,100:{Y}{\ctlptrcc{I_Y}} +\ctet0,100,150,100:\alpha +\ctnohead\cteb0,0,150,0:{\beta'}\cthead +\ctel0,100,0,0:{\alpha'} +\cter150,100,150,0:\beta +\ctelr0,100,150,0:11{\alpha';\beta'}{\alpha;\beta} +}$$ + +$$\ctdiagram{ +\ctv0,0:{Y'} +\ctvg150,0:{Z=Z_0}{\ctgl{Z}} +\ctvg0,100:{X_0=X}{\ctgr{X}} +\ctv150,100:Y +\ctetg0,100,150,100;50:\alpha +\ctebg0,0,150,0;50:{\beta'} +\ctelg0,100,0,0;30:{\alpha'} +\cterg150,100,150,0;30:\beta +\ctetbg0,100,150,0;50,100:10{\rho}{\rho'} +\ctelrg0,0,150,100;70,30:01{\theta}{\theta'} +}$$ + +\newpage + +%The next two diagrams are further tests of the macros for drawing +%double edges. + +$$\ctdiagram{ +\ctv0,0:X +\ctv-100,100:Y\ctv-100,0:Y\ctv-100,-100:Y +\ctv100,100:Z\ctv100,0:Z\ctv100,-100:Z +\ctetb0,0,-100,100:10\alpha\beta +\ctdash\ctetb0,0,-100,0:00\alpha\beta\ctsolid +\ctetb0,0,-100,-100:01\alpha\beta +\ctetb0,0,100,100:10\alpha\beta +\ctdash\ctetb0,0,100,0:11\alpha\beta\ctsolid +\ctetb0,0,100,-100:01\alpha\beta +}$$ + +$$\ctdiagram{ +\ctv0,0:X +\ctv-100,100:Y\ctv0,100:Y\ctv100,100:Y +\ctv-100,-100:Z\ctv0,-100:Z\ctv100,-100:Z +\ctelr0,0,-100,100:10\alpha\beta +\ctelr0,0,0,100:00\alpha\beta +\ctelr0,0,100,100:01\alpha\beta +\ctelr0,0,-100,-100:10\alpha\beta +\ctelr0,0,0,-100:11\alpha\beta +\ctelr0,0,100,-100:01\alpha\beta +}$$ + +\newpage + +%These two diagrams test the usage of \ctinnermid and \ctoutermid. + +$$\ctdiagram{\ctv 0,0:{ +{\displaystyle\sum_{i=0}^{100}x_i\cdot y_i}\over +{\displaystyle\sqrt{\biggl(\sum_{i=0}^{100}x_i^2\biggr) ++\biggl(\sum_{i=0}^{100}y_i^2\biggr)}}} +\ctv0,150:A\ctv150,150:B\ctv150,0:C\ctv150,-150:D +\ctv0,-150:E\ctv-150,-150:F\ctv-150,0:G\ctv-150,150:H +\cter0,0,0,150:A\ctinnermid\cter0,0,0,150:a\ctoutermid +\cter150,150,0,0:B\ctinnermid\cter150,150,0,0:b\ctoutermid +\cteb0,0,150,0:C\ctinnermid\cteb0,0,150,0:c\ctoutermid +\cteb150,-150,0,0:D\ctinnermid\cteb150,-150,0,0:d\ctoutermid +\ctel0,0,0,-150:E\ctinnermid\ctel0,0,0,-150:e\ctoutermid +\ctel-150,-150,0,0:F\ctinnermid\ctel-150,-150,0,0:f\ctoutermid +\ctet0,0,-150,0:G\ctinnermid\ctet0,0,-150,0:g\ctoutermid +\ctet-150,150,0,0:H\ctinnermid\ctet-150,150,0,0:h +}$$ + +$$\ctdiagram{\ctv 0,0:{ +{\displaystyle\sum_{i=0}^{100}x_i\cdot y_i}\over +{\displaystyle\sqrt{\biggl(\sum_{i=0}^{100}x_i^2\biggr) ++\biggl(\sum_{i=0}^{100}y_i^2\biggr)}}} +\ctv-150,150:A\ctv0,150:C\ctv150,150:E\ctv150,0:G +\ctelr0,0,-150,150:11AB\ctinnermid +\ctelr0,0,-150,150:11ab\ctoutermid +\ctelr0,150,0,0:11CD\ctinnermid +\ctelr0,150,0,0:11cd\ctoutermid +\ctetb0,0,150,150:11EF\ctinnermid +\ctetb0,0,150,150:11ef\ctoutermid +\ctetb150,0,0,0:11GH\ctinnermid +\ctetb150,0,0,0:11gh +}$$ + +\newpage + +%This is a ``real'' diagram, relating directed complete relations to +%Scott's inverse limit construction. It is sufficiently crowded +%that it has been necessary to place some of the abutted expressions +%carefully to avoid ambiguity. + +$$\ctdiagram{ +\ctvg0,0:{D_0}{\border{2pt}{0pt}} +\ctv72,0:{D_1} +\ctv144,0:{D_2} +\ctv216,0:{\quad\cdots} +\ctvg288,144:{D_\infty}{\advance\ycenter by 5pt\border{50pt}{10pt}} +\ctv234,36:{\cdots} +\ctetbg0,0,72,0;48,48:10{\phi_0}{\psi_0} +\ctetbg72,0,144,0;114,114:10{\phi_1}{\psi_1} +\ctetb144,0,216,0:10{\phi_2}{\psi_2} +\ctelrg0,0,288,144;42,30:10{\Phi_0}{\Psi_0} +\ctelrg72,0,288,144;42,30:10{\Phi_1}{\Psi_1} +\ctelrg144,0,288,144;42,30:10{\Phi_2}{\Psi_2} +\ctvg0,-72:{D'_0}{\border{2pt}{0pt}} +\ctv72,-72:{D'_1} +\ctv144,-72:{D'_2} +\ctv216,-72:{\quad\cdots} +\ctvg288,-216:{D'_\infty}{\advance\ycenter by -5pt\border{50pt}{10pt}} +\ctv234,-108:{\cdots} +\ctetbg0,-72,72,-72;48,48:10{\phi'_0}{\psi'_0} +\ctetbg72,-72,144,-72;114,114:10{\phi'_1}{\psi'_1} +\ctetb144,-72,216,-72:10{\phi'_2}{\psi'_2} +\ctelrg0,-72,288,-216;-114,-102:10{\Phi'_0}{\Psi'_0} +\ctelrg72,-72,288,-216;-114,-102:10{\Phi'_1}{\Psi'_1} +\ctelrg144,-72,288,-216;-114,-102:10{\Phi'_2}{\Psi'_2} +\cter0,0,0,-72:{\alpha_0} +\cter72,0,72,-72:{\alpha_1} +\cter144,0,144,-72:{\alpha_2} +\ctv216,-36:{\cdots} +\ctdash +\cter288,144,288,-216:{\alpha_\infty} +}$$ + +\newpage + +%This shows how a macro can be defined and then used to give two different +%views of the same diagram. + +\def\testcube#1#2#3#4#5#6#7#8{ +$$\ctdiagram{ +\ctv#1,#3:{A_1} +\ctv#2,#3:{B_1} +\ctv#1,#4:{A_2} +\ctv#2,#4:{B_2} +\ctv#5,#7:{A'_1} +\ctv#6,#7:{B'_1} +\ctv#5,#8:{A'_2} +\ctv#6,#8:{B'_2} +\ctet#1,#3,#2,#3:{\gamma_1} +\ctet#1,#4,#2,#4:{\gamma_2} +\cter#1,#3,#1,#4:{\alpha} +\cter#2,#3,#2,#4:{\beta} +\ctet#5,#7,#6,#7:{\gamma'_1} +\ctet#5,#8,#6,#8:{\gamma'_2} +\cter#5,#7,#5,#8:{\alpha'} +\cter#6,#7,#6,#8:{\beta'} +\cter#1,#3,#5,#7:{a_1} +\cter#2,#3,#6,#7:{b_1} +\cter#1,#4,#5,#8:{a_2} +\cter#2,#4,#6,#8:{b_2} +}$$} + +\testcube{0}{200}{200}{0}{50}{150}{150}{50} + +\testcube{0}{150}{150}{0}{100}{250}{200}{50} + +\newpage + +%An example of a partial ordering with a limit point. + +$${\def\diagramunit{0.25in} +\ctdiagram{\ctnohead +\ctv0,0:{\geq 0} +\ctv2,2:{\geq 1} +\ctv4,4:{\geq 2} +\ctv7,7:\infty +\ctv-2,2:{=0} +\ctv0,4:{=1} +\ctv2,6:{=2} +\cten0,0,2,2: +\cten2,2,4,4: +\cten0,0,-2,2: +\cten2,2,0,4: +\cten4,4,2,6: +\ctdot +\cten4,4,7,7: +}}$$ + +%An example of a binary tree, produced by user macros. + +\newcount\cnx\newcount\cny\newcount\cnxx\newcount\cnyy + +\def\treea#1{\cnxx=\cnx\cnyy=\cny +\ctv\cnx,\cny:{\scriptstyle #1} +\advance\cnx by -1\advance\cny by 4 +\ctdot +\cten\cnxx,\cnyy,\cnx,\cny: +\advance\cnx by 2 +\cten\cnxx,\cnyy,\cnx,\cny: +\ctsolid +\cnx=\cnxx\cny=\cnyy} + +\def\treeb#1{\ctv\cnx,\cny:{\scriptstyle #1} +\advance\cnx by -2\advance\cny by 4 +\treea{#10} +\cnxx=\cnx\advance\cnxx by 2\cnyy=\cny\advance\cnyy by -4 +\cten\cnxx,\cnyy,\cnx,\cny: +\advance\cnx by 4 +\treea{#11} +\cnxx=\cnx\advance\cnxx by -2\cnyy=\cny\advance\cnyy by -4 +\cten\cnxx,\cnyy,\cnx,\cny: +\advance\cnx by -2\advance\cny by -4} + +\def\treec#1{\ctv\cnx,\cny:{\scriptstyle #1} +\advance\cnx by -4\advance\cny by 4 +\treeb{#10} +\cnxx=\cnx\advance\cnxx by 4\cnyy=\cny\advance\cnyy by -4 +\cten\cnxx,\cnyy,\cnx,\cny: +\advance\cnx by 8 +\treeb{#11} +\cnxx=\cnx\advance\cnxx by -4\cnyy=\cny\advance\cnyy by -4 +\cten\cnxx,\cnyy,\cnx,\cny: +\advance\cnx by -4\advance\cny by -4} + +\def\treed#1{\ctv\cnx,\cny:{\scriptstyle #1} +\advance\cnx by -8\advance\cny by 4 +\treec{#10} +\cnxx=\cnx\advance\cnxx by 8\cnyy=\cny\advance\cnyy by -4 +\cten\cnxx,\cnyy,\cnx,\cny: +\advance\cnx by 16 +\treec{#11} +\cnxx=\cnx\advance\cnxx by -8\cnyy=\cny\advance\cnyy by -4 +\cten\cnxx,\cnyy,\cnx,\cny: +\advance\cnx by -8\advance\cny by -4} + +\def\tree{\ctv\cnx,\cny:\bot\def\centerheight{2pt} +\advance\cnx by -16\advance\cny by 4 +\treed{0} +\cnxx=\cnx\advance\cnxx by 16\cnyy=\cny\advance\cnyy by -4 +\cten\cnxx,\cnyy,\cnx,\cny: +\advance\cnx by 32 +\treed{1} +\cnxx=\cnx\advance\cnxx by -16\cnyy=\cny\advance\cnyy by -4 +\cten\cnxx,\cnyy,\cnx,\cny: +\advance\cnx by -16\advance\cny by -4} + +$${\def\diagramunit{7.5pt} +\ctdiagram{\ctnohead\cnx=0\cny=0\tree}}$$ + +\end{document} |