\chapter{Drawing Lines} \section{The \ddrawline\ Command} The principal command for drawing a particle line in \FEYNMAN\ is via the {\it \ddrawline}\ statement. The syntax is:\\ %\ddrawline$<$particle type$>$[$<$particle direction$> <$configuration$>$] %$<$co-ordinates of the beginning of the line of form $(x,y)>$[$<$length$>$\\ \begin{verbatim} \drawline[] <(x,y), the co-ordinates of the beginning of the line>[] \end{verbatim} where it is understood that the triangular brackets do not appear in the syntax but their contents describe what goes in that position. An example would be: \begin{verbatim} \drawline\photon[\NW\FLIPPED](1500,-18000)[6] \end{verbatim} which would draw a photon (or W,Z) starting from the point (1500,-18000), with distances in centipoints (section 1.5), in the NorthWesterly direction (\ie towards the upper lefthand corner), in a {\it flipped} configuration (described below) for an extent of 6 `units'. In the case of photons each unit is a `half-wiggle'. The actual distance traversed on the page is not specified since photons come in distinct `quanta', in this case half-wiggles. When the above statement is issued a number of parameters are returned. These may be used by the artiste (this is `Feyn Art'). These are: \begin{verbatim} \photonfrontx,\photonfronty: The (x,y) co-ordinates of the front of the line. \photonbackx,\photonbacky: The (x,y) co-ordinates of the back of the line. \photonlengthx,\photonlengthy: The (x,y) extent of the line. \photoncount The number of photons printed thus far. \particlefrontx,\particlefronty: The (x,y) co-ords of the front of the line. \particlemidx,\particlemidy: The (x,y) co-ordinates of the middle of the line. \particlebackx,\particlebacky: The (x,y) co-ordinates of the back of the line. \particlelengthx,\particlelengthy: The (x,y) extent of the line. \end{verbatim} All of the terms beginning with `\bs particle' may be abbreviated to `\bs p' plus the rest of the name. For instance, \bs pmidx is the same as \bs particlemidx \etc You may be wondering why there is some duplication, such as \bs pfrontx and \bs photonfrontx. The reason is that terms beginning with {\it \bs particle} refer to the last particle line drawn by \ddrawline\ whereas those commencing with {\it \bs photon} refer only to the last {\em photon line} drawn. This distinction often proves to be most convenient. To illustrate this point consider the following combination of statements: \begin{verbatim} \input FEYNMAN \vskip -0.7in \begin{picture}(18000,8000) \drawline\gluon[\S\CURLY](0,0)[8] \drawline\fermion[\SW\REG](\particlebackx,\particlebacky)[2500] \drawline\fermion[\SE\REG](\particlebackx,\particlebacky)[2500] \drawline\gluon[\S\CURLY](10000,0)[8] \drawline\fermion[\SW\REG](\particlebackx,\particlebacky)[2500] \drawline\fermion[\SE\REG](\gluonbackx,\gluonbacky)[2500] \end{picture} \vskip 1.5in \end{verbatim} which produces the following results: %\input FEYNMAN % MOVED TO FEYNMANDOC.TEX calling routine. \vskip -0.7in \begin{picture}(18000,8000) \drawline\gluon[\S\CURLY](0,0)[8] \drawline\fermion[\SW\REG](\particlebackx,\particlebacky)[2500] \drawline\fermion[\SE\REG](\particlebackx,\particlebacky)[2500] \drawline\gluon[\S\CURLY](10000,0)[8] \drawline\fermion[\SW\REG](\particlebackx,\particlebacky)[2500] \drawline\fermion[\SE\REG](\gluonbackx,\gluonbacky)[2500] \end{picture} \vskip 1.5in We see immediately what the difference is. In the first diagram \bs particlebackx,y refers, first to the gluon after it is drawn, and then to the \bs SW fermion after that has been drawn. If a third reference to those co-ordinates had been made it would have referred to the final co-ordinates of the \bs SE fermion. The second gluon line begins 10000 centipoints to the right of the first. This time the \bs SE fermion commences at \bs gluonbackx,y which is the end co-ordinates of the gluon and will remain so until the end of the picture of until another gluon is drawn. We now proceed to detail the various parameters of the \ddrawline\ command given above. \section{The Types of Particle Lines} The four basic kinds of particle lines which \FEYNMAN\ can draw are given by the following input parameters: \begin{verbatim} \fermion \scalar \photon \gluon \end{verbatim} A fifth category, {\tt \bs especial}, which allows the user to define his own non-standard style of line, has not yet been implemented. As we have seen this is the first input parameter of \ddrawline\ and is implemented as \begin{verbatim} \drawline\fermion... \drawline\scalar... \drawline\photon... \drawline\gluon... \end{verbatim} Note that this is the only argument of \ddrawline\ which is in lowercase letters. The specifics of each of these will be given in separate sections. \section{Particle Direction} Each particle line may be drawn in any of eight possible directions, specified by the points of the compass with North always understood as being at the top of the page: \begin{verbatim} \N \NE \E \SE \S \SW \W \NW \end{verbatim} This is the second argument of \ddrawline. \begin{verbatim} \drawline\photon[\NW... \end{verbatim} and so forth. The line is drawn from the specified {\it front} point in the indicated direction. Note that all directions are in uppercase and don't forget the backslash! \section{Line Configuration} Many different particle types are available in a variety of forms. For the specific forms available see the section on the appropriate particle type below. The current configurations available are: \begin{verbatim} \REG \FLIPPED \CURLY \FLIPPEDCURLY \FLAT \FLIPPEDFLAT \CENTRAL \FLIPPEDCENTRAL \LONGPHOTON \FLIPPEDLONG \SQUASHEDGLUON \end{verbatim} The shape of the individual particle types will be presented shortly but a few general words might be given here. First, not every particle type exists in all configurations in all directions. \bs REG is something of a default (although \verb@\REG@\ must still be included as a parameter in \ddrawline) and is defined for all particles in all directions. It may not be your own \ae sthetic favourite, however. \bs FLIPPED lines are inverted about their axes with respect to the \bs REG lines, but are otherwise the same. Obviously, since fermions and scalars are merely straight lines, the \bs REG and \bs FLIPPED configurations will be the same. In fact for these two particle types \bs REG is the {\bf only} legitimate configuration (but see \bs THICKLINES at the conclusion of this section). The other configurations do not, in general, exist for all of the directions available to a particle. \bs FLIPPEDCURLY, \bs FLIPPEDCENTRAL, \bs FLIPPEDLONG and \bs FLIPPEDFLAT are flipped along the direction axis when compared with \bs CURLY, \bs CENTRAL, \bs LONGPHOTON and \bs FLAT respectively. \bs CENTRAL and \bs FLIPPEDCENTRAL exist only for gluons (\bs gluon), \bs LONGPHOTON and \bs FLIPPEDLONG exist only for photons, while \bs SQUASHED exists only for gluons in the \bs E direction! The line configuration is the third input to \ddrawline\ and follows the direction within the square brackets with no intervening spaces: \begin{verbatim} \drawline\scalar[\S\REG]... \drawline\gluon[\NW\FLIPPED]... \end{verbatim} and so forth. A final word concerns the \bs THINLINES and \bs THICKLINES commands. When issued prior to a \ddrawline\ or \ddrawvertex\ statement they cause the particle lines to be drawn in a thin or thick fashion This remains in effect until superseded or the end of the picture is encountered. The default is \bs THINLINES. % The only lines which %cannot be drawn in a thick mode are photons in the These commands are identical to the \LaTeX\ \bs thinlines and \bs thicklines commands except when drawing photons in the \verb@\E@ and \verb@\W@ directions (any configuration). These will not be emboldened by \bs thicklines but will be by \bs THICKLINES. Photons in the \verb@\N@ and \verb@\S@ directions (any configuration) are unaffected since they are drawn in an intermediate state. The following illustrates the difference: \begin{verbatim} \begin{picture}(8000,2000) \drawline\gluon[\E\FLIPPEDCENTRAL](0,0)[5] \end{picture} \begin{picture}(8000,2000) \THICKLINES \drawline\gluon[\E\FLIPPEDCENTRAL](0,0)[5] \end{picture} \end{verbatim} which produces \begin{picture}(8000,2000) \drawline\gluon[\E\FLIPPEDCENTRAL](0,0)[5] \end{picture} \begin{picture}(8000,2000) \thicklines \drawline\gluon[\E\FLIPPEDCENTRAL](0,0)[5] \end{picture} \section{Line Co-ordinate Parameters} \subsection{Input Parameters} The fourth and fifth arguments of the \ddrawline\ command are the (x,y) co-ordinates of the {\em beginning} of the particle line. these are as measured in {\em centipoints} on the grid which \FEYNMAN\ has established. They are entered in the format $(x\ co$-$ordinate,y\ co$-$ordinate)$\ where $x$ and $y$ may be integer numbers (between, roughly, -30,000 and +30,000) or variables (counters) with numerical values. A number of variables have been pre-defined and available for use. The user may also define his own (see section 4.2.1 on storing information). Some samples may be illustrative: \begin{verbatim} \drawline\fermion[\SE\REG](-1500,12000)[2000] \drawline\gluon[\E\SQUASHEDGLUON](\photonbackx,\photonbacky)[2] \drawline\scalar[\N\REG](\Xone,\Yone)[11] \drawline\photon[\S\FLIPPEDFLAT](3000,\pmidy)[7] \end{verbatim} In the above \bs photonbackx and \bs photonbacky are co-ordinates, presumably returned from a previously drawing a photon. \bs Xone and \bs Yone are some values stored by the user and \bs pmidy is the ordinate of the midpoint of the previously drawn line. Note in the last example how the x co-ordinate is a number while the y co-ordinate is a variable. \subsection{Output Parameters} As discussed in section 2.1 a number of useful positional parameters are returned when \ddrawline\ is called. For the appropriate lines the following are defined, each definition superseding the previous value of the variable. For all lines: \begin{verbatim} \particlefrontx,\particlefronty: The (x,y) co-ords of the front of the line. \particlemidx,\particlemidy: The (x,y) co-ordinates of the middle of the line. \particlebackx,\particlebacky: The (x,y) co-ordinates of the back of the line. \end{verbatim} For photons: \begin{verbatim} \photonfrontx,\photonfronty: The (x,y) co-ordinates of the front of the photon. \photonbackx,\photonbacky: The (x,y) co-ordinates of the back of the photon. \end{verbatim} For gluons: \begin{verbatim} \gluonfrontx,\gluonfronty: The (x,y) co-ordinates of the front of the gluon. \gluonbackx,\gluonbacky: The (x,y) co-ordinates of the back of the gluon. \end{verbatim} For fermions: \begin{verbatim} \fermionfrontx,\fermionfronty: The (x,y) co-ordinates of the front of the fermion. \fermionbackx,\fermionbacky: The (x,y) co-ordinates of the back of the fermion. \end{verbatim} For scalars: \begin{verbatim} \scalarfrontx,\scalarfronty: The (x,y) co-ordinates of the front of the scalar. \scalarbackx,\scalarbacky: The (x,y) co-ordinates of the back of the scalar. \end{verbatim} As an illustration the command\\ \verb@ \drawline\fermion[\E\REG](0,0)[2000] @ \\ has \\ \begin{verbatim} \pfrontx=0 \pfronty=0 \pmidx=1000 \pmidy=0 \pbackx=2000 \pbacky=0 \end{verbatim} \section{Line Length} \subsection{Input Parameters} The final parameter to be given to \ddrawline\ is the {\it length} of the particle to be drawn. The units in which the length is given vary between the four particle types. For fermions it is simply the actual length in centipoints. For scalars it is the number of line segments, each of which is separated by a {\it gap}. The scalar line always begins and ends on a line segment, never a gap. The default lengths of both segments and gaps may be overruled by the user (see the section on scalars). For gluons the length parameter is the {\em number of loops}. The actual length of each loop depends upon which style is selected and whether the gluon is drawn diagonally (at a slant) or not. For photons the unit of measure is not a `wiggle', but a `half-wiggle'. This enables one to produce a photon which both begins and ends on the `upward' (or `downward') part of its oscillation. This is illustrated in the following program: \begin{verbatim} \begin{document} \input FEYNMAN \textheight 600pt \textwidth 400pt \begin{picture}(20000,10000) \drawline\fermion[\E\REG](0,10000)[5000] \drawline\gluon[\E\REG](10000,10000)[5] \drawline\photon[\E\REG](0,5000)[5] \drawline\photon[\E\REG](0,0)[6] \drawline\scalar[\E\REG](10000,5000)[5] \seglength=1200 \gaplength=300 % Changes the \scalar defaults. \drawline\scalar[\E\REG](10000,0)[5] \end{picture} \end{document} \end{verbatim} \newpage which produces: \vskip 0.5in \begin{picture}(20000,10000) \drawline\fermion[\E\REG](0,10000)[5000] \drawline\gluon[\E\REG](10000,10000)[5] \drawline\photon[\E\REG](0,5000)[5] \drawline\photon[\E\REG](0,0)[6] \drawline\scalar[\E\REG](10000,5000)[5] \seglength=1200 \gaplength=300 % Changes the \scalar defaults. \drawline\scalar[\E\REG](10000,0)[5] \end{picture} \vskip 0.5in \subsection{Output Parameters} In the same way as co-ordinates related to the particle are returned, the {\em displacements} of the line are also given. These are essentially the $(\Delta x,\Delta y)$\ between the ending and beginning co-ordinates. Thus a particle drawn in the \bs N direction will have positive \bs particlelengthy (\bs plengthy for short) and zero \bs particlelengthx. A particle in the \bs SE direction will have a positive \bs particlelengthx but a negative \bs particlelengthy (and these will be of the same magnitude). The full list is: \begin{verbatim} \particlelengthx,\particlelengthy: The (x,y) extent of the line. \photonlengthx,\photonlengthy: The (x,y) extent of the photon. \gluonlengthx,\gluonlengthy: The (x,y) extent of the gluon. \scalarlengthx,\scalarlengthy: The (x,y) extent of the scalar. \fermionlengthx,\fermionlengthy: The (x,y) extent of the fermion. \fermionlength The total length of the fermion. \end{verbatim} A common usage of this parameter would be to draw a fermion line to be the same length as, say, a gluon line. \begin{verbatim} \begin{picture}(9000,4500) \drawline\gluon[\E\REG](0,4000)[7] \drawline\fermion[\E\REG](0,0)[\gluonlengthx] \end{picture} \end{verbatim} \vskip 0.5in \begin{picture}(9000,4500) \drawline\gluon[\E\REG](0,4000)[7] \drawline\fermion[\E\REG](0,0)[\gluonlengthx] \end{picture} \vskip 0.2in Each time that a new line is drawn \bs plengthx,y are re-assigned. \bs photonlengthx,y is only re-assigned when the next photon is drawn, \bs gluonlengthx,y is only re-assigned when the next gluon is drawn \etc \newpage \section{Fermions} The format for drawing a fermion with \ddrawline\ is: \begin{verbatim} \drawline\fermion[\REG] <(x,y), the co-ordinates of the beginning of the fermion>[] \end{verbatim} where the arguments have been described in the preceding sections. When \FEYNMAN draws a fermion it uses the \bs line facility of \LaTeX. This places a restriction upon diagonally drawn fermions (those in the \bs NE, \bs SE, \bs NW and \bs SW directions). The minimum length of such lines is 1416 centipoints. The user can draw fermions in other directions (NNE \etc) by utilizing the \bs put command of \LaTeX. The minimum extension of these lines is at least as great. Fermions drawn in the N,S,E and W directions have no such restrictions. If attempts are made to draw a fermion of sub-minimal length it will be as though it were invisible. That is all of the spacing and parameters will be as before but a blank space will appear on the page. This effect may be deliberately achieved for {\em any} particle or vertex using the {\em phantom} facility discussed under advanced features. The only option really available for fermions is the use of \bs thicklines and \bs thinlines as described at the conclusion of section 2.4. The parameters returned by \ddrawline\ when a fermion is drawn are: \begin{verbatim} \fermionfrontx,\fermionfronty: The (x,y) co-ordinates of the front of the line. \fermionbackx,\fermionbacky: The (x,y) co-ordinates of the back of the line. \fermionlengthx,\fermionlengthy: The (x,y) extent of the line. \fermionlength The total length of the line. \fermioncount The number of fermions printed thus far. \particlefrontx,\particlefronty: The (x,y) co-ords of the front of the line. \particlemidx,\particlemidy: The (x,y) co-ordinates of the middle of the line. \particlebackx,\particlebacky: The (x,y) co-ordinates of the back of the line. \particlelengthx,\particlelengthy: The (x,y) extent of the line. \end{verbatim} As an example: \begin{verbatim} \begin{picture}(8000,8000) \drawline\fermion[\E\REG](0,0)[6000] \THICKLINES \drawline\fermion[\N\REG](\fermionbackx,\pbacky)[5000] \drawline\fermion[\SW\REG](\fermionbackx,\fermionbacky)[4000] \THINLINES \drawline\fermion[\NW\REG](\pmidx,\pmidy)[2000] \drawline\fermion[\W\REG](\pfrontx,\pfronty)[\fermionlengthy] \end{picture} \end{verbatim} which produces: \vskip -0.25in \hskip 1.75in \begin{picture}(8000,8000) \drawline\fermion[\E\REG](0,0)[6000] \thicklines \drawline\fermion[\N\REG](\fermionbackx,\pbacky)[5000] \drawline\fermion[\SW\REG](\fermionbackx,\fermionbacky)[4000] \thinlines \drawline\fermion[\NW\REG](\pmidx,\pmidy)[2000] \drawline\fermion[\W\REG](\pfrontx,\pfronty)[\fermionlengthy] \end{picture} %\bigskip \newpage \section{Scalars} \subsection{The Anatomy of a Scalar} A scalar particle (eg. a Higgs) is generally drawn as a dashed line, that is a series of line {\it segments} separated by {\it gaps}. The format for drawing a scalar with \ddrawline\ is: \begin{verbatim} \drawline\scalar[\REG] <(x,y), the co-ordinates of the beginning of the scalar> [] \end{verbatim} We note the similarity between the fermion and scalar lines. The principal difference is in the units of length. Here the extent of the line is given by an integral number of segments. These will be of equal length and equally spaced. The same limitations on the length of fermion lines applies here: diagonal (NW, NE \etc) line segments may not be shorter than 1416 centipoints, which is the default length for all directions. The default gap length is 850 centipoints and there is no innate limitation on this value (other than it must not be negative or larger than the page size). Attempts to draw a slanted scalar with segments of under 1416 centipoints will lead to a blank space in the diagram. The parameters returned by \ddrawline\ when a scalar is drawn are: \begin{verbatim} \scalarfrontx,\scalarfronty: The (x,y) co-ordinates of the front of the scalar. \scalarbackx,\scalarbacky: The (x,y) co-ordinates of the back of the scalar. \scalarlengthx,\scalarlengthy: The (x,y) extent of the scalar. \scalarcount: The total number of scalars printed thus far. \particlefrontx,\particlefronty: The (x,y) co-ords of the front of the line. \particlemidx,\particlemidy: The (x,y) co-ordinates of the middle of the line. \particlebackx,\particlebacky: The (x,y) co-ordinates of the back of the line. \particlelengthx,\particlelengthy: The (x,y) extent of the line. \end{verbatim} \subsection{Adjusting the Spacing} The segment length is given, in centipoints, by the variable \bs seglength and the gap length by \bs gaplength. Both are adjustable by the user by issuing the commands: \begin{verbatim} \global\seglength= \global\gaplength= \end{verbatim} just prior to a \ddrawline\bs scalar command. After this they automatically return to their default values of 1416 and 850. If only one is adjusted the other will retain its default value. The modifiers \bs thicklines and \bs thinlines work normally here with \bs thinlines the default (but \bs thicklines will remain in effect until \bs thinlines or \eendpic\ is encountered). \bigskip These features are illustrated by the following sample file: \newpage \begin{verbatim} \begin{picture}(25000,25000) % First picture: \drawline\scalar[\E\REG](6000,0)[5] \drawline\fermion[\SE\REG](\scalarbackx,\scalarbacky)[2000] \drawline\fermion[\NE\REG](\scalarbackx,\scalarbacky)[2000] \drawline\fermion[\SW\REG](\scalarfrontx,\scalarfronty)[2000] \drawline\fermion[\NW\REG](\scalarfrontx,\scalarfronty)[2000] \global\seglength=1000 \global\gaplength=250 \drawline\scalar[\W\REG](\pbackx,\pbacky)[5] \drawline\fermion[\SW\REG](\particlebackx,\particlebacky)[2000] \drawline\fermion[\NW\REG](\scalarbackx,\scalarbacky)[2000] \drawline\fermion[\NE\REG](\scalarfrontx,\scalarfronty)[2000] \put(4000,0){1} % Second Picture: \THICKLINES \drawline\fermion[\N\REG](23000,3000)[12345] \global\gaplength=473 \drawline\scalar[\NW\REG](\pbackx,\fermionbacky)[3] \global\gaplength=473 \drawline\scalar[\NE\REG](\fermionbackx,\pfronty)[3] \drawline\scalar[\SW\REG](\fermionfrontx,\fermionfronty)[3] \drawline\scalar[\SE\REG](\fermionfrontx,\fermionfronty)[3] \put(23500,14000){2} % Third Picture: \THINLINES \global\gaplength=0 \global\seglength=1000 \drawline\scalar[\S\REG](1000,16000)[4] \drawline\fermion[\NW\REG](1000,16000)[1500] \drawline\fermion[\NE\REG](1000,16000)[1500] \drawline\fermion[\SW\REG](1000,12000)[1500] \drawline\fermion[\SE\REG](1000,12000)[1500] \put(1500,14000){3} % Fourth Picture: \global\gaplength=500 \global\seglength=500 \thicklines \drawline\scalar[\NE\REG](6000,7000)[10] \put(\pmidx,\pmidy){4} \drawline\fermion[\W\REG](\scalarfrontx,\scalarfronty)[2500] \drawline\fermion[\S\REG](\scalarfrontx,\scalarfronty)[2500] \drawline\fermion[\E\REG](\scalarbackx,\scalarbacky)[2500] \drawline\fermion[\N\REG](\scalarbackx,\scalarbacky)[2500] \end{picture} \end{verbatim} which produces: \begin{picture}(25000,25000) % First picture: \drawline\scalar[\E\REG](6000,0)[5] \drawline\fermion[\SE\REG](\scalarbackx,\scalarbacky)[2000] \drawline\fermion[\NE\REG](\scalarbackx,\scalarbacky)[2000] \drawline\fermion[\SW\REG](\scalarfrontx,\scalarfronty)[2000] \drawline\fermion[\NW\REG](\scalarfrontx,\scalarfronty)[2000] \global\seglength=1000 \global\gaplength=250 \drawline\scalar[\W\REG](\pbackx,\pbacky)[5] \drawline\fermion[\SW\REG](\particlebackx,\particlebacky)[2000] \drawline\fermion[\NW\REG](\scalarbackx,\scalarbacky)[2000] \drawline\fermion[\NE\REG](\scalarfrontx,\scalarfronty)[2000] \put(4000,0){1} % Second Picture: \thicklines \drawline\fermion[\N\REG](23000,3000)[12345] \global\gaplength=473 \drawline\scalar[\NW\REG](\pbackx,\fermionbacky)[3] \global\gaplength=473 \drawline\scalar[\NE\REG](\fermionbackx,\pfronty)[3] \drawline\scalar[\SW\REG](\fermionfrontx,\fermionfronty)[3] \drawline\scalar[\SE\REG](\fermionfrontx,\fermionfronty)[3] \put(23500,14000){2} % Third Picture: \thinlines \global\gaplength=0 \global\seglength=1000 \drawline\scalar[\S\REG](1000,16000)[4] \drawline\fermion[\NW\REG](1000,16000)[1500] \drawline\fermion[\NE\REG](1000,16000)[1500] \drawline\fermion[\SW\REG](1000,12000)[1500] \drawline\fermion[\SE\REG](1000,12000)[1500] \put(1500,14000){3} % Fourth Picture: \global\gaplength=500 \global\seglength=500 \thicklines \drawline\scalar[\NE\REG](6000,7000)[10] \put(\pmidx,\pmidy){4} \drawline\fermion[\W\REG](\scalarfrontx,\scalarfronty)[2500] \drawline\fermion[\S\REG](\scalarfrontx,\scalarfronty)[2500] \drawline\fermion[\E\REG](\scalarbackx,\scalarbacky)[2500] \drawline\fermion[\N\REG](\scalarbackx,\scalarbacky)[2500] \end{picture} \bigskip \bigskip \bigskip There are a few points of note. In the first picture observe how \bs pbackx,y ($\equiv$\bs particlebackx,y) is used as opposed to \bs scalarbackx,y. For more on this point see section 2.1. Also note that the \bs seglength assignments stay in effect until the next scalar is drawn. The second figure illustrates how \bs gaplength automatically reverts to its default and further demonstrates the assignments of the co-ordinate variables. The third diagram shows that a zero gaplength creates a fermion and that numerical co-ordinates may be used just as easily as the pre-assigned positioning variables. This is also illustrated by the positioning of the numerical labels (1,2,3,4) which was accomplished with minimal thought. Finally the fourth diagram shows what happens when the value of \bs seglength is reduced to less that 1416 for a diagonal (non-horizontal, non-vertical) scalar. It also demonstrates how to locate the central point of a line. \section{Photons} \subsection{The Anatomy of a Photon} A photon line consists of a series of `wiggles' or undulations. The length of each undulation depends upon the style of photon selected. The format for drawing a photon with \ddrawline\ is: \begin{verbatim} \drawline\photon[] <(x,y), the co-ordinates of the beginning of the photon> [] \end{verbatim} The reason for specifying the number of {\it half}-wiggles is that it is sometimes convenient to have a line begin and end in either orientation (see section 2.6.1 for an example of this). The styles currently available for photons are: