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1. In Unix Research System Papers volume II pages 109 146 AT amp T Bell Laboratories Murray Hill New Jersey tenth edition 1990 2 William S Cleveland The Elements of Graphing Data Hobart Press Summit New Jersey 1985 revised 1994 3 William S Cleveland Visualizing Data Hobart Press Summit New Jersey 1993 4 William S Cleveland A model for studying display methods of statistical graphics with dis cussion Journal of Computational and Statistical Graphics 3 to appear 5 John D Hobby A user s manual for MetaPost Computing Science Technical Report no 162 AT amp T Bell Laboratories Murray Hill New Jersey April 1992 Available as http cm bell labs com cs cstr 162 ps gz 6 Donald E Knuth METAFONT the Program Addison Wesley Reading Massachusetts 1986 Volume D of Computers and Typesetting 7 Leslie Lamport HMTpX A Document Preparation System Addison Wesley Reading Mas sachusetts 1986 8 U S Bureau of the Census Statistical Abstracts of the United States 1992 Washington D C 112th edition 1992 9 Edward R Tufte Visual Display of Quantitative Information Graphics Press Box 430 Cheshire Connecticut 06410 1983 172. 40 60 80 Age in years Figure 3 An improved version of the 1991 age distribution graph All flavors of T X can handle multi line labels via the hbox within vbox arrangement used above but TFX users will find it more natural to use the tabular environment 7 Troff user s can use nofill mode btex nf Population in millions etex 2 2 Coordinate Systems The graph macros automatically shift and rescale coordinates from data files gdraw paths and glabel locations to fit the graph Whether the range of y coordinates is 0 64 to 4 6 or 640 000 to 4 600 000 they get scaled to fill about 88 of the height specified in the begingraph statement Of course line widths labels and plotting symbols are not rescaled The setrange command controls the shifting and rescaling process by specifying the minimum and maximum graph coordinates setrange coordinates coordinates where coordinates pair expression numeric or string expression numeric or string expression The first coordinates give min Ymin and the second give Lmax Ymax The lines z Tmin Tmax Y Ymin and y Ymax define the rectangular frame around the graph in Figures 1 3 For example an adding a statement setrange origin whatever whatever to the input for Figure 3 yields Figure 4 The first coordinates are given by the predefined pair constant origin and the other coordinates are left unspecified Any unknown value would wo
3. A number x in Mlog form represents 16 24 u where u e Any value between 1 61 x 107 8 and 3 88 x 1055 can be represented this way There is a constant Mten such that k Mten represents 10 for any integer k in the interval 29 55 The main reason for mentioning Mlog form is that it allows graph data to be manipulated as a MetaPost path The function Mreadpath file name reads a data file and returns a path where all the coordinates are in Mlog form An internal variable Gpaths determines whether gdraw and gfill expect paths to be given in Mlog form For example this graphs the data in agepop91 d with y coordinates divided by one million interim Gpaths log gdraw Mreadpath agepop91 d shifted 0 6 Mten 4 Typesetting Numbers The graph package needs to compute axis labels and then typeset them The macro format string expression numeric or string expression does this You must first input graph or input format to load the macro file The macro takes a format string and a number to typeset and returns a picture containing the typeset result Thus format g 2 2 yields 4 and format 3g 6 022e23 yields 6 02x10 3 A format string consists of TThe argument to a unary operator need not be parenthesized unless it is an expression involving binary operators 13 e an optional initial string not containing a percent sign a percent sign an optional numeric precision p e one o
4. Drawing Graphs with MetaPost John D Hobby Abstract This paper describes a graph drawing package that has been implemented as an extension to the MetaPost graphics language MetaPost has a powerful macro facility for implementing such extensions There are also some new language features that support the graph macros Existing features for generating and manipulating pictures allow the user to do things that would be difficult to achieve in a stand alone graph package Contents 4 Typesetting Numbers 13 1 Introduction 1 5 Conclusion 14 2 Using the Graph Macros 2 A Summary of the Graph Package 14 2 1 Basic Graph Drawing Commands 3 A 1 Graph Administration 15 2 2 Coordinate Systems 5 A 2 Drawing and Labeling 15 2 3 Explicit Grids and Framing 8 A 3 Grids Tick Marks and Framing 15 2 4 Processing Data Files 10 A 4 Miscellaneous Commands 16 A 5 Arithmetic on Numeric Strings 16 3 Manipulating Big Numbers 11 A 6 Internal Variables and Constants 16 1 Introduction MetaPost is a batch oriented graphics language based on Knuth s METAFONT but with PostScript output and numerous features for integrating text and graphics The author has tried to make this paper as independent as possible of the user s manual 5 but fully appreciating all the material requires some knowledge of the MetaPost language We concentrate on the mechanics of producing particular kinds of graphs because the question of what type
5. U ae Ad f 1900 1920 1940 1960 Figure 9 A graph of U S annual energy production and the commands that generated it 12 of computers Fixed point numbers are seldom a problem in MetaPost because all computations are based on coordinates that are limited by the size the paper on which the output is to be printed This does not hold for the input data in a graph drawing application Although graphs look best when coordinate axes are labeled with numbers of reasonable magnitude the strict limits of fixed point arithmetic would be inconvenient A simple way to handle large numbers is to include the line input sarith and then use binary operators Sadd Ssub Smul and Sdiv in place of x and These operators are inefficient but very flexible They accept numbers or strings and return strings in exponential notation with the exponent marked by e e g 6 7e 11 means 6 7 x 1071 The unary operator Sabs string finds a string the represents the absolute value Binary operators Sleq and Sneq perform numeric comparisons on strings and return boolean results The operation Scvnum string finds the numeric value for a string if this can be done without overflowing MetaPost s fixed point number system If the string does not contain e it is much more efficient to use the primitive operation scantokens string The above operators are based on a low level package that manipulates numbers in Mlog form
6. bottom and the left side of the frame using the coordinate system 100 4 Standard 4 draw begingraph 2 3in 2in setcoords log log 104 glabel lft btex Seconds etex QUT glabel bot btex Matrix size etex Seconds 1 OUT gdraw matmul d dashed evenly 0 1 glabel ulft btex Standard etex 8 gdraw matmul d 0 01 4 glabel 1rt btex Strassen etex 7 endgraph 50 100 200 Matrix size 500 Figure 5 Timings for two matrix multiplication algorithms with the corresponding MetaPost input draw begingraph 6 5cm 4 5cm setrange 80 0 90 whatever glabel bot btex Year etex OUT glabel 1ft btex vbox hbox Emissions in hbox thousands of hbox metric tons hbox heavy line etex OUT gdraw lead d withpen pencircle scaled 1 5pt autogrid otick 1ft setcoords linear linear setrange 80 0 90 whatever glabel rt btex vbox hbox Micrograms hbox per cubic hbox meter of air hbox thin line etex OUT gdraw lead d autogrid otick bot otick rt endgraph L 0 5 60 0 4 Emissions in 40 0 3 Micrograms thousands of per cubic metric tons 0 2 meter of air heavy line 20 thin line 0 1 0 l l l l 0 80 82 84 86 88 90 Year Figure 6 Annual lead emissions and average level at atmospheric monitoring stations in the United States The MetaPost input is shown above the graph in effect when the endgraph command is interpreted Figure 6 uses t
7. cation and offset as specified by the label suffix The location can be x and y coordinates a pair giving x and y a numerc value giving a time on the last path drawn or OUT to label the outside of the graph A 3 Grids Tick Marks and Framing auto x or y Generate default x or y coordinates for tick marks autogrid axis label command axis label command Draw default axis labels using the speci fied commands for the x and y axes An axis label command may be empty or it may be itick otick or grid followed by a label suffix frame label suffix option list Draw a frame around the graph or draw the part of the frame specified by the label suffix grid label suffix f z Draw a grid line across the graph from the side specified by the label suffix and label it there using format string f and coordinate value z If f is a picture it gives the label itick label suffix f z This is like grid except it draws an inward tick mark otick label suffix f z This is like grid except it draws an outward tick mark 15 A 4 Miscellaneous Commands augment variable coordinates Append coordinates to the path stored in variable format f x Typeset x according to format string f and return the resulting picture gdata f variable commands Read the file named by string f and execute commands for each input line using the variable as an array to store data fields init_numbers s m x t e Provi
8. cede the exponent when typesetting a power of ten Calling init_numbers initializes Fe_plus to an empty picture and constructs Fe_base from its second and third arguments 5 Conclusion The graph package makes it convenient to generate graphs from within the MetaPost language The primary benefits are the power of the MetaPost language and its ability to interact with T X or troff for typesetting labels Typeset labels can be stored in picture variables and manipulated in various ways such measuring the bounding box and providing a white background We have seen how to generate shaded regions and control line width color and styles of dashed lines Numerous other variations are possible The full MetaPost language 5 provides many other potentially useful features It also has enough computing power to be useful for generating and processing data A Summary of the Graph Package In the following descriptions italic letters such as w and h denote expression parameters and words in angle brackets denote other syntactic elements Unless specified otherwise expression parameters can be either numerics or strings An option list is a list of drawing options such as withcolor 5white or dashed evenly a label suffix is one of 1ft rt top bot ulft urt 11ft lrt 14 A 1 Graph Administration begingraph w h Begin a new graph with the frame width and height given by numeric parameters w and h endgraph End a graph and return the result
9. de five pictures as templates for future format operations s is a leading minus m is a sample mantissa x follows the mantissa t is a leading minus for the exponent e Mreadpath f Read a path for the data file named by string f and return it in Mlog form A 5 Arithmetic on Numeric Strings It is necessary to input sarith before using the following macros Sabs x Compute z and return a numeric string x Sadd y Compute y and return a numeric string Scvnum x Return the numeric value for string x x Sdiv y Compute z y and return a numeric string x Sleq y Return the boolean result of the comparison x lt y x Smul y Compute x y and return a numeric string x Sneq y Return the boolean result of the comparison x y x Ssub y Compute x y and return a numeric string A 6 Internal Variables and Constants Autoform Format string used by autogrid Default Zg Fe_base What precedes the exponent when typesetting a power of ten Fe_plus Picture of the leading plus sign for positive exponents Gmarks Minimum number of tick marks per axis for auto and autogrid Default 4 Gminlog Minimum largest smallest ratio for logarithmic spacing with auto and autogrid Default 3 0 Gpaths Code for coordinates used in gdraw and gfill paths linear for standard form log for Mlog form Mten The Mlog form for 10 0 16 References 1 Jon L Bentley and Brian W Kernighan Grap a language for typesetting graphs
10. e 5 graphs execution times for two matrix multiplication algorithms using setcoords log log to specify logarithmic spacing on both axes The data file matmul d gives timings for both algo rithms 20 007861 standard MM size seconds 30 022051 40 050391 60 15922 80 4031 120 1 53 160 3 915 240 18 55 320 78 28 480 279 24 20 006611 Strassen size seconds 30 020820 40 049219 60 163281 80 3975 120 1 3125 160 3 04 240 9 95 320 22 17 480 72 60 A blank line in a data file ends a data set Subsequent gdraw commands access additional data sets by just naming the same data file again Since each line gives one x coordinate and one y coordinate commentary material after the second data field on a line is ignored Placing a setcoords command between two gdraw commands graphs two functions in different coordinate systems as shown in Figure 6 Whenever you give a setcoords command the interpreter examines what has been drawn selects appropriate x and y ranges and scales everything to fit Everything drawn afterward is in a new coordinate system that need not have anything in common with the old coordinates unless setrange commands enforce similar coordinate ranges For instance the two setrange commands force both coordinate systems to have x ranging from 80 to 90 and y starting at 0 When you use multiple coordinate systems you have to specify where the axis labels go The default is to put tick marks on the
11. e as in autogrid and label format is either a format string like g or a picture containing the typeset numeric label The axis label commands use a macro format format string numeric or string expression to typeset numeric labels Full details appear in Section 4 but when the format string is g it uses decimal notation unless the number is large enough or small enough to require scientific notation The example in Figure 7 invokes format g y explicitly so that grid lines can be placed at transformed coordinates It defines the transformation newy y y 75 lny and shows that this function increases almost linearly This is a little like using logarithmic y coordinates except that y is mapped to y 75 ln y instead of just In y Figure 7 uses the command frame label suffix option list to draw a special frame around the graph In this case the label suffix is 11 t to draw just the bottom and left sides of the frame Suffixes lrt ulft and urt draw other combinations of two sides suffixes 1ft rt top bot draw one side and empty draws the whole frame For example frame dashed evenly The manual 5 explains how vardef defines functions and mlog computes logarithms vardef newy expr y 256 75 y mlog y enddef draw begingraph 3in 2in glabel 1ft btex vbox hbox Population hbox in millions etex OUT path p gdata timepop d augment p 1 newy Scvnum 2 gdraw p wi
12. ed in the User s Manual 5 In addition to the standard drawing options the option list in a gdraw statement can contain plot picture expression The picture expression gives a plotting symbol to be drawn at each path knot The plot option suppresses line drawing so that gdraw agepop91 d plot btex bullet etex generates only bullets as shown in Figure 2 Following the plot option with a withpen option would cause the line to reappear superimposed on the plotting symbols 6 e 4x 10 a e of 6 3x10 2x106 4 10 ie ne 0 20 40 60 80 Figure 2 The 1991 age distribution plotted with bullets Watch out for the following the picture expression is placed with the lower left corner at the path knot not its center If you want it to be dead center you have to correct the placement yourself For the example above you need something like this instead def MPbullet btex lower fontdimen22 cmsy hbox to Opt hss cmsy chari5 hss etex enddef 4See the User s Manual 5 for explanations of draw commands and syntactic elements like picture expression 5 Troff users should replace btex bullet etex with btex bu etex followed by gdraw agepop91 d plot MPbullet The glabel and gdotlabel commands add labels to a graph The syntax for glabel is glabel label suffix string or picture expression location option list where location identifies the location being labeled a
13. f the conversion letters e f g G e an optional final string p The initial and final strings are typeset in the default font usually cmr10 and the typeset number is placed between them For the e and g formats the precision p is the number of significant digits allowed after rounding for f and G the number is rounded to the nearest multiple of 107 If the precision is not specified the default is p 3 The e format always uses scientific notation and the f format uses ordinary decimal notation but reverts to scientific notation if the number is at least 10000 The g and G formats also revert to scientific notation for non zero numbers of magnitude less than 0 001 The format macro needs a set of templates to determine what font to use how to position the exponent etc The templates are normally initialized automatically but it is possible to set them explicitly by passing five picture expressions to init_numbers For instance the default definition for TEX users is init_numbers btex etex btex ifetex btex times 10 etex btex etex btex 2 etex The first argument tells how to typeset a leading minus sign the second argument is an example of a 1 digit mantissa third comes whatever to put after the mantissa in scientific notation next come a leading minus sign for the exponent and a sample 1 digit exponent Picture variable Fe_plus gives a leading plus sign for positive numbers and Fe_base gives whatever should pre
14. for autogrid otick bot 2 4 Processing Data Files The most general tool for processing data files is the gdata command gdata string expression variable commands It takes a file name a variable v and a list of commands to be executed for each line of the data file The commands are executed with i set to the input line number and strings v1 v2 v3 set to the input fields on the current line A null string marks the end of the v array Using a glabel command inside of gdata generates a scatter plot as shown in Figure 8 The data file countries d begins 20 910 75 7 US 1 831 66 7 Alg where the last field in each line gives the label to be plotted Setting defaultfont in the first line of input selects a small font for these labels Without these labels no gdata command would be needed Replacing the gdata command with gdraw countries d plot btex circ etex would change the abbreviated country names to open circles Both gdraw and gdata ignore an optional initial on each input line parse data fields separated by white space and stop if they encounter an input line with no data fields Leading percent signs make graph data look like MetaPost comments so that numeric data can be placed at the beginning of a MetaPost input file It is often useful to construct one or more paths when reading a data file with gdata The augment command is designed for this augment path variable coordinates If
15. he autogrid otick 1lft to label the left side of the graph with the y coordinates in effect before the setcoords command This suppresses the default axis labels so another autogrid command is needed to label the bottom and right sides of the graph using the new coordinate system The general syntax is autogrid axis label command axis label command option list where axis label command empty grid or tick label suffix grid or tick grid itick otick The label suffix should be 1ft rt top or bot The first argument to autogrid tells how to label the x axis and the second argument does the same for y An empty argument suppresses labeling for that axis Otherwise the label suffix tells which side of the graph gets the numeric label Be careful to use bot or top for the x axis and lft or rt for the y axis Use otick for outward tick marks itick for inward tick marks and grid for grid lines The option list tells how to draw the tick marks or grid lines Grid lines tend to be a little overpowering so it is a good idea to give a withcolor option to make them light gray so they do not make the graph too busy 2 3 Explicit Grids and Framing In case autogrid is not flexible enough axis label commands generate grid lines or tick marks one at a time The syntax is grid or tick label suffix label format numeric or string expression option list where grid or tick and label suffix ar
16. he Graph Macros A MetaPost input file that uses the graph macros should begin with input graph This reads a macro file graph mp and defines the graph drawing commands explained below The rest of the file should be one or more instances of beginfig figure number graphics commands endfig followed by end The following graphics commands suffice to generate the graph in Figure from the data file agepop91 d draw begingraph 3in 2in gdraw agepop91 d endgraph Each line of agepop91 d gives an age followed the estimated number of Americans of that age in 1991 8 4x108 5 3x 10 4 2x106 4 106 0 20 40 60 80 Figure 1 A graph of the 1991 age distribution in the United States 2 1 Basic Graph Drawing Commands All graphs should begin with begingraph width height and end with endgraph This is syntactically a picture expression so it should be preceded by draw and followed by a semicolon as in the example The width and height give the dimensions of the graph itself without the axis labels The command gdraw expression option list draws a graph line If the expression is of type string it names a data file otherwise it is a path that gives the function to draw The option list is zero or more drawing options withpen pen expression withcolor color expression dashed picture expression that give the line width color or dash pattern as explain
17. ing picture setcoords tz ty Set up a new coordinate system as specified by numeric flags tz ty Flag values are linear and 1log setrange coordinates coordinates Set the lower and upper limits for the current coordinate system Each coordinates can be a single pair expression or two numeric or string expressions A 2 Drawing and Labeling All of the drawing and labeling commands can be followed by an option list In addition to the usual MetaPost drawing options the list can contain a plot picture clause to plot a specified picture at each data point The drawing and labeling commands are closely related to a set of similarly named commands in plain MetaPost The gdrawarrow and gdrawdblarrow commands are included to maintain this relationship gdotlabel label suffix p location This is like glabel except it also puts a dot at the location being labeled gdraw p Draw path p or if p is a string read coordinate pairs from file p and draw a polygonal line through them gdrawarrow p This is like dgraw p except it adds an arrowhead at the end of the path gdrawdblarrow p This is like dgraw p except it adds an arrowheads at each end of the path gfill p Fill cyclic path p or read coordinates from the file named by string p and fill the resulting polygonal outline glabel label suffix p location If p is not a picture it should be a string Typeset it using defaultfont then place it near the given lo
18. nd label suffix tells how the label is offset relative to that location The gdotlabel command is identical except it marks the location with a dot A label suffix is as in plain MetaPost empty centers the label on the location 1ft rt top bot offset the label horizontally or vertically and ulft urt 11ft lrt give diagonal offsets The location can be a pair of graph coordinates a knot number on the last gdraw path or the special location OUT Thus gdotlabel top btex 50 0 etex 50 0 would put a dot at graph coordinates 50 0 and place the typeset text 50 0 above it Alter natively glabel ulft Knot3 3 typesets the string Knot3 and places it above and to the left of Knot 3 of the last gdraw path The knot number 3 the path s time parameter 5 Section 8 2 The location OUT places a label relative to the whole graph For example replacing gdraw agepop91 d with glabel 1ft btex vbox hbox Population hbox in millions etex OUT glabel bot btex Age in years etex OUT gdraw agepopm d in the input for Figure 1 generates Figure 3 This improves the graph by adding axis labels and using a new data file agepopm d where the populations have been divided by one million to avoid large numbers We shall see later that simple transformations such as this can be achieved without generating new data files 4 3 4 Population in millions 2 1 0 20
19. ng labels The expression image unfill bbox lab j draw lab j executes the given drawing commands and returns the resulting picture unfill bbox lab j puts down a white background and draw lab j puts the label on the background The gfill command is just like gdraw except it takes a cyclic path and fills the interior with a solid color The color is black unless a withcolor clause specifies another color See the manual 5 for explanations of for loops arrays colors and path construction operators like cycle and reverse 3 Manipulating Big Numbers MetaPost inherits a fixed point number system from Knuth s METAFONT 6 Numbers are expressed in multiples of 2716 and they must have absolute value less than 32768 Knuth chose this system because it is perfectly adequate for font design and it guaranteed to give identical results on all types 11 draw begingraph 3in 2in glabel 1ft btex vbox hbox Quadrillions hbox of BTU etex OUT path p numeric t gdata energy d t 0 augment p1 1 0 for j 2 upto 5 t ttscantokens j augment p j 1 t endfor picture lab lab2 btex coal etex lab3 btex crude oil etex lab4 btex natural gas etex labbd btex hydroelectric etex for j 5 downto 2 gfill p j reverse p j 1 cycle withcolor 16j white glabel 1ft image unfill bbox lab j draw lab j 7 length p j endfor endgraph 60 hydroelectric gA natural gas Quadrillions 7 o BT
20. of graph is best in a given situation is covered elsewhere e g Cleveland 2 3 4 and Tufte 9 The goal is to provide at least the power of UNIX grap 1 but within the MetaPost language Hence the package is implemented using MetaPost s powerful macro facility The graph macros provide the following functionality l 2 GE ae 6 Automatic scaling Automatic generation and labeling of tick marks or grid lines Multiple coordinate systems Linear and logarithmic scales Separate data files Ability to handle numbers outside the usual range IMETAFONT is a trademark of Addison Wesley Publishing Company PostScript is a registered trademark of Adobe Systems Inc SUNIX is a registered trademark of UNIX System Laboratories Inc 7 Arbitrary plotting symbols 8 Drawing filling and labeling commands for graphs In addition to these items the user also has access to all the features described in the MetaPost user s manual 5 These include access to almost all the features of PostScript ability to use and manipulate typeset text ability to solve linear equations and data types for points curves pictures and coordinate transformations Section 2 describes the graph macros from a user s perspective and presents several examples Sections 3 and 4 discuss auxiliary packages for manipulating and typesetting numbers and Section 5 gives some concluding remarks Appendix A summarizes the graph drawing macros 2 Using t
21. rk as well but whatever is the standard MetaPost representation for an anonymous unknown value draw begingraph 3in 2in glabel 1ft btex vbox hbox Population hbox in millions etex OUT glabel bot btex Age in years etex OUT setrange origin whatever whatever gdraw agepopm d endgraph 4 4 354 Population in millions 2 1s 0 I I I I 0 20 40 60 80 Age in years Figure 4 The 1991 age distribution graph and the input that creates it Notice that the syntax for setrange allows coordinate values to be given as strings Many commands in the graph package allow this option It is provided because the MetaPost language uses fixed point numbers that must be less than 32768 This limitation is not as serious as it sounds because good graph design dictates that coordinate values should be of reasonable magnitude 2 9 If you really want x and y to range from 0 to 1 000 000 setrange origin 1e6 1e6 does the job Any fixed or floating point representation is acceptable as long as the exponent is introduced by the letter e Coordinate systems need not be linear The setcoords command allows either or both axes to have logarithmic spacing coordinate setting setcoords coordinate type coordinate type coordinate type log linear log Linear A negative coordinate type makes x or y run backwards so it is largest on the left side or bottom of the graph Figur
22. the path variable does not have a known value it becomes a path of length zero at the given coordinates otherwise a line segment to the given coordinates is appended to the path The coor dinates may be a pair expression or any combination of strings and numerics as explained at the beginning of Section 2 2 10 defaultfont cmr7 draw begingraph 3in 2in glabel 1ft btex vbox hbox Life hbox expectancy etex OUT glabel bot btex Per capita G N P thousands of dollars etex OUT setcoords log linear gdata countries d s glabel s3 s1 s2 endgraph 80 J wi aS Gre Spn Por Tai rele TEMS Ree Sr Ar 70 4 Chn Eo SKo USS al Por Inpgart Life K expectancy 60 4 Anri Pey Pik Bar mati Sud Uga ep 50 Nig Moz 0 1 0 2 05 1 2 5 10 20 Per capita G N P thousands of dollars Figure 8 A scatter plot and the commands that generated it If a file timepop d gives t p pairs augment can be used like this to graph newy p versus t path p gdata timepop d s augment p s1 newy scantokens s2 gdraw p MetaPost s scantokens primitive interprets a string as if it were the contents of an input file This finds the numeric value of data field s2 Figure 9 shows how to use augment to read multiple column data and make multiple paths Paths p2 p3 p4 p5 give cumulative totals for columns 2 through 5 and pictures lab2 through lab5 give correspondi
23. thpen nullpen for y 5 10 20 50 100 150 200 250 grid lft format Zg y newy y withcolor 85white endfor autogrid grid bot withcolor 85white gdraw p frame 1llft endgraph 250 200 150 100 Population in millions 50 20 10 5 1800 1850 1900 1950 2000 Figure 7 Population of the United States in millions versus time with the population re expressed as p 75 lnp The MetaPost input shown above the graph assumes a data file ttimepop d that gives year p 75 In p pairs draws all four sides with dashed lines The default four sided frame is drawn only when there is no explicit frame command To label an axis as autogrid does but with the labels transformed somehow use auto x or auto y for positioning tick marks or grid lines These macros produce comma separated lists for use in for loops Any x or y values in these lists that cannot be represented accurately within MetaPost s fixed point number system are given as strings A standard macro package that is loaded via input sarith defines arithmetic operators that work on numbers or strings Binary operators Sadd Ssub Smul and Sdiv do addition subtraction multiplication and division One possible application is rescaling data Figure 4 used a special data file agepopm d that had y values divided by one million This could be avoided by replacing gdraw agepopm d by gdraw agepop91 d for u auto y otick 1ft format g u Sdiv 1e6 u end
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