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Learning METAPOST by Doing

1. f drawit pic slanted 0 5 en drawit pic slanted 0 5 eve Se 1 drawit pic xscaled 2 Sus drawit pic yscaled 1 Learning MetaPost by doing VOORJAAR 2005 25 C 71 2 drawit pic zscaled 2 0 5 endfig a _ I end The effect of rotated 0 is rotation of O degrees about the origin counter clockwise The transformation rotatedaround p rotates degrees counter clockwise around point p Accordingly it is defined in METAPOST as follows def rotatedaround expr p theta rotates theta degrees around p shifted p rotated theta shifted p enddef x When you identify a point x y with the 3 vector y each of the above op 1 erations is described by an affine matrix For example the rotation of O degrees around the origin counter clockwise and the translation with a b have the fol lowing matrices cos sin 0 1 Oa rotated 0 sinO cos Oj translated a b 0 1 bj 0 0 1 0 O0 1 It is easy to verify that rotatedaround a b 9 10a cos sin 0 1 0 a O 1 bj sind cos 0 O 10 1 b 0 0 1 0 0 1 0 o0 1 The matrix of zscaled a b is as follows a b 0 zscaled a b b a O 0 0 1 Thus the effect of zscaled a b is to rotate and scale so as to map 1 0 into a b The operation zscaled can also be thought of as multiplication of complex numbers The picture on the next page illustrates this The general form of an affine matrix T is Tex Tey Tx T Ty
2. The errmessage command is for displaying an error message if something goes wrong and interrupting the program at this point The show statement is used here for debugging purposes When you run the METAPOST program froma shell show puts its results on the standard output device In our example the shell window looked like 36 MAPS 32 Andr Heck heck remote 1 mpost barycenter This is MetaPost Version 0 641 Web2C 7 3 1 barycenter mp gt gt location 13 14597 80 09227 gt gt weight 1 0579 gt gt location 19 7488 43 93861 gt gt weight 1 39641 gt gt location 43 89838 7 07126 gt gt weight 1 37593 gt gt location 2 69252 9 70473 gt gt weight 0 48659 gt gt location 24 17944 25 14096 gt gt weight 2 22429 gt gt location 67 98569 55 73247 gt gt weight 0 73831 gt gt location 20 28859 76 48691 gt gt weight 0 00526 gt gt location 67 07672 18 69904 gt gt weight 0 68236 1 1 output file written barycenter 1 Transcript written on barycenter log heck remote 2 The boolean expression that forms the condition can be built up with the following relational and logical operators Relational Operators Operator Meaning equal lt gt unequal lt less than lt less than or equal gt greater than gt greater than or equal Logical Operators Operator Meaning and
3. label btex lbrace etex xscaled 1 5 yscaled 5 7 V3 rotated 60 1 2 z0 z2 dir 120 2mm labeloffset 3 5mm label bot btex sqrt 3 etex 1 2 z0 z1 label rt btex 1 etex 1 2 z1 z2 label btex 2 etex 1 2 z0 z2 dir 120 5mm endfig end Until now we have only used plain T X commands But what if you want to run another TrX version The following example shows how you can use a verbatimtex etex block to specify that LaTgX is used and which style and or packages are chosen verbatimtex amp latex documentclass article begin document etex beginfig 1 zO 0 0 z1 sqrt 3 cm 0 z2 sqrt 3 cm 1cm V3 draw z0 z1 z2 cycle label bot btex sqrt 3 etex 1 2 z0 z1 label rt btex frac 1 2 etex 1 2 z1 z2 label top btex 1 etex 1 2 z0 z2 endfig N end One last remark about using LalpX Between btex and etex you can not use displayed math mode such as frac x x 1 You must use displaystyle frac x x 1 instead Let us use what we have learned so far in this chapter in a more practical ex X from 0 to 5 with the vertical ample drawing the graph of the function x gt i x axis in a logarithmic scale The picture is generated by the following METAPOST code verbatimtex Zelatex documentclass article begin document etex some function definitions vardef exp expr x mexp 256 x enddef vardef In expr x mlog x 2
4. 0 1cm and 1icm 0 endfig pl beginfig 8 drawsquare drawarrow pO curl 80 0 1cm curl 8 p1 endfig beginfig 9 drawsquare drawarrow p0 tension 2 0 1cm p1 endfig end and have been defined in terms of curl and tension directives as follows def curl i curl 1 enddef def tension infinity enddef def tension atleast 1 enddef The meaning of is choose an inflection free path between the points unless the endpoint directions make this impossible The meaning of is get a smooth connection between a straight line and the rest of the curve pair p p0 0 0 pi 1cm icm def drawsquare draw unitsquare scaled 1cm withcolor 0 7white enddef beginfig 1 drawsquare endfig drawarrow p0 1 5cm 0 p1 beginfig 2 drawsquare endfig drawarrow pO 1 5cm 0 p1 end The above examples were also meant to give you the impression that you can draw in METAPOST almost any curve you wish EXERCISE 15 Draw an angle of 40 degrees that looks like 14 MAPS 32 Andr Heck EXERCISE 16 Draw a graph that looks like EXERCISE 17 Draw the Yin Yang symbol that looks like Angle and Direction Vector In a previous exercise you have already seen that the dir command generates a pair that is a point on the unit circle at a given angle with the horizontal axis The inverse of dir is angle which takes a pair interprets it
5. 90 dir 150 cycle endfig end gt t gt RSA If you want arrowheads of different size you can change the arrowhead length through the variable ahlength 4bp by default and you can control the angle at the tip of the arrowhead with the variable ahangle 45 degrees by default You can also completely change the definition of the arrowhead procedure In the example below we draw a curve with an arrow symbol along the path As a matter of fact the path is drawn in separate pieces that are joined together with the amp operator beginfig 1 save arrowhead vardef arrowhead expr p save A u a b pair A u path a b A point length p 2 of p u unitvector direction length p 2 of p a A u A ahlength u rotated 30 b A u A ahlength u rotated 30 a amp reverse a amp b amp reverse b cycle enddef u 2cm ahlength 0 3cm drawarrow 0 0 u u u u endfig end Circle Ellipse Square and Rectangle You have already seen that you can draw a circle through three points z0 z1 and z2 that do not lie on a straight line with the statement draw z0 z1 z2 cycle But METAPOST also provides predefined paths to build circles and circular disks or parts of them Similarly you can draw a rectangle once the four corner points say z0 z1 z2 and z3 are known with the statement draw z0 z1 z2 z3 cycle The path 0 0 1 0 1 1 0 1 cycleis in METAPOST predefin
6. When typesetting is successful the device independent file sample pdf is gen erated The Structure of aMETAPOST Document We shall use the above examples to explain the basic structure of a METAPOST document We start with a closer look at the slightly modified METAPOST code in the file example mp of our first example beginfig 1 draw a square draw 0 0 10 0 10 10 0 10 0 0 endfig end This example illustrates the following list of general remarks about regular META POST files It is recommended to end each METAPOST program in a file with extension mp so that this part of the name can be omitted when invoking METAPOST Each statement in a METAPOST program is ended by a semicolon Only in cases where the statement has a clear endpoint e g in the end and endfig statement you may omit superfluous semicolons We shall not do this in this tutorial You can put two or more statements on one line as long as they are separated by semicolons You may also stretch a statement across several lines of code and you may insert spaces for readability You can add comments in the document by placing a percentage symbol in front of the commentary A METAPOST document normally contains a sequence of beginfigand endfig pairs with an end statement after the last one The numeric argument to the beginfig macro determines the name of the output file that will contain the PostScript code gener
7. p2 cycle for i 0 upto length q drawpoint point i of q black p3 precontrol i of q p4 postcontrol i of q draw p3 p4 withcolor gray drawpoint p3 gray drawpoint p4 gray endfor draw q endfig end Do not worry when you do not understand all details of the above METAPOST program It contains features and programming constructs that will be dealt with later in the tutorial There are various ways of controlling curves Vary the angles at the start and end of the curve with one of the keywords up down left and right or with the dir command Specify the requested control points manually Vary the inflection of the curve with tension and curl tension influences the curvature whereas curl influences the approach of the starting and end points pair p p0 0 0 pi 1cm 1cm def drawsquare draw unitsquare scaled 1cm withcolor 0 7white enddef beginfig 1 drawsquare endfig beginfig 2 drawsquare endfig beginfig 3 drawsquare endfig beginfig 4 drawsquare endfig Dogs drawarrow p0 p1 drawarrow pO right p1 drawarrow pO up p1 drawarrow pO left p1 Learning MetaPost by doing Duuo The METAPOST operators VOORJAAR 2005 13 beginfig 5 drawsquare endfig drawarrow pO down p1 beginfig 6 drawsquare endfig drawarrow pO dir 45 p1 beginfig 7 drawsquare drawarrow p0 controls
8. with its three medians 2 The dotlabel command allows you to mark a point with a dot and to position some text around it For instance dotlabel 1ft A 0 0 generates a dot with the label A to the left of the point Other dotlabel suffixes and their meanings are shown in the picture below eae ulft eurt bot llft lrt Use the dotlabel command to put labels in the picture in part 1 so that it looks like Ce 5The A median of a triangle ABC is the line from A to the midpoint of the opposite edge BC Learning MetaPost by doing VOORJAAR 2005 9 3 Recall that 1 2 z1 z2 denotes the point halfway between the points z1 and z2 Similarly 1 3 z1 z2 denotes the point on the line connecting the points z1 and z2 one third away from z1 For a numeric variable which is possibly unknown yet c z1 z2 is c times of the way from z1 to z2 If you do not want to waste a name for a variable use the special name whatever to specify a general point on a line connecting two given points whatever z1 z2 denotes some point on the line connecting the points z1 and z2 Use this fea ture to define the intersection point of the medians also known as the centre of gravity and extend the above picture to the one below Use the label com mand which is similar to the dotlabel command except that it does not draw a dot to position the character G around the centre of gravity If necessary assign labeloffset another value so that the la
9. xa ya MM xm ym projection axes AA 0 5 1 7 projection axes MM 0 5 1 7 label bot btex a etex rpoint xa 0 label bot btex x etex rpoint xm 0 label 1ft btex f a etex rpoint 0 ya label 1ft btex f x etex rpoint 0 ym hh fin des lectures draw c withpen pencircle scaled 1 5pt withcolor 2 4 9 draw droite A M 1 2 draw tangente 0 6 9 withpen pencircle scaled 1 5pt dotlabel ulft btex A etex scaled 1 5 A dotlabel M label btex displaystyle frac f x f a x a etex scaled 1 25 M shifted 1cm icm label ulft btex f a etex scaled 1 25 5cm 5cm label bot btex graph of f etex scaled 1 25 point 2 of c hh Cartouche fill cartouche withcolor 8white draw cartouche label rt btex textbf Derivative at a point etex 3cm 5mm hh fin du cartouche decoupe repere etiquette axes endfig end References GRS94 Michel Goossens Sebastian Rahtz Frank Mittelbach The LaTex Graphics Companion Addison Wesley 1994 ISBN 0 201 85469 4 Hag02 Hans Hagen The Metafun Manual 2002 downloadable as www pragma ade com general manuals metafun p pdf Hob92a John D Hobby A User s manual for MetaPost AT amp T Bell Laboratories Computing Science Technical Report 162 1992 Hob92b John D Hobby Drawing Graphs with MetaPost AT amp T Bell Laboratories Computing Science Technical Report 164 1992
10. as a vector and computes the two argument arctangent i e it gives the angle corresponding with the vector In the example below we use it to draw a bisector of a triangle C pair A B C u 1cm A 0 0 B 5u 0 C 2u 3u C whatever A B C whatever dir 1 2 angle A C 1 2 angle B C beginfig 1 draw A B C cycle draw C C dotlabel 1ft A A dotlabel urt B B dotlabel top C C dotlabel bot C C endfig A B end C EXERCISE 18 Change the above picture to the following geometrical diagram which illustrates better that a bisector is actually drawn for the acute angled triangle C B A EXERCISE 19 Draw a picture that shows all the bisectors of a acute angled triangle Your picture should look like 7See www chinesefortunecalendar com YinYang htm for details about the symbol Learning MetaPost by doing VOORJAAR 2005 15 B In this way it illustrates that the bisectors of a triangle go through one point the so called incenter which is the centre of the inner circle of the triangle Arrow The drawarrow command draws the given path with an arrowhead at the end For double headed arrows simply use the drawdblarrow command A few examples beginfig 1 drawarrow 0 0 60 0 drawarrow reverse 0 20 60 20 withpen pencircle scaled 2 drawdblarrow 0 40 60 40 drawarrow 0 65 dir 30 dir 30 60 65 drawarrow 0 90 dir 30 up 60
11. by elseif in which case the corresponding fi must be omitted For example nesting of two basic if operations looks as follows if 1st condition 1st tokens elseif 2nd condition 2nd tokens fi Let us give an example computing the centre of gravity also called barycentre of a number of objects with randomly generated weight and position The example contains many more programming constructs some of which will be covered in sections later on in the tutorial so you may ignore them if you wish The nested conditional statement is easily found in the METAPOST code below With the commands numeric x and pair x we test whether x is a number or a point respectively beginfig 1 vardef centerofgravity text t save x wght G X pair G X numeric wght w G origin wght 0 for x t if numeric x show weight amp decimal x G G x X wght wght x elseif pair x show location amp decimal xpart x amp amp 2 22429 a decimal ypart x amp 1 37593 0 48659 X x store pair else errmessage should not happen 0 68236 a endfor G wght 1 39641 enddef 0 73831 numeric w pair A e n 8 1 0579 0 00526 for i 1 upto n Afi 1 5cm normaldeviate normaldeviate w i abs normaldeviate dotlabel bot decimal w i A i endfor draw centerofgravity A 1 w 1 for i 2 upto n A i w i endfor withpen pencircle scaled 4bp withcolor 0 7white endfig end
12. possible but there are no abbreviations balancing with respect to for and endfor is obligatory You must use both endfor keywords in 38 MAPS 32 EXERCISE 27 EXERCISE 28 EXERCISE 29 EXERCISE 30 Andr Heck for i 0 upto 10 for j 0 upto 10 show i amp decimal i amp j amp decimal j endfor endfor Create the following picture Create the picture below which illustrates the upper and lower Riemann sum for the area enclosed by the horizontal axis and the graph of the function f x 4 x Create the following piece of millimetre paper A graph is bipartite when its vertices can be partitioned into two disjoint sets A and B such that each of its edges has one endpoint in A and the other in B The most famous bipartite graph is Ks show below to the left Write a program that draws the Kn n graph for any natural number n gt 1 Show that your program indeed creates the graph Ks s which is shown below to the right Learning MetaPost by doing VOORJAAR 2005 39 a3 b3 a3 a2 b2 a2 SOT XS LPS al bl al Another popular type of repetition is the conditional loop METAPOST does not have a pre or post conditional loop while loop or until loop built in You must create one by an endless loop and an explicit jump outside this loop First the endless loop this is created by forever loop text endfor To terminate such a loop when a boolean condition becomes true use an exit cla
13. t t ci Pp where t 0 1 METAPOST automatically calculates the control points such that the segments have the same direction at the interior knots In the figure below the additional control points are drawn as gray dots and connected to their parent point with gray line segments The curve moves from the starting point in the direction of the post control point but possibly bends after a while in another direction The further away the post control point is the longer the curve keeps this direction Similarly the curve arrives at a point coming from the direction of the pre control point The further away the pre control point is the earlier the curve gets this direction It is as if the control points pull their parent point in a certain direction and the further away a control point is the stronger it pulls By default in META POST the incoming and outgoing direction at a point on the curve are the same so that the curve is smooth u 1 25cm color gray gray 0 6white pair p pO 0 0 p1 2u 3u p2 3u 2u def drawpoint expr z c draw z withpen pencircle scaled 3 withcolor c enddef beginfig 1 path q q pO pi p2 for i 0 upto length q drawpoint point i of q black p3 precontrol i of q p4 postcontrol i of q draw p3 p4 withcolor gray drawpoint p3 gray drawpoint p4 gray endfor draw q endfig 12 MAPS 32 beginfig 2 path q q Andr Heck pO pi
14. test if all conditions hold or test if one of many conditions hold not negation of condition One final remark on the use of semicolons in the conditional statement Where the colons after the if and else part are obligatory semicolons are optional depending on the context For example the statement if cycle p fill p elseif path p draw p else errmessage what fi fills or draws a path depending on the path p being cyclic or not You may omit whatever semicolon in this example and rewrite it even as if cycle p fill p elseif path p draw p else errmessage what fi Learning MetaPost by doing VOORJAAR 2005 37 However when you use the conditional clause to build up a single statement then you must be more careful with placing or omitting semicolons In draw p withcolor if cycle p red else blue fi withpen pensquare you cannot add semicolons after the colour specifications nor omit the final semi colon that marks the end of the statement unless it is a statement that is recognised as finished because of another keyword e g endfor Repetition Numerous examples in previous section have used the for loop of the form for counter start step stepsize until finish loop text endfor where counter is a counting variable with initial value start The counter is incre mented at each step in the repetition by the value of stepsize until it passes the value of finish Then the repetition stops and METAP
15. two parts a preprocessor and a PostScript generator The preprocessor only reads from the input stream and prepares input for the PostScript generator EXERCISE 20 Draw the graph of the function x gt y on the interval 0 2 Your picture should look like Your METAPOST code should be such that only a minimal change in the code is required to draw the graph on a different domain say 0 3 EXERCISE 21 Draw the following picture in METAPOST The dashed lines can be drawn by adding dashed evenly at the end of the draw statement C ib a ib z EXERCISE 22 The annual beer consumption in the Netherlands in the period 1980 2000 is listed below year 1980 1985 1990 1995 2000 litre 86 85 91 86 83 Draw the following graph in METAPOST Learning MetaPost by doing Style Directives beer consumption liter VOORJAAR 2005 21 Kej N Kej n GO o0 o0 aD o0 A CO N A Lv 1980 1985 1990 1995 2000 year In this section we explain how you can alter the appearance of graphics primitives e g allowing certain lines to be thicker and others to be dashed using different colours and changing the type of the drawing pen Dashing Examples show you best how the specify a dash pattern when drawing a line or curve beginfig 1 path p p 0 0 102 0 def drawit suffix p expr pattern draw p dashed pattern p p shifted 0 13 enddef dra
16. with the exponential curve as solution curve vardef f expr x y y enddef define routine to compute function values def compute_curve suffix g expr xmin xmax xinc C xmin g xmin for x xmint xinc step xinc until xmax x g x endfor enddef is i tags ag eg ea age i ig ag a ag aaa T compute and draw exponential curve vardef exp expr x mexp 256 x enddef path p p compute_curve exp xmin xmax xinc scaled u draw p draw direction field pair vec path v for x xmin step 0 5 until xmax for y ymin 0 5 step 0 5 until ymax 0 5 vec unitvector 1 f x y scaled 1 2u v 0 0 vec shifted 1 2vec drawarrow v shifted x u y u withcolor blue endfor endfor draw directions along the exponential curve for x 0 5 step 0 5 until xmax vec unitvector 1 f x exp x scaled 1 2u v 0 0 vec shifted 1 2vec drawarrow v shifted x u exp x u withcolor red endfor draw ticks and labels for x round xmin upto xmax draw x 0 05 u x 0 05 u endfor for y round ymin upto ymax draw 0 05 y u 0 05 y u endfor label bot btex x etex xmax 0 5 0 u label 1ft btex y etex 0 ymax 0 5 u 50 MAPS 32 Andr Heck label btex y e x etex xmax exp xmax 0 5 u endfig end Now we shall show some other examples of directional fields corresponding with ODE s and solution curves using the macro package courbe from Jean Michel Sarlat whi
17. xmin ymax u define routine to compute function values def compute_curve suffix g expr xmin xmax xinc C xmin g xmin for x xmintxinc step xinc until xmax x g x endfor 0 5 enddef 0 5 1 compute and draw cosine curve numeric pi pi 3 1415926 numeric radian radian 180 pi 2pi radian 360 vardef cos primary x cosd x radian enddef path p p compute_curve cos xmin xmax xinc scaled u draw p draw identity graph draw xmin ymin u xmax xmax u withcolor 0 5white orbit 0 5 Learning MetaPost by doing 2 9 3 4 3 9 VOORJAAR 2005 53 compute the orbit starting from some point numeric x initial orbitlength x 1 0 the starting point initial 1 some initial iterations orbitlength 15 number of iterations for i 1 upto initial do initial iterations x cos x endfor dotlabel x cos x u mark starting point for i 1 upto orbitlength draw hooks draw x cos x u cos x cos x u cos x cos cos x u withcolor blue x cos x next value endfor draw axis labels labeloffset 0 25cm label bot decimal xmin xmin ymin u label bot decimal xmax xmax ymin u label 1ft decimal ymin xmin ymin u label 1ft decimal ymax xmin ymax u endfig beginfig 2 numeric rmin rmax r dr n ux uy rmin 2 9 rmax 3 9 r rmin n 175 dr rmax rmin n
18. 0 vardef tand primary x sind x cosd x enddef vardef cotd primary x cosd x sind x enddef vardef sin primary x sind x radian enddef vardef cos primary x cosd x radian enddef vardef tan primary x sin x cos x enddef vardef cot primary x cos x sin x enddef hyperbolic functions vardef sinh primary x save xx xx exp xx xx 1 xx 2 enddef vardef cosh primary x save xx xx exp xx xx 1 xx 2 enddef vardef tanh primary x sinh x cosh x enddef vardef coth primary x cosh x sinh x enddef inverse trigonometric and hyperbolic functions vardef arcsind primary x angle 1 x x enddef vardef arccosd primary x angle x 1 x enddef vardef arcsin primary x arcsind x radian enddef vardef arccos primary x arccosd x radian enddef vardef arccosh primary x 1n x xt 1 enddef vardef arcsinh primary x 1n x xt 1 enddef 46 MAPS 32 Andr Heck Most definitions speak for themselves except that you may not be familiar with Pythagorean addition and subtraction a b Va2 b a b Va b More Examples The examples in this section are meant to give you an idea of the strength of META POST Electronic Circuits mpcirc is a macro package for drawing electronic circuits developed by Tomasz Cholewo and downloadable from http ci uof1l edu tom software LaTeX mpcirc Let us use it to create some dia
19. 50i 1503 z 3i 9j 1 150i 50 1503 50 Z 3i 9j 2 150i 100 1503 endfor endfor drawoptions withpen pencircle scaled 24 withcolor 0 75white linejoin mitered linecap butt draw z0 z1i z2 linejoin mitered linecap rounded draw z3 z4 z5 linejoin mitered linecap squared draw z6 z7 z8 linejoin rounded linecap butt draw z9 z10 z11 linejoin rounded linecap rounded draw z12 z13 z14 Learning MetaPost by doing VOORJAAR 2005 29 linejoin rounded linecap squared draw z15 z16 z17 linejoin beveled linecap butt draw z1i8 z19 z20 linejoin beveled linecap rounded draw z21 z22 z23 linejoin beveled linecap squared draw z24 z25 z26 drawoptions for i 0 upto 26 dotlabel bot decimal i z i endfor labeloffset 25pt label bot linejoin mitered z1 labeloffset 40pt label bot linecap butt z1 labeloffset 25pt label bot linejoin mitered z4 labeloffset 40pt label bot linecap rounded z4 labeloffset 25pt label bot linejoin mitered z7 labeloffset 40pt label bot linecap squared z7 h labeloffset 25pt label bot linejoin rounded z10 labeloffset 40pt label bot linecap butt z10 labeloffset 25pt label bot linejoin rounded z13 labeloffset 40pt label bot linecap rounded 213 labeloffset 25pt label bot linejoin rounded z16 labeloffset 40pt label bot lin
20. 56 enddef vardef log expr x 1n x 1n 10 enddef vardef f expr x exp x 1 x enddef ux 1cm uy 4cm Learning MetaPost by doing VOORJAAR 2005 19 beginfig 1 numeric xmin xmax ymin ymax xmin 0 xmax 6 ymin 0 ymax 2 draw axes draw xmin 0 ux xmax 1 2 0 ux 100 draw 0 ymin uy 0 ymax 1 10 uy draw tickmarks and labels on horizontal axis for i 0 upto xmax draw i 0 05 ux i 0 05 ux label bot decimal i i 0 ux endfor draw tickmarks and labels on vertical axis for i 2 upto 10 draw 0 01 log i uy 0 01 log i uy x 50 graph of 7 4 draw 0 01 log 10 i uy 0 01 log 10 i uy S endfor o for i 0 upto 2 label lft decimal 10 i 0 i uy endfor a 10 for i 0 upto 1 label 1ft decimal 5 10 i a 0 log 5 10 i uy endfor E compute and draw the graph of the function D xinc 0 1 2 5 path pts_f pts_f xmin ux log f xmin uy for x xmin xinc step xinc until xmax x ux log f x uy endfor draw pts_f withpen pencircle scaled 2 draw title label btex graph of displaystyle x mapsto frac e x 1 x etex 2ux 1 7uy 1 t t draw axis explanation 0 1 2 3 4 5 6 labeloffset 0 5cm linear scale label bot btex linear scale etex 3 0 ux label 1ft btex logarithmic scale etex rotated 90 0 1 uy endfig end The above code needs some explanation First of all METAPOST does
21. 6u 2u rotated 45 fill quarterdisk rotated 90 scaled 2u shifted 8u 3u fill unitsquare scaled u shifted 0 2u fill unitsquare xscaled u yscaled 3 2u shifted 2u 2u endfig end Text You have already seen how the dotlabel command can be used to draw a dot and a label in the neighbourhood of the dot If you do not want the dot simply use the Learning MetaPost by doing VOORJAAR 2005 17 label command label suffix string expression pair It uses the same suffixes as the dotlabel command to position the label relative to the given pair No suffix means that the label is printed with its centre at the specified location Available directives for the specification of the label relative to the given pair are see also page 8 top top ulft upper left lft left urt upper right rt right lrt lower right bot bot 11ft lower left The distance from the pair to the label is set by the numeric variable labeloffset The commands label and dotlabel both use a string expression for the label text and typeset it in the default font which is likely to be cmr10 and which can changed through the variables defaultfont and defaultscale For example defaultfont ptmr defaultscale 12pt fontsize defaultfont makes labels come out as Adobe Times Roman at about 12 points Until now the string expression in a text command has only been a string delimited by double quotes optionally joined to another string v
22. 980 1990 1990 2000 2000 Compare this example with your own code in exercise 22 Learning MetaPost by doing VOORJAAR 2005 45 A few changes in the code draws the data point as bullets changes the frame style limits the number of tick marks on the horizontal axis and puts labels near the axes e n 90 beginfig 1 2 draw begingraph 4cm 4cm a gdraw consumption dat plot btex bullet etex a 88 5 Gmarks 3 limit number of ticks g frame llft Q e e glabel 1ft btex consumption in litres etex S 86 5 rotated 90 OUT g 2 glabel bot btex year etex OUT 7 84 4 endgraph a e endfig o end 1980 1990 2000 year With the graph package you can easily change in a plot the ticks the scales the grid used an or displayed the title of the plot and so on Mathematical functions The following METAPOST code defines some mathematical functions that are not built in Note that METAPOST contains two trigonometric functions sind and cosd but they expect their argument in degrees not in radians vardef sqr primary x x x enddef vardef log primary x if x 0 0 else mlog x mlog 10 fi enddef vardef ln primary x if x 0 0 else mlog x 256 fi enddef vardef exp primary x mexp 256 x enddef vardef inv primary x if x 0 0 else x 1 fi enddef vardef pow expr x p x p enddef trigonometric functions numeric pi pi 3 1415926 numeric radian radian 180 pi 2pixradian 36
23. Andr Heck VOORJAAR 2005 Learning METAPOST by Doing Introduction 1 Building Cycles 30 Clipping 31 A Simple Example 2 Dealing with Paths Parametrically 31 Running METAPOST 2 Using the Generated PostScript in a LaTeX Control Structures 34 Document 3 Conditional Operations 34 The Structure of aMETAPOST Document 4 Repetition 37 Numeric Quantities 5 If Processing Goes Wrong 6 Macros 39 Defining Macros 39 Basic Graphical Primitives 7 Grouping and Local Variables 40 Pair 7 Vardef Definitions 41 Path 9 Defining the Argument Syntax 41 Angle and Direction Vector 14 Precedence Rules of Binary Operators 42 Arrow 15 Recursion 42 Circle Ellipse Square and Rectangle 15 Using Macro Packages 44 Text 16 Mathematical functions 45 Style Directives 21 More Examples 46 Dashing 21 Electronic Circuits 46 Colouring 22 Marking Angles and Lines 47 Specifying the Pen 23 Vectorfields 49 Setting Drawing Options 23 Riemann Sums 51 Iterated Functions 52 Transformations 24 A Surface Plot 54 Miscellaneous 55 Advanced Graphics 27 Joining Lines 27 Solutions to the exercises 57 Introduction TEX is the well known typographic programming language that allows its users to produce high quality typesetting especially for mathematical text METAPOST is the graphic companion of TEX It is a graphic programming language developed by John Hobby that allows its user to produce high quality graphics It is based on Donald Knuth s METAFONT but with PostScript o
24. F y 1 y Riemann Sums VOORJAAR 2005 51 beginfig 1 repere 10cm 10cm 5cm 5cm 2cm 2cm trace axes 0 5pt marque unites 1mm Champs de directions vardef F expr x y 1 y 2 enddef champ segments 0 0 0 2 0 1 0 5white Courbes int grales for n 0 upto 16 a n 2 4 draw ftrace 1 5 a 1 5 a 50 en_place withcolor 0 5 0 6 0 1 endfor decoupe repere etiquette axes etiquette unites label btex y 1 y 2 etex scaled 2 5 rpoint 0 3 endfig end Another example of the courbes mp macro package is the following illustration of Riemann sums We show two pictures for different number of segments Riemann Sum E ei y f x Ya FE S FErz i 1 Riemann Sum S DS F amp i 24 1 verbatimtex Z amp latex documentclass article everymath displaystyle begin document etex input courbes vardef fx expr t vardef fy expr t t enddef t 5 sin t cos t 2 enddef beginfig 1 numeric a b n h a 1 b 7 n 6 h b a n color aubergine aubergine 37 256 2 256 29 256 repere 10cm 10cm 2cm 3cm 1cm 1cm fill a 0 ftrace a b 200 b 0 cycle en_place withcolor 0 7red for i 1 upto n path cc aa a i 1 h bb aa h ff fy aa h 2 cc rpoint aa 0 rpoint aa ff rpoint bb ff rpoint bb 0 fill cc cycle withcolor 0 8white draw cc endfor trace axes 0 5pt
25. I prepare L R C Vac mention your elements z0 0 0 lower left node ht 6u height of circuit zi z0 0 ht upper left node Vac 5 z0 z1 location of voltage Vac t T u rotated 90 degrees L t R t C t T r default orientation set equal distances equally_spaced 5u 0 z1 R L C z2 Learning MetaPost by doing VOORJAAR 2005 47 edraw draw elements of circuits draw wires connecting nodes the first ones rotated 90 degrees wire v Vac a z0 wire v Vac b z1 wire v z2 z0 wire z1 R a wire R b L a wire L b C a wire C b z2 endfig end R u 10bp unit of length input mpcirc L beginfig 1 ANA c prepare L R C Vac mention your elements z0 0 0 lower left node ht 6u height of circuit zi z0 0 ht upper left node Vac 5 z0 z1 location of voltage Vac t T u rotated 90 degrees L t R t C t T r default orientation set equal distances equally_spaced 7 5u 0 z1 z2 z3 L 0 5 z1 z2 location of spool c 0 5 z2 23 0 2u R 0 5 z2 23 0 2u z4 23 2 5u 0 edraw draw elements of circuits draw wires connecting nodes wire v Vac a z0 wire v Vac b z1 wire z1 L a wire L b z2 wire v z2 C a wire v z2 R a wire v z3 C b wire v z3 R b wire z3 z4 wire v z4 z0 endfig end Marking Angles and Lines In geometric pictures line segments of equal length are often marked by an equal number
26. OST continues with what comes after the endfor part The loop text is usually any valid METAPOST state ment or sequence of statements separated by semicolons that are executed at each step in the repetition Instead of step 1 until we can also use the keyword upto downto is another word for step 1 until This counted for loop is an example of an unconditional repetition in which a predetermined set of actions are carried out Below we give another example of a counted for loop generating a Bernoulli walk We use the normaldeviate operator to generate a random number with the standard normal distribution mean 0 and standard deviation 1 beginfig 1 n 60 pair p pO 0 0 for i 1 upto n pli pfi 1 3 if normaldeviate gt 0 3 else 3 fi endfor draw pO for i 1 upto n p i endfor endfig end The last example in the previous section in which we computed the centre of grav ity of randomly generated weighted points contained another unconditional repe tition viz the for loop over a sequence of zero or more expressions separated by commas Another example of this kind is 80 120 20 100 beginfig 1 j fill for p 0 0 40 0 80 120 20 100 p endfor cycle withcolor 0 8white for p 0 0 40 0 80 120 20 100 dotlabel bot amp decimal xpart p amp amp decimal ypart p amp p endfor endfig end 0 0 40 0 Nesting of counted for loops is of course
27. a bump a shown below You can then repeat the process of breaking a line segment into four pieces of length one fourth of the segment that is broken up Below you see the next iteration Write a program that can compute the picture after n iterations After six iteration the Koch snowflake should look like 44 MAPS 32 Andr Heck Using Macro Packages METAPOST comes with built in macro packages which are nothing more than files containing macro definitions and constants The most valuable macro package is graph which contains high level utility macros for easy drawing graphs and data plots The graph package is loaded by the statement input graph A detailed description including examples of the graph package can be found in Hob92b GRS94 Here we just show one example of its use We represent the following data about the annual beer consumption in the Netherlands in the period 1980 2000 graphically year 1980 1985 1990 1995 2000 litre 86 85 91 86 83 Suppose that the data are stored columnwise in a file say consumption dat as 1980 86 1985 85 1990 91 1995 86 2000 83 The following piece of code produces a line plot of the data The only drawback of the graph package appears years are by default marked by decade This explains the densely populated labels 90 88 5 86 5 84 input graph beginfig 1 draw begingraph 4cm 4cm gdraw consumption dat endgraph endfig end 1
28. a point I VOORJAAR 2005 55 dx xmax xmin nx dy ymax ymin ny for i 0 upto nx xt xminti dx for j 0 upto ny yt ymin j dy zt f xt yt Z i j project xt yt zt factor i j brightnessfactor f xt yt zt endfor endfor for i 0 upto nx 1 for j 0 upto ny 1 fill Z i j Z i j 1 Z i 1 j 1 Z i 1 j cycle withcolor factor i j base_color endfor endfor enddef beginfig 1 xp 3 yp 3 zp 10 bf 100 numeric pi pi 3 14159 vardef cos primary x cosd x pi 180 enddef vardef f expr x y cos x y enddef z_surface f 3 3 3 3 100 100 frameXYZ 5 label btex z cos xy etex scaled 1 25 0 4cm endfig end verbatimtex A amp latex documentclass article begin document etex input courbes input grille vardef droite expr a b t t a b t b a enddef path c cartouche c 2 3cm 1cm 4 5cm 4cm 9cm 3cm cartouche 3cm 2mm 7cm 2mm 7cm 8mm 3cm 8mm cycle pair A M A point 0 6 of c M point 1 5 of c vardef tangente expr t x pair X Y X point t 05 of c Y point t 05 of c 56 MAPS 32 Andr Heck droite X Y x enddef beginfig 1 grille 1cm 0 10cm 0 7cm repere 10cm 7cm 2cm 2cm 1cm 1cm trace axes 5pt marque unites 0 1 lectures sur la grille numeric xa ya xm ym xa 1 25 ya 1 1 xm 4 9 ym 2 25 pair AA MM AA
29. act this is done automatically by the beginfig macro 24 MAPS 32 Andr Heck Transformations A very characteristic technique with METAPOST which we applied already in many of the previous examples is creating a graphic and then using it several times with different transformations METAPOST has the following built in operators for scaling rotating translating reflecting and slanting x y shifted a b x a y b x y rotated 8 x cos y sin 0 x sin 8 y cos 0 x y rotatedaround a b 8 xcos y sin 0 a 1 cos 0 b sin 9 xsin y cos 0 b 1 cos 0 a sin 0 x y slanted a x ay y x y scaled a ax ay x y xscaled a ax y x y yscaled a x ay x y zscaled a b ax by bx ay The effect of the translation and most scaling operations is obvious The following playful example in which the formula e 1 is drawn in various shapes serves as an illustration of most of the listed transformations beginfig 1 pair s s 0 2cm def drawit expr p draw p shifted s s s shifted 0 2cm enddef e 1 picture pic draw btex e pi i 1 etex draw bbox currentpicture withcolor 0 6white eTt pic currentpicture draw pic shifted 1cm 1cm pic pic scaled 1 5 drawit pic e a work with the enlarged base picture I ou drawit pic scaled 1 A A drawit pic rotated 30 oTi
30. after cutbefore direction of directionpoint of directiontime of intersectionpoint intersectiontimes length point of subpath Arguments and Result left numeric path path numeric pair pair path path numeric pair right path path path path path path path path path path path path VOORJAAR 2005 33 Meaning result numeric arc length of a path numeric time ona path where arc length from the start reaches a given value path left argument with part after the intersection dropped path left argument with part before the intersection dropped path the direction of a path at a given time pair point where a path has a given direction numeric time when a path has a given direction pair an intersection point numeric times t t2 on paths p and p2 when the paths intersect numeric number of arcs in a path pair point on a path given a time value path portion of a path for given range of time values times Let us apply what we have learned in this subsection to a couple of examples In the first example we draw a tangent line at a point of a curve beginfig 1 path c c 0 0 dir 60 100 50 dir 10 pair p pO point 0 1 of c pi point 0 5 of c p2 point 0 9 of c p3 direction 0 5 of c dotlabel 1rt btex T_ i 1 etex p0 dotlabel 1rt btex U_i etex p1 dotlabel 1rt btex T_i etex p2 draw c draw p3 p3 scaled 40 xpart p3 s
31. ary pen change The statement pickup pen expression causes the given pen to be used in subsequent draw statements The default pen is circular with a diameter of 0 5 bp If you want to change the line thickness simply use the following pen expression pencircle scaled numeric expression You can create an elliptically shaped and rotated pen by transforming the circular pen An example beginfig 1 pickup pencircle xscaled 2bp yscaled 0 25bp rotated 60 withcolor red for i 10 downto 1 draw 5 i 0 5 0 i 5 i 0 5 0 i 1 5 4 1 0 endfor endfig end In the following example we define a triangular shaped pen It can be used to plot data points as triangles instead of dots For comparison we draw a large triangle with both the triangular and the default circular pen beginfig 1 path p p dir 30 dir 90 dir 210 cycle pen pentriangle pentriangle makepen p draw origin withpen pentriangle scaled 2 draw p scaled 1cm withpen pentriangle scaled 4 draw p scaled 2cm withpen pencircle scaled 8 endfig end Setting Drawing Options The function drawoptions allows you to change the default settings for drawing For example if you specify drawoptions dashed evenly withcolor red then all draw statements produce dashed lines in red colour unless you overrule the drawing setting explicitly To turn off drawoptions all together just give an empty list drawoptions As a matter of f
32. ated by the next graphic statements before the correspond ing endfig command In the above case the output of the draw statement between beginfig 1 en endfig is written in the file example 1 In general a METAPOST document consists of one or more instances of beginfig figure number graphic commands endfig followed by end The draw statement with the points separated by two hyphens draws straight lines that connect the neighbouring points In the above case for example the point 0 0 is connected by straight lines with the point 10 0 and 0 10 The picture is a square with edges of size 10 units where a unit is of an inch We shall refer to this default unit as a PostScript point or big point bp to distinguish it from the standard printer s point pt which is ae of an inch Other units of measure include in for inches cm for centimetres and mm for millimetres For example draw 0 0 1icm 0 1cm 1cm 0 1cm 0 0 generates a square with edges of size 1cm Here 1cm is shorthand for 1 cm You may use 0 instead of Ocm because cm is just a conversion factor and Ocm just multiplies the conversion factor by zero Learning MetaPost by doing VOORJAAR 2005 5 EXERCISE 3 Create a METAPOST file say exercise3 mp that generates a circle of diameter 2cm using the fullcircle graphic object EXERCISE 4 1 Create a METAPOST file say exercise4 mp that generates an equilateral t
33. aw axes draw 0 5 0 u vline0 5 intersectionpoint pts_f zr draw 4 0 u vline4 intersectionpoint pts_f draw pts_f withpen pencircle scaled 2 label bot btex x etex 0 9xmax 0 u label lft btex y etex 0 0 9ymax u label urt btex f x etex 0 5 f 0 5 u endfig end Learning MetaPost by doing VOORJAAR 2005 31 EXERCISE 25 Create the following picture Clipping Clipping is a process to select just those parts of a picture that lie inside an area that is determined by a cyclic path and to discard the portion outside this is area The command to do this in METAPOST is clip picture variable to path expression You can use it to shade a picture element beginfig 1 pair p path cf c0 500 dir 40 500 dir 40 for i 0 upto 25 draw c0 shifted 0 10 i draw cO shifted 0 10 i endfor p1 100 0 c1 20 0 120 0 c2 pi 100 infinity c3 20 220 dir 45 120 140 right c4 0 0 0 infinity c5 buildcycle c1 c2 c3 c4 clip currentpicture to c5 p3 c4 intersectionpoint c3 p2 100 ypart p3 draw 0 0 p1 p2 p3 cycle draw c1 draw c3 withpen pencircle scaled 2 endfig end Dealing with Paths Parametrically In METAPOST a path is a continuous curve that is composed of a chain of seg ments Each segment is a cubic B zier curve which is determined by 4 control points The points on the curved segment from points po to p
34. bel is further away from the centre of gravity Cc EXERCISE 12 The dir command is a simple way to define a point on the unit circle For ex ample dir 30 generates the pair 0 86603 0 5 E 4v3 1 Use the dir command to generate a regular pentagon EXERCISE 13 Use the dir command to draw a line in northwest direction through the point 1 1 and a line segment through this point that makes an angle of 30 degrees with the line Your picture should look like Path Open and Closed Curves METAPOST can draw straight lines as well as curved ones You have already seen that a draw statement with points separated by draws straight lines connecting one point with another For example the result of draw p0 p1 p2 after defining three points by pair p pO 0 0 pil 2cm 3cm p2 3cm 2cm is the following picture 10 MAPS 32 EXERCISE 14 Andr Heck Closing the above path is done either by extending it with p0 or by connecting the first and last point via the cycle command Thus the path pO p1 p2 cycle when drawn looks like The difference between these two methods is that the path extension with the start ing point only has the optical effect of closing the path This means that only with the cycle extension it really becomes a closed path Verify that you can only fill the interior of a closed curve with some colour or shade of gray using the fill command when the path is really closed wit
35. ch we downloaded from http melusine eu org syracuse metapost courbes verbatimtex Z amp latex documentclass article begin document etex input courbes X vardef fx expr t t enddef vardef fy expr t ita t 1 t enddef y S beginfig 1 y r repere 10cm 10cm 5cm 5cm 2cm 2cm p y ys eee ETE trace axes 0 5pt M R et a marque unites 1mm i 7 a aR ers Champs de vecteurs by yaxana Vardef Flexpr x y y 1 x x 2 enddef yyy WW champ vecteurs 0 1 0 1 0 2 0 15 0 5white re ee aaa A Courbes int grales 4 y MEN i RA color la_couleur y N DIR cae la_couleur 0 9 0 1 0 9 ee a oo in for n 0 upto 20 a n 8 1 25 a7 o draw ftrace 0 995 2 5 50 en_place withcolor la_couleur a x y SYS draw ftrace 2 5 1 1 50 en_place withcolor la_couleur endfor h draw rpoint r_xmin 1 rpoint r_xmax 1 withcolor la_couleur decoupe repere etiquette axes etiquette unites label btex x x72 y y 1 etex scaled 2 5 rpoint 0 3 endfig end In the next example we added a macro to the courbes mp package for drawing a directional field with line segments instead of arrows verbatimtex A amp latex documentclass article begin document etex input courbes vardef fx expr t t enddef vardef fy expr t tan t a enddef Qo figure fonctions Learning MetaPost by doing FE
36. compute the dot product of two pairs You can render a point x y as a dot at the specified location with the statement draw x y Because the drawing pen has by default a circular shape with a diameter of 1 Post Script point a hardly visible point is rendered You must explicitly scale the drawing pen to a more appropriate size either locally in the current statement or globally for subsequent drawing statements You can resize the pen for example with a scale factor 4 by draw x y withpen pencircle scaled 4 temporary change of pen or by pickup pencircle scaled 4 new drawing pen is chosen draw x y EXERCISE 9 Explain the following result beginfig 1 draw unitsquare scaled 70 draw 10 20 draw 10 15 scaled 2 e draw 30 40 withpen pencircle scaled 4 pickup pencircle scaled 8 draw 40 50 draw 50 60 endfig end Assignment of pairs is often not done with the usual operator but with the equa tion symbol As a matter of fact METAPOST allows you to use linear equations to define a pair in a versatile way A few examples will do for the moment Using a name that consists of the character z followed by a number a statement such as z0 1 2 not only declares that the left hand side is equal to the right hand side but it also implies that the variables x0 and y0 exist and are equal to 1 and 2 respectively Alternatively you can assign values to the numeric variable x1 an
37. d withdots draw 1 0 1 1 0 1 zscaled 6pt unitvector u1 shifted u4 labeloffset 4pt label rt btex u_1 etex u1 label bot btex u_2 etex u2 label bot btex 2u_2 etex 2 u2 label bot btex u_3 etex u3 label 1ft btex u_4 etex u4 endfig end Recursion A macro is defined recursively if in its definition it makes a call to itself Recursive definition of a macro is possible in METAPOST We shall illustrate this with the computation of a Pythagorean tree beginfig 1 u 1cm branchrotation 60 offset 180 branchrotation thinning 0 7 shortening 0 8 def drawit expr p linethickness draw p withpen pencircle scaled linethickness enddef vardef tree expr A B n size save C D thickness pair C D thickness size Learning MetaPost by doing VOORJAAR 2005 43 C shortening B A rotatedaround B offset uniformdeviate branchrotation D shortening B A rotatedaround B offset uniformdeviate branchrotation if n gt 0 drawit A B thickness thickness thinning thickness tree B C n 1 thickness tree B D n 1 thickness else drawit A B thickness thickness thinning thickness drawit B C thickness drawit B D thickness fi enddef tree 0 0 0 u 10 2mm endfig end EXERCISE 31 The Koch snowflake is constructed as follows start with an equilateral triangle Break each edge into four straight pieces by adding
38. d y1 with the result that the pair x1 y1 is defined and can be referred to by the name z1 A statement like zi z2 3 4 4If you have not specified another editor in the EDITOR environment variable then the vi editor will be started You can leave this editor by entering ZZ In the c shell you can add in the file cshrc the line setenv MPEDIT xemacs d s so that XEMACS is used 8 MAPS 32 EXERCISE 10 EXERCISE 11 Andr Heck is equivalent to zi 3 4 z2 3 4 If two pairs say z1 and z2 are given you can define the pair say z3 right in the middle between these two points by the statement z3 1 2 z1 z2 When you have declared a pair say P then xpart P and ypart P refer to the first and second coordinate of P respectively For example pair P P 10 20 is the same as pair P xpart P 10 ypart P 20 Verify that when you use a name that begins with z followed by a sequence of alphabetic characters and or numbers a statement such asz P 1 2 not only declares that the left hand side is equal to the right hand side but it also implies that the variables x P and y P exist and are equal to 1 and 2 respectively Alternatively you can assign values to the numeric variable x P and y P with the result that the pair x P y P is defined and can be referred to by the name z P 1 Create the following geometrical picture of an acute angled triangle together
39. e main purpose of the vardef definition of the form def name replacement text returned text enddef By using vardef instead of def we hide the replacement text but the last statement which returns a value Below we given an example of a macro that generates a random point in the region 1 r x b u We use the uniformdeviate to generate a random number with the uniform distribution between 0 and the given argument A validity test on the actual arguments is carried out in case a region is not defined properly we use errmessage to display some text and exit from the macro call returning the origin as default point when computing is continued beginfig 1 vardef randompoint expr 1 r b u if r lt 1 or u lt b errmessage not a proper region origin else numeric x y x l uniformdeviate r 1 y btuniformdeviate u b a x y is fi e e enddef for i 0 upto 10 dotlabel randompoint 10 100 10 100 s endfor ad endfig end Do not place a semicolon after origin or x y In that case the statement be comes part of the hidden replacement text and an empty value is returned This causes a runtime error Defining the Argument Syntax In METAPOST you can explicitly define the argument syntax and construct unary binary or tertiary operators Let us look at the code of a predefined unary operator vardef unitvector primary z z abs z enddef As the example suggests the keyword primary is e
40. ecap squared 216 h labeloffset 25pt label bot linejoin beveled z19 labeloffset 40pt label bot linecap butt z19 labeloffset 25pt label bot linejoin beveled z22 labeloffset 40pt label bot linecap rounded 222 labeloffset 25pt label bot linejoin beveled z25 labeloffset 40pt label bot linecap squared 225 endfig end By setting the variable miterlimit you can influence the mitering of joints The next example demonstrates that the value of this variable acts as a trigger i e when the value of miterlimit gets smaller than some threshhold value then the mitered join is replaced by a beveled value beginfig 1 for i 0 upto 2 z 3i 150i 0 z 3i 1 150i 50 50 z 3i 2 150i 100 0 endfor drawoptions withpen pencircle scaled 24pt labeloffset 25pt linejoin mitered linecap butt for i 0 upto 2 miterlimit i draw z 3i z 3i 1 z 3i 2 label bot miterlimit amp decimal miterlimit z 3i 1 endfor endfig end 30 MAPS 32 Andr Heck miterlimit 0 miterlimit 1 miterlimit 2 Building Cycles In previous examples you have seen that intersection points of straight lines can be specified by linear equations A more direct way to deal with path intersection is via the operator intersectionpoint So given four points z1 z3 z3 and z4 in general position you can specify the intersection point z5 of the line between z1 and z2 a
41. ed as unitsquare 16 MAPS 32 Andr Heck Path Definition fullcircle circle with diameter 1 and centre 0 0 halfcircle upper half of fullcircle quartercircle first quadrant of fullcircle unitsquare 0 0 1 0 1 1 cyle You can construct from these basic paths any circle ellipse square or rectangle by rotating rotated operator by scaling operators scaled xscaled and yscaled and or by translating the graphic object shifted operator Keep in mind that the ordering of operators has a strong influence on the final shape But pictures say more than words The diagram eis y ED P PEERY a k 4 is drawn with the following METAPOST code beginfig 1 u 24 24 24bp 1 3 inch for i 1 upto 9 draw i u 4u i u 3u withcolor 0 7white endfor for i 3 upto 4 draw u i u 9u i u withcolor 0 7white endfor dotlabel origin the grid with reference point 0 0 has been drawn draw fullcircle scaled u draw halfcircle scaled u shifted 2u 0 draw quartercircle scaled u shifted 4u 0 draw fullcircle xscaled 2u yscaled u shifted 6u 0 draw fullcircle xscaled 2u yscaled u rotated 45 shifted 8u 0 fill fullcircle scaled u shifted 0 2u fill halfcircle cycle scaled u shifted 2u 2u path quarterdisk quarterdisk quartercircle origin cycle fill quarterdisk scaled u shifted 4u 2u fill quarterdisk scaled u rotated 45 shifted 6u 2u fill quarterdisk scaled u shifted
42. eral ways to proceed after the interrupt Enter a question mark and you see your options read Type lt return gt to proceed S to scroll future error messages R to run without stopping Q to run quietly I to insert something E to edit your file 1 or or 9 to ignore the next 1 to 9 tokens of input H for help X to quit 3 Press RETURN METAPOST will continue processing and tries to make the best of it Logging continues 1 1 output file written example 1 Transcript written on example log 4 Verify that only the following path is generated newpath 0 0 moveto 10 O lineto 10 10 lineto stroke 5 Format the METAPOST document again but this time enter the character e Learning MetaPost by doing VOORJAAR 2005 7 Your default editor will be opened and the cursor will be at the location where METAPOST spotted the error Correct the source filet by adding a semicolon at the right spot and give the METAPOST processing another try Basic Graphical Primitives In this section you will learn how to build up a picture from basic graphical prim itives such as points lines and text objects Pair The pair data type is represented as a pair of numeric quantities in METAPOST On the one hand you may think of a pair say 1 2 as a location in two dimensional space On the other hand it represents a vector From this viewpoint it is clear that you can add or subtract two pairs apply a scalar multiplication to a pair and
43. erative graphic process beginfig 1 pair A B C u 3cm A u dir 30 B u dir 90 C u dir 210 transform T A transformed T 1 6 A B B transformed T 1 6 B C C transformed T 1 6 C A path p p A B C cycle for i 0 upto 60 draw p p p transformed T endfor endfig end EXERCISE 24 Using transformations construct the following picture Advanced Graphics In this section we deal with fine points of drawing lines and with more advanced graphics This will allow you to create more professional looking graphics and more complicated pictures Joining Lines In the last example of the section on pen styles you may have noticed that lines are joined by default such that line joints are normally rounded You can influence the appearances of the lines by the two internal variables linejoinand linecap The picture below shows the possibilities 28 MAPS 32 18 20 19 linejoin beveled linecap butt 1 10 linejoin rounded linecap butt 2 kre 21 23 22 linejoin beveled linecap rounded 12 14 13 linejoin rounded linecap rounded 3 5 4 Andr Heck J4 26 25 linejoin beveled linecap squared 15 iy 16 linejoin rounded linecap squared Qe Coe 7 linejoin mitered linejoin mitered linejoin mitered linecap butt linecap rounded linecap squared This picture can be produced by the following METAPOST code beginfig 1 for i 0 upto 2 for j 0 upto 2 z 3i 9j 1
44. ext into a PostScript document that you can preview and print In this section we shall describe the basics of this process Running METAPOST Do the following steps 1 Create a text file say example mp that contains the following very simple META POST document beginfig 1 draw 0 0 10 0 10 10 0 10 0 0 endfig end For example you can use the editor XEmacs xemacs example mp The above UNIX command starts the editor and creates the source file example mp It is common and useful practice to always give a METAPOST source file a name with extension mp This will make it easier for you to distin guish the source document from files with other extensions which METAPOST will create during the formatting 2 Generate from this file PostScript code Here the METAPOST program does the job mpost example It is not necessary to give the filename extension here METAPOST now creates some additional files example 1 a PostScript file that can be printed and previewed example log METAPOST s log file 3 Check that the file example 1 contains the following normal Encapsulated Post Script code Notice that the bounding box is larger than you might expect due to the default width of the line drawing the square Learning MetaPost by doing VOORJAAR 2005 3 A PS BoundingBox 1 1 11 11 A4Creator MetaPost A4CreationDate 2003 05 11 2203 hhPages 1 4EndProlog hhPage 1 1 0 0 5 dtransform truncate
45. fb cycle draw unitvector endofa common endofa common rotated tn 90 unitvector endofb common scaled length shifted common withcolor 0 3white enddef w drawangle 1 0 cm z 0 0 0 4cm drawangle w zw 0 0 0 4cm z drawangle 0 0 z 1 0 cm 0 2cm drawangle 0 0 z 1 0 cm 0 15cm drawangle 0 0 zw w 0 2cm drawangle 0 0 zw w 0 15cm 0 1 label 11ft btex 0 etex Ocm Ocm label 1rt btex 1 etex 1cm 0cm label rt btex z etex z label rt btex w etex w label rt btex zw etex zw endfig end Specifying the images of three points It is possible to apply an unknown transform to a known pair and use the result in a linear equation For example transform T 1 0 transformed T 1 0 0 1 transformed T 0 1 0 0 transformed T 1 1 defines the reflection in the line through 1 0 and 0 1 The built in transformation reflectedabout p q which reflects about the line connecting the points p and q is defined by a combination of the last two techniques def reflectedabout expr p q transformed begingroup transform T_ Learning MetaPost by doing VOORJAAR 2005 27 p transformed T_ p q transformed T_ q xxpart T_ yypart T_ xypart T_ yxpart T_ T_ is a reflection Te endgroup enddef Given a transformation T the inverse transformation is easily defined by inverse T We end with another playful example of an it
46. g end EXERCISE 26 Create the following picture which has to do with Kepler s law of areas At At Control Structures In this section we shall look at two commonly used control structures of imperative programming languages condition and repetition Conditional Operations The basic form of a conditional statement is if condition balanced tokens else balanced tokens fi where condition represents an expression that evaluates to a boolean value i e true or false and the balanced tokens normally represent any valid METAPOST statement or sequence of statements separated by semicolons The if statement is used for branching depending on some condition one sequence of METAPOST statements is executed or another The keyword fi is the reverse of if and marks the end of the conditional statement The keyword fi separates the statement sequence of the preceding command clause so that the semicolon at the end of the last command in the else part can be omitted It also marks the end of the Learning MetaPost by doing VOORJAAR 2005 35 conditional statement so that you do not need a semicolon after the keyword to separate the conditional statement from then next statement Other forms of conditional statements are obtained from the basic form by Omitting the else part when there is nothing to be said Nesting of conditional operations The following shortcut can be used else if can be replaced
47. g an empty replacement text In other words this command allows you to run code silently Grouping often occurs automatically in METAPOST For example the beginfig macro starts with begingroup and the replacement text for endfig ends with endgroup vardef macros are always grouped automatically too You may want to go one step further not only treating values of a variable locally but also its name For example in a macro definition you may want to use a so called local variable i e a variable that only has meaning inside that definition and does not interfere with existing variables In general variables are made local by the statement save name sequence For example the macro whatever has the replacement text begingroup save endgroup This macro returns an unknown If the save statement is used outside of a group the original values are simply discarded This explains the following definition of the built in macro clearxy in fact save is a vardef macro which has the begingroup and begingroup automatically placed around the replacement text Thus the begingroup and endgroup are superfluous here Learning MetaPost by doing VOORJAAR 2005 41 def clearxy save x y enddef Vardef Definitions Sometimes we want to return a value from a macro as if it is a function or sub routine In this case we want to make sure that the calculations inside the macro do not interfere with the expected use of the macro This is th
48. grams We show the diagrams and the code to create them The basic idea of mpcirc is that you have at your disposal a set of predefined electronic components such as resistor capacity diode and so on Each component has some connection points referred to by a b for wires You place the elements using predefined orientations and then connect them with wires This mode of operating with mpcirc is referred to as turtle based Another programming style for drawing diagrams is node based In this approach the node locations are determined first and then the elements are put between them using betw x macros We shall use the turtle mode in our examples and hope that the comments speak for themselves u 10bp unit of length L input mpcirc LN beginfig 1 prepare L C Vac mention your elements z0 10u 10u lower right node C ht 6u height of circuit z1 z0 0 ht upper right node c 5 z0 z1 location of capacitor L t T r use default orientation C t Vac t T u components rotated 90 degrees set the distance between Voltage and Capacitor equally_spaced 5u 0 Vac C L z1 0 5 C Vac location of spool edraw draw components of the circuit draw wires connecting components the first ones rotated 90 degrees wire v Vac a z0 wire v Vac b L a wire v L b z1 wire C a z0 wire C b z1 endfig end u 10bp unit of length L C input mpcirc MN beginfig 1
49. h the cycle command The gray shading is obtained by the directive withcolor c white where c is a number between 0 and 1 Straight and Curved Lines Compare the pictures from the previous subsection with the following ones which show curves through the same points u 1cm pair p pO 0 0 p1 2u 3u p2 3u 2u beginfig 1 draw pO p1 p2 draw open curve for i 0 upto 2 draw p i withpen pencircle scaled 3 endfor draw defining points endfig beginfig 2 draw pO p1 p2 cycle draw closed curve for i 0 upto 2 draw p i withpen pencircle scaled 3 endfor draw defining points endfig end Sthe curves through the three points are a circle or a part of the circle Learning MetaPost by doing VOORJAAR 2005 11 Just use where you want straight lines and where you want curves beginfig 1 u 1cm pair p pO 0 0 p1 2u 3u p2 3u 2u draw p0 p1 p2 cycle for i 0 upto 2 draw p i withpen pencircle scaled 3 endfor endfig end Construction of Curves When METAPOST draws a smooth curve through a se quence of points each pair of consecutive points is connected by a cubic B zier curve which needs in order to be determined two intermediate control points in addition to the end points The points on the curved segment from points po to p with post control point co and pre control point c are determined by the formula p t 1 t po 3 1 t tco 3 1
50. hifted p1 endfig end The second example is the trefoil knot i e the torusknot of type 1 3 The META POST code has been written such that assigning n 5 and n 7 draws the torusknot of type 1 5 and 1 7 respectively in a nice way beginfig 1 pair A B path p numeric n n 3 A 0 2cm B A rotated 2 360 n pO A dir 180 tension n 1 2 B dir 180 2 360 n numeric a a whatever pO intersectiontimes pO rotated 360 n pi subpath 0 a 04 of p0 p2 subpath a 04 1 of p0 drawoptions withpen pencircle scaled 2 for i 0 upto n 1 draw p1 rotated i 360 n draw p2 rotated i 360 n endfor endfig end 34 MAPS 32 Andr Heck The third example shows a few steps in the Newton iterative method of finding zeros beginfig 1 u icm draw 5u 0 5u 0 draw 0 5u 0 4u label 11ft btex 0 etex 0 0 label rt btex x etex 5u 0 label top btex y etex 0 4u path f f 25u 5u right 5u 4u dir 70 y draw f withpen pencircle scaled 1 2pt x0 4 6 u numeric t for i 0 upto 1 t i whatever f intersectiontimes x i infinity x i infinity z i point t i of f x i 1 0 z i whatever direction t i of f draw x i 0 z i x i 1 0 fill fullcircle scaled 3 6pt shifted z i endfor label bot btex x_0 etex x0 0 label bot btex x_1 etex x1 0 O T2 T To label bot btex x_2 etex x2 0 endfi
51. ia the concatenation operator amp But you can also bracket the text with btex and etex do not put it in quotes this time and pass it to TEX for typesetting This allows you to use METAPOST in combination with TeX for building complex labels Let us begin with a simple example beginfig 1 zO 0 0 z1 sqrt 3 cm 0 9 z2 sqrt 3 cm 1cm draw z0 z1 z2 cycle label bot btex sqrt 3 etex 1 2 z0 z1 label rt btex 1 etex 1 2 z1 z2 v3 label top btex 2 etex 1 2 z0 z2 endfig end Whenever the METAPOST program encounters btex typesetting commands etex it suspends the processing of the input in order to allow TEX to typeset the com mands and the dvitomp preprocessor to translate the typeset material into a picture expression that can be used in a label or dotlabel statement The generated low level METAPOST code is placed in a file with extension mpx Hereafter META POST resumes its work We speak about a picture expression that is created by typesetting commands because it is a graphic object to which you can apply transformation This is il lustrated by the following example in which we use diagonal curly brackets and text 18 MAPS 32 Andr Heck beginfig 1 zO 0 0 z1 sqrt 3 cm 0 z2 sqrt 3 cm 1cm draw z0 z1 z2 cycle label bot btex lbrace etex rotated 90 xscaled 5 yscaled 1 4 1 2 z0 z1 2 label rt btex rbrace etex xscaled 1 3 1 yscaled 3 1 2 z1 z2
52. idtransform setlinewidth pop 0 setdash 1 setlinecap 1 setlinejoin 10 setmiterlimit newpath 0 0 moveto 10 O lineto 10 10 lineto O 10 lineto O 10 lineto 0 O lineto stroke showpage EOF 4 Preview the PostScript document on your computer screen e g by typing gs example 1 You will notice that gs does not use the picture s bounding box alternatively try ggv instead of gs or 5 Convert the PostScript document into a printable PDF document epstopdf example i It creates the file example pdf that you can you can view on the computer screen with the Adobe Acrobat Reader by entering the command acroread example pdf You can print this file in the usual way The picture should look like the following small square Using the Generated PostScript in a LaTkxX Document EXERCISE 2 Do the following steps 1 Create a file say sample tex that contains the following lines of LaTpX com mands that will include the image documentclassf article usepackage graphicx DeclareGraphicsRule mps begin document includegraphics example 1 end document Above we use the extended graphicx package for including the external graphic file that was prepared by METAPOST The DeclareGraphicsRule statement causes all file extensions that are not associated with a well known graphic format to be treated as Encapsulated PostScript files 2 Typeset the LaTkX file 4 MAPS 32 Andr Heck pdflatex sample
53. me scalar between 0 and 1 The syntax of using a colour in a graphic statement is withcolor colour expression Let us draw two colour charts RGB r g 0 beginfig 1 u 1 2cm defaultscale 10pt fontsize defaultfont beginfig 1 path sqr sqr unitsquare scaled u for i 0 upto 10 label bot decimal i 10 it 1 2 u 0 label 1ft decimal i 10 0 i 1 2 u for j 0 upto 10 fill sqr shifted i u j u withcolor i 0 1 j 0 1 0 draw sqr shifted i u j u draw grid endfor endfor label bot r 6u 2 3u label 1ft g u 6u label top RGB r g 0 6u 11u endfig end 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 r beginfig 2 draw u u 11u u 11u u u u cycle bounding box pickup pencircle scaled 1 2u a for i 0 upto 10 EE EE EE E draw i u 0 withcolor i 0 1 white endfor EXERCISE 23 endfig end Compare the linear conversion from colour to gray defined by the function b r g Ls b r g b gt 5 1 1 1 with the following conversion formula used in black and white television r g b 0 30r 0 59g 0 11b x 1 1 1 Learning MetaPost by doing VOORJAAR 2005 23 Specifying the Pen In METAPOST you can define your drawing pen for specifying the line thickness or for calligraphic effects The statement draw path withpen pen expression causes the chosen pen to be used to draw the specified path This is only a tempor
54. nd the line between z3 and z4 by z5 zi z2 intersectionpoint z3 z4 You do not need to rely on setting up linear equations with z5 whatever z1 z2 whatever z3 z4 The strength of the intersection operator is that it also works for curved lines We use this operator in the next example of a filled area beneath the graph of a function The closed curve that forms the border of the filled area is constructed with the buildcycle command When given two or more paths the buildcycle macro tries to piece them together so as to form a cyclic path In case there are more intersection points between paths the general rule is that buildcycle pi p2 Pn chooses the intersection between each p and p to be as late as possible on the path p i and as early as possible on p 1 In practice it is more convenient to choose the path arguments such that consecutive ones have a unique intersection beginfig 1 numeric xmin xmax ymin ymax xmin 1 4 xmax 6 ymax 1 xmin u 1cm compute the graph of the function vardef f expr x 1 x enddef xinc 0 1 path pts_f pts_f xmin f xmin u for x xmintxinc step xinc until xmax x x u endfor path hline vline hlineO 0 0 u xmax 0 u vlineO 0 0 u 0 ymax u vlineO 5 0 5 0 u 0 5 ymax u vline4 4 0 u 4 ymax u fill buildcycle hlineO vlineO 5 pts_f vline4 withcolor 0 8 blue white draw hlineO draw vlineO dr
55. ngcheck 0 The numeric data type is used so often that it is the default type of any non declared variable This explains why n 10 is the same as numeric n n 10 and 6 MAPS 32 EXERCISE 8 Andr Heck why you cannot enter p 0 0 norp 0 0 to define the point but must use pair p p 0 0 orpair p p 0 0 If Processing Goes Wrong If you make a mistake in the source file and METAPOST cannot process your doc ument without any trouble the code generation process is interrupted In the fol lowing exercise you will practice the identification and correction of errors Deliberately make the following typographical error in the source file example mp Change the line draw 0 0 10 0 10 10 0 10 0 0 into the following two lines draw 0 0 10 0 10 10 draw 10 10 0 10 0 0 1 Try to process the document METAPOST will be unable to do this and the processing would be interrupted The terminal window where you entered the mpost command looks like example mp Extra tokens will be flushed lt to be read again gt addto draw gt addto currentpicture if picture EXPRO also EXPRO else doubl lt to be read again gt 1 3 draw 10 10 0 10 0 0 In a rather obscure way the METAPOST program notifies the location where it signals that something goes wrong viz at line number 3 However this does not mean that the error is necessarily there 2 There are sev
56. not know about the exponential or logarithmic function But you can easily define these functions with the help of the built in functions mexp x exp x 256 and mlog x 256Inx Note that we have re served the name 1log for the logarithm with base 10 in the above program As you will see later in this tutorial METAPOST has several repetition control structures Here we apply the for loop to draw tick marks and labels on the axes and to compute the path of the graph The basic form is for counter start step stepsize until finish loop text endfor Instead of step 1 until you may use the keyword upto downto is another word for step 1 until In the following code snippet for i 0 upto xmax draw i 0 05 ux i 0 05 ux label bot decimal i 1 0 ux endfor the input lines are two statements one to draw tick marks and the other to put a label We use the decimal command to convert the numeric variable i into a string 20 MAPS 32 Andr Heck so that we can use it in the label statement The following code snippet pts_f xmin ux log f xmin uy for x xmint txinc step xinc until xmax xtux log f x uy endfor shows that you can also use the for loop to build up a single statement The input lines within the for loop are pieces of a path definition This mode of creating a statement may look strange at first sight but it is an opportunity given by the fact that METAPOST consists more or less of
57. nough to specify the macro as a unary vardef operator Other keywords are secondary and tertiary The advantage of specifying an n ary operator is that you do not need to place brackets around arguments in compound statements METAPOST will sort out which tokens are the arguments For example unitvector v rotated angle v is understood to be equivalent to unitvector v rotated angle v You can also define a macro to play the role of an of operation For example the direction of macro is predefined by Andr Heck vardef direction expr t of p postcontrol t of p precontrol t of p enddef Precedence Rules of Binary Operators METAPOST provides the classifiers primarydef secondarydef and tertiarydef for def macros not for vardef macros to set the level of priority of a binary operator In the example below the orthogonal projection of a vector v along another vector w is defined as a secondary binary operator beginfig 1 secondarydef v projectedalong w if pair v and pair w v dotprod w w dotprod w w else errmessage arguments must be vectors fi enddef pair u u1 20 80 u2 60 15 drawarrow origin ul drawarrow origin u2 drawarrow origin 2 u2 u3 u1 projectedalong u2 u4 2 u2 projectedalong ul drawarrow origin u3 withcolor blue draw ui u3 dashed withdots draw 1 0 1 1 0 1 zscaled 6pt unitvector u2 shifted u3 drawarrow origin u4 withcolor blue draw 2 u2 u4 dashe
58. of ticks and equal angles are often marked the same too In the following example the macros tick mark_angle and mark_right_angle mark lines and angles When dealing with angles we use the macro turningnumber to find the direction of a cyclic path 1 means counter clockwise 1 means clockwise We use it to make our macros mark_angle and mark_right_angle independent of the order in which the non common points of the angle are specified 48 MAPS 32 Andr Heck set some user adjustable constants angle_radius 4mm angle_delta 0 5mm mark_size 2mm def mark_angle expr A common B n draw 1 2 3 or 4 arcs draw_angle A common B angle_radius if n gt 1 draw_angle A common B angle_radiustangle_delta fi if n gt 2 draw_angle A common B angle_radius angle_delta fi if n gt 3 draw_angle A common B angle_radius 2 angle_delta fi enddef def draw_angle expr endofa common endofb r begingroup save tn tn turningnumber common endofa endofb cycle draw unitvector endofa common endofa common rotated tn 90 unitvector endofb common scaled r shifted common endgroup enddef def mark_right_angle expr endofa common endofb begingroup save tn tn turningnumber common endofa endofb cycle draw 1 0 1 1 0 1 zscaled mark_size unitvector 1 tn endofat 1 tn endofb 2 common shifted common endgroup enddef def tick expr
59. ound expands into in line code The expr in this definition means that a formal parameter here z or d can be an arbitrary expression Each occurrence of a formal parameter will be replaced by the corresponding actual argument this is referred to as call by value Thus the line rotatedaround p q atb will be replaced by the line shifted ptq rotated a b shifted ptq Macro parameters need not always be expressions Another argument type is text indicating that the parameters are just past as an arbitrary sequence of tokens Grouping and Local Variables In METAPOST all variables are global by default But you may want to use in some piece of METAPOST code a variable that has temporarily inside that portion of code a value different from the one outside the program block The general form of a program block is begingroup statements endgroup where statements is a sequence of one or more statements separated by semicolons For example the following piece of code x 1 y 2 begingroup x 3 y x 1 show x y endgroup show x y will reveal that the x and y values are 3 and 4 respectively inside the program block But right after this program block the values will be 1 and 2 as before The program block is used in the definition of the hide macro def hide text t exitif numeric begingroup t endgroup enddef It takes a text parameter and interprets it as a sequence of statements while ulti mately producin
60. p n begingroup save midpnt midpnt 0 5 arclength p find the time when half way the path for i n 1 2 upto n 1 2 draw_mark p midpnttmark_size i 2 place n tick marks endfor endgroup enddef def draw_mark expr p m begingroup save t dm pair dm t arctime m of p find a vector orthogonal to p at time t dm mark_size unitvector direction t of p rotated 90 draw 1 2dm 1 2dm shifted point t of p draw tick mark endgroup enddef beginfig 1 pair A B C D A 0 0 B 3cm 0 C 1 5cm 4cm D 1 5cm 0 draw A B C cycle draw C D draw triangle and altitude Learning MetaPost by doing VOORJAAR 2005 49 label bot A A label bot B B label top C C label bot D D tick A D 1 tick D B 1 tick A C 2 tick B C 2 mark_angle C A B 2 mark_angle A B C 2 mark_angle B C A 1 mark_right_angle C D B endfig end Vectorfields In the first example below we show the directional field corresponding with the ordinary differential equation y y For clarity we show a vectorfield instead of a directional field with small line segments beginfig 1 some constants numeric xmin xmax ymin ymax xinc u xmin 1 5 xmax 1 5 ymin 0 ymax 4 5 xinc 0 05 u 1cm draw axes draw xmin 0 5 0 u xmax 0 5 0 u draw 0 ymin 0 5 u 0 ymaxt0 5 u define f making up the ODE y f x y Here we take y y
61. ri angle with edges of size 2cm 2 Extend the METAPOST document such that it generates in a separate file the PostScript code of an equilateral triangle with edges of size 3cm EXERCISE 5 Define your own unit say 0 5cm by the statement u 0 5cm and use this unit u to generate a regular hexagon with edges of size 2 units EXERCISE 6 Create the following two pictures Numeric Quantities Numeric quantities in METAPOST are represented in fixed point arithmetic as in teger multiples of 4 2 and with absolute value less or equal to 4096 2 Since METAPOST uses fixed point arithmetic it does not understand exponential notation such as 1 23E4 It would interpret this as the real number 1 23 followed by the symbol E followed by the number 4 Assignment of numeric values can be done with the usual operator Numeric values can be shown via the show command EXERCISE 7 1 Create a METAPOST file say exercise mp that contains the following code numeric p q n n 11 p 2 n q 2 nt 1 show p q end Find out what the result is when you run the above METAPOST program 2 Replace the value of n in the above METAPOST document by 12 and see what happens in this case Hint press Return to get processing as far as possible Explain what goes wrong 3 Insert at the top of the current METAPOST document the following line and see what happens now when you process the file warni
62. row O Yr scaled s drawarrow O Zr scaled s label 11ft btex X etex scaled 1 25 Xr scaled s label rt btex Y etex scaled 1 25 Yr scaled s label top btex Z etex scaled 1 25 Zr scaled s enddef from 3D to 2D coordinates vardef project expr x y z x Xr y Yr z Zr enddef numerical derivatives by central differences vardef diffx suffix f expr x y numeric h h 0 01 xth y f x h y 2 h enddef vardef diffy suffix f expr x y numeric h h 0 01 x yt h f x y h 2 h enddef Compute brightness factor at a point vardef brightnessfactor suffix f expr x y z numeric dfx dfy ca cb cc dfx diffx f x y dfy diffy x y ca zp z dfy yp y dfx xp x cb sqrt 1 dfx dfxtdfy dfy cc sqrt z zp z zp y yp y yp x xp x xp bf ca cb cc cc cc enddef compute the colors and draw the patches vardef z_surface suffix f expr xmin xmax ymin ymax nx ny numeric dx dy xt yt zt factor pair Z Learning MetaPost by doing 2 Miscellaneous We adopt another example from Jean Michel Sarlat that uses his macro package courbe and another one called grille The example has been downloaded and slightly adapted from http melusine eu org syracuse metapost cours sarlat derivation f a f x f a THa F a fla i graph of f a Derivative at
63. trace courbe a b 200 2pt aubergine decoupe repere etiquette axes 52 MAPS 32 Andr Heck abel bot btex x_ i 1 etex rpoint atn 2 h 0 abel bot btex x_ O etex rpoint a 0 abel bot btex x_ n etex rpoint b 0 abel ulft btex mathbf a etex rpoint a 0 abel urt btex mathbf b etex rpoint b 0 abel top btex mathbf y f x etex f b en_place abel bot btex x_ i etex rpoint atn 2 h th 0 projection axes f at n 1 2 h 0 5pt 2 abel 1ft btex f xi_i etex rpoint 0 fy at n 1 2 h abel bot btex xi_i etex rpoint a n 1 2 h 0 4 drawarrow rpoint at n 1 2 h 0 4 rpoint at n 1 2 h 0 1 abel btex textit Riemann Sum etex scaled 2 rpoint 4 5 abel btex S sum_ i i 7n xi_i x_i x_ i 1 etex scaled 1 5 rpoint 5 2 endfig end Iterated Functions The following diagrams are standard in the theory of iterative processes The cobweb graph of applying the cosine function iteratively The bifurcation diagram of the logistic function f x rx 1 x forO lt r lt 4 The code that produced these diagrams is shown below verbatimtex 1 A amp latex documentclass article begin document etex beginfig 1 some constants u 10cm numeric xmin xmax ymin ymax xinc xmin 0 5 xmax 1 0 ymin xmin ymax xmax xinc 0 02 draw axes draw xmin ymin u xmax ymin u draw xmin ymin u
64. uivalent to subpath xpart p1 intersectiontimes p2 length p1 of p1 except that it also sets the path variable cuttings to the parts of p1 that gets cut off With multiple intersections it tries to cut off as little as possible Similarly pi cutafter p2 tries to cut off the part of p1 after its last intersection with p2 We have seen that for a time parameter t we can find the corresponding point on the curve p by the statement point t of p Another statement of the general form direction t of path allows you to obtain a direction vector at the point of the path that corresponds with time t The magnitude of the direction vector is somewhat arbitrary The directiontime operation is the inverse of the direction of operation Given a direction vector a pair and a path directiontime direction vector of path return a numeric value that gives the first time t when the path has the indicated direction directionpoint direction vector of path returns the first point on the path where the given direction is achieved The more familiar concept of arc length is also provided for in METAPOST arclength path returns the arc length of the given path If p is a path and a is a number between 0 and arclength p then arctime a of p gives the time t such that arclength subpath 0 t of p a A summary of the path operators is listed in the table below Learning MetaPost by doing Name arclength arctime of cut
65. use exitif boolean expression When the exit clause is encountered METAPOST evaluates the boolean expression and exits the current loop if the expression is true Thus METAPOST s version of a until loop is forever loop text exitif boolean expression endfor If it is more convenient to exit the loop when an expression becomes false then use the predefined macro exitunless Thus METAPOST s version of a while loop is forever exitunless boolean expression loop text endfor Macros Defining Macros In the section about the repetition control structure we introduced upto as a short cut of step 1 until This is also how it is is internally defined in METAPOST def upto step 1 until enddef It is a definition of the form def name replacement text enddef It calls for a macro substitution of the simplest kind subsequent occurrences of the token name will be replaced by the replacement text The name in a macro is a variable name the replacement text is arbitrary and may for example consist of a sequence of statements separated by semicolons It is also possible to define macros with arguments so that the replacement text can be different for different calls of the macro An example of a built in paramet rised macro is 40 MAPS 32 Andr Heck def rotatedaround expr z d rotates d degrees around z shifted z rotated d shifted z enddef Although it looks like a function call a use of rotatedar
66. utput and facilities for in cluding typeset text This course is only meant as a short hands on introduction to METAPOST for newcomers who want to produce rather simple graphics The main objective is to get students started with METAPOST on a UNIX platform A more thorough but also much longer introduction is the Metafun manual of Hans Hagen Hag02 For complete descriptions we refer to the METAPOST Manual and the Introduction to METAPOST of its creator John Hobby Hob92a Hob92b We have followed a few didactical guidelines in writing the course Learning is best done from examples learning is done from practice The examples are often formatted in two columns as follows TYou can also run METAPOST on a windows platform e g using MikT X and the WinEdt shell 20n the left is printed the graphic result of the METAPOST code on the right Here a square is drawn 1 2 MAPS 32 EXERCISE 1 Andr Heck beginfig 1 draw unitsquare scaled 1cm endfig The exercises give you the opportunity to practice METAPOST instead of only reading about the program Compare your answers with the ones in the section Solutions to the Exercises A Simple Example METAPOST is not a WYSIWYG drawing tool like xfig or xpaint It is a graphic document preparation system First you write a plain text containing graphic formatting commands into a file by means of your favourite editor Next the META POST program converts this t
67. ux 8cm uy 8cm for i 1 upto n x 0 5 our starting point for j 1 upto 75 initial iterations x r x 1 x endfor for j 1 upto 150 the next 100 iterations x r x 1 x draw r ux x uy withpen pencircle scaled 5pt endfor r r dr endfor draw axes and labels draw rmin ux 0 rmax ux 0 draw rmin ux 0 rmin ux uy labeloffset 0 25cm label bot decimal rmin rmin ux 0 label bot decimal rmintrmax 2 rmint rmax 2 ux 0 label bot decimal rmax rmax ux 0 label 1ft decimal 0 rmin ux 0 label 1ft decimal 0 5 rmin ux 0 5 uy label 1ft decimal 1 rmin ux uy label bot btex r etex rmax 0 2 ux 0 label 1ft btex orbit etex rotated 90 rmin ux 0 75 uy endfig end 54 MAPS 32 A Surface Plot Andr Heck You can draw surface plots from basic principles We give one example z cos xy verbatimtex Y amp latex documentclass article begin document etex u dimensional unit Xp yp zp coordinates of light source bf brightness factor base_color base color numeric u xp yp zp bf color base_color u icm xp 3 yp 3 zp 5 bf 30 base_color redt green 0 Xr Yr Zr reference frame pair 0 Xr Yr Zr 0 0 0 Xr 7 7 scaled u Yr 1 0 scaled u Zr 0 1 scaled u for drawing the reference frame vardef frameXYZ expr s drawarrow O Xr scaled s drawar
68. wit p withdots drawit p withdots scaled 2 drawit p evenly drawit p evenly scaled 2 drawit p evenly scaled 4 drawit p evenly scaled 6 p 0 150 102 150 def shiftit suffix p expr s draw p dashed evenly scaled 4 shifted s dotlabel point 0 of p dotlabel point 1 of p p p shifted 0 13 enddef shiftit p 0 0 shiftit p 4bp 0 shiftit p 8bp 0 shiftit p 12bp 0 shiftit p 16bp 0 shiftit p 20bp 0 picture dd dd dashpattern on 6bp off 2bp on 2bp off 2bp draw 0 283 102 283 dashed dd draw 0 296 102 296 dashed dd scaled 2 endfig end In general the syntax for dashing is draw path dashed dash pattern 22 MAPS 32 Andr Heck You can define a dash pattern with the dashpattern function whose argument is a sequence of on off distances Predefined patterns are evenly dashpattern on 3 off 3 equal length dashes withdots dashpattern off 2 5 on 0 off 2 5 dotted lines Colouring The color data type is represented as a triple r g b that specifies a colour in the RGB colour system Each of r g and b must be a number between 0 and 1 inclusively representing fractional intensity of red green or blue respectively Predefined colours are red 1 0 0 green 0 1 0 blue 0 0 1 black 0 0 0 white 1 1 1 A shade of gray can be specified most conveniently by multiplying the white col our with so
69. with post control point co and pre control point c are determined by the formula p t 1 t po 3t 1 t co 3t 1 t ci tp where t 0 1 If the path consists of two arcs i e consists of three points po pi and p2 then the time parameter t runs from 0 to 2 If the path consists of n curve segments then t runs normally from 0 to n At t 0 it starts at point po and at intermediate time t 1 the second point p is reached a third point p2 in the path if present is reached at t 2 and so on You can get the point on a path at any time t with the construction point t of path For a cyclic path with n arcs through the points po pi Pn 1 the normal para meter range is O lt t lt n but point t of path can be computed for any t by first 32 MAPS 32 Andr Heck reducing t modulo n The number of arcs in a path is available through length path The correspondence between the time parameter and a point on the curve is also used to create a subpath of the curve The command has the following syntax subpath pair expression of path expression If the value of the pair expression is t1 t2 and the path expression equals p then the result is a path that follows p from point t1 of p to point t2 of p If t1 gt t2 then the subpath runs backward along p Based on the subpath operation are the binary operators cutbefore and cutafter For intersecting paths p1 and p2 pi cutbefore p2 is eq
70. x Tyy Ty 0 0 1 The corresponding transformation in the two dimensional space is x y Txxx TxyY Tx TyxX T TyyY Ty ys This mapping is completely determined by the sextuple Tx Ty Txx Ixy Tyx Tyy The information about the mapping can be stored in a variable of data type transform and then be applied in a transformed statement There are three ways to define a transform In terms of basic transformations For example 26 MAPS 32 Andr Heck transform T T identity shifted 1 0 rotated 60 shifted 1 0 defines the transformation T as a composition of translating with vector 1 0 rotating around the origin over 60 degrees and translating with a vector 1 0 Specifying the sextuple Tx Ty Txx Ixy Tyx Tyy The six parameters than define a transformation T can be referred to directly as xpart T ypart T xxpart T xypart T yxpart T and yypart T Thus transform T xpart T ypart T 1 xxpart T yypart T 0 xypart T yxpart T 1 defines a transformation viz the reflection in the line through 1 0 and 0 1 beginfig 1 pair z z 2 1 cm pair w w 7 4 3 2 cm pair zw zw z zscaled w cm draw 0 5 0 cm 3 0 cm draw 0 0 5 cm 0 5 5 cm draw 1 0 cm z draw 0 0 z draw 0 0 w draw 0 0 zw draw w zw ZW def drawangle expr endofa endofb common length save tn tn turningnumber common endofa endo

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