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+% This file gives the macros for plain MetaPost
+% It contains all the features of plain METAFONT except those specific to
+% font-making. (See The METAFONTbook by D.E. Knuth).
+% There are also a number of macros for labeling figures, etc.
+string base_name, base_version; base_name="plain"; base_version="0.64";
+
+message "Preloading the plain mem file, version "&base_version;
+
+delimiters (); % this makes parentheses behave like parentheses
+def upto = step 1 until enddef; % syntactic sugar
+def downto = step -1 until enddef;
+def exitunless expr c = exitif not c enddef;
+let relax = \; % ignore the word `relax', as in TeX
+let \\ = \; % double relaxation is like single
+def ]] = ] ] enddef; % right brackets should be loners
+def -- = {curl 1}..{curl 1} enddef;
+def --- = .. tension infinity .. enddef;
+def ... = .. tension atleast 1 .. enddef;
+
+def gobble primary g = enddef;
+primarydef g gobbled gg = enddef;
+def hide(text t) = exitif numeric begingroup t;endgroup; enddef;
+def ??? = hide(interim showstopping:=1; showdependencies) enddef;
+def stop expr s = message s; gobble readstring enddef;
+
+warningcheck:=1;
+tracinglostchars:=1;
+
+def interact = % sets up to make "show" commands stop
+ hide(showstopping:=1; tracingonline:=1) enddef;
+
+def loggingall = % puts tracing info into the log
+ tracingcommands:=3; tracingtitles:=1; tracingequations:=1;
+ tracingcapsules:=1; tracingspecs:=2; tracingchoices:=1; tracinglostchars:=1;
+ tracingstats:=1; tracingoutput:=1; tracingmacros:=1; tracingrestores:=1;
+ enddef;
+
+def tracingall = % turns on every form of tracing
+ tracingonline:=1; showstopping:=1; loggingall enddef;
+
+def tracingnone = % turns off every form of tracing
+ tracingcommands:=0; tracingtitles:=0; tracingequations:=0;
+ tracingcapsules:=0; tracingspecs:=0; tracingchoices:=0; tracinglostchars:=0;
+ tracingstats:=0; tracingoutput:=0; tracingmacros:=0; tracingrestores:=0;
+ enddef;
+
+
+
+%% dash patterns
+
+vardef dashpattern(text t) =
+ save on, off, w;
+ let on=_on_;
+ let off=_off_;
+ w = 0;
+ nullpicture t
+enddef;
+
+tertiarydef p _on_ d =
+ begingroup save pic;
+ picture pic; pic=p;
+ addto pic doublepath (w,w)..(w+d,w);
+ w := w+d;
+ pic shifted (0,d)
+ endgroup
+enddef;
+
+tertiarydef p _off_ d =
+ begingroup w:=w+d;
+ p shifted (0,d)
+ endgroup
+enddef;
+
+
+
+%% basic constants and mathematical macros
+
+% numeric constants
+newinternal eps,epsilon,infinity,_;
+eps := .00049; % this is a pretty small positive number
+epsilon := 1/256/256; % but this is the smallest
+infinity := 4095.99998; % and this is the largest
+_ := -1; % internal constant to make macros unreadable but shorter
+
+newinternal mitered, rounded, beveled, butt, squared;
+mitered:=0; rounded:=1; beveled:=2; % linejoin types
+butt:=0; rounded:=1; squared:=2; % linecap types
+
+
+% pair constants
+pair right,left,up,down,origin;
+origin=(0,0); up=-down=(0,1); right=-left=(1,0);
+
+% path constants
+path quartercircle,halfcircle,fullcircle,unitsquare;
+fullcircle = makepath pencircle;
+halfcircle = subpath (0,4) of fullcircle;
+quartercircle = subpath (0,2) of fullcircle;
+unitsquare=(0,0)--(1,0)--(1,1)--(0,1)--cycle;
+
+% transform constants
+transform identity;
+for z=origin,right,up: z transformed identity = z; endfor
+
+% color constants
+color black, white, red, green, blue, background;
+black = (0,0,0);
+white = (1,1,1);
+red = (1,0,0);
+green = (0,1,0);
+blue = (0,0,1);
+background = white; % The user can reset this
+
+% picture constants
+picture blankpicture,evenly,withdots;
+blankpicture=nullpicture; % `display blankpicture...'
+evenly=dashpattern(on 3 off 3); % `dashed evenly'
+withdots=dashpattern(off 2.5 on 0 off 2.5); % `dashed withdots'
+
+% string constants
+string ditto, EOF;
+ditto = char 34; % ASCII double-quote mark
+EOF = char 0; % end-of-file for readfrom and write..to
+
+% pen constants
+pen pensquare,penrazor,penspeck;
+pensquare = makepen(unitsquare shifted -(.5,.5));
+penrazor = makepen((-.5,0)--(.5,0)--cycle);
+penspeck=pensquare scaled eps;
+
+% nullary operators
+vardef whatever = save ?; ? enddef;
+
+% unary operators
+let abs = length;
+
+vardef round primary u =
+ if numeric u: floor(u+.5)
+ elseif pair u: (round xpart u, round ypart u)
+ else: u fi enddef;
+
+vardef ceiling primary x = -floor(-x) enddef;
+
+vardef byte primary s =
+ if string s: ASCII fi s enddef;
+
+vardef dir primary d = right rotated d enddef;
+
+vardef unitvector primary z = z/abs z enddef;
+
+vardef inverse primary T =
+ transform T_; T_ transformed T = identity; T_ enddef;
+
+vardef counterclockwise primary c =
+ if turningnumber c <= 0: reverse fi c enddef;
+
+vardef tensepath expr r =
+ for k=0 upto length r - 1: point k of r --- endfor
+ if cycle r: cycle else: point infinity of r fi enddef;
+
+vardef center primary p = .5[llcorner p, urcorner p] enddef;
+
+
+
+% binary operators
+
+primarydef x mod y = (x-y*floor(x/y)) enddef;
+primarydef x div y = floor(x/y) enddef;
+primarydef w dotprod z = (xpart w * xpart z + ypart w * ypart z) enddef;
+
+primarydef x**y = if y=2: x*x else: takepower y of x fi enddef;
+def takepower expr y of x =
+ if x>0: mexp(y*mlog x)
+ elseif (x=0) and (y>0): 0
+ else: 1
+ if y=floor y:
+ if y>=0: for n=1 upto y: *x endfor
+ else: for n=_ downto y: /x endfor
+ fi
+ else: hide(errmessage "Undefined power: " & decimal x&"**"&decimal y)
+ fi fi enddef;
+
+vardef direction expr t of p =
+ postcontrol t of p - precontrol t of p enddef;
+
+vardef directionpoint expr z of p =
+ a_:=directiontime z of p;
+ if a_<0: errmessage("The direction doesn't occur"); fi
+ point a_ of p enddef;
+
+secondarydef p intersectionpoint q =
+ begingroup save x_,y_; (x_,y_)=p intersectiontimes q;
+ if x_<0: errmessage("The paths don't intersect"); origin
+ else: .5[point x_ of p, point y_ of q] fi endgroup
+enddef;
+
+tertiarydef p softjoin q =
+ begingroup c_:=fullcircle scaled 2join_radius shifted point 0 of q;
+ a_:=ypart(c_ intersectiontimes p); b_:=ypart(c_ intersectiontimes q);
+ if a_<0:point 0 of p{direction 0 of p} else: subpath(0,a_) of p fi
+ ... if b_<0:{direction infinity of q}point infinity of q
+ else: subpath(b_,infinity) of q fi endgroup enddef;
+newinternal join_radius,a_,b_; path c_;
+
+
+path cuttings; % what got cut off
+
+tertiarydef a cutbefore b = % tries to cut as little as possible
+ begingroup save t;
+ (t, whatever) = a intersectiontimes b;
+ if t<0:
+ cuttings:=point 0 of a;
+ a
+ else: cuttings:= subpath (0,t) of a;
+ subpath (t,length a) of a
+ fi
+ endgroup
+enddef;
+
+tertiarydef a cutafter b =
+ reverse (reverse a cutbefore b)
+ hide(cuttings:=reverse cuttings)
+enddef;
+
+
+
+% special operators
+vardef incr suffix $ = $:=$+1; $ enddef;
+vardef decr suffix $ = $:=$-1; $ enddef;
+
+def reflectedabout(expr w,z) = % reflects about the line w..z
+ transformed
+ begingroup transform T_;
+ w transformed T_ = w; z transformed T_ = z;
+ xxpart T_ = -yypart T_; xypart T_ = yxpart T_; % T_ is a reflection
+ T_ endgroup enddef;
+
+def rotatedaround(expr z, d) = % rotates d degrees around z
+ shifted -z rotated d shifted z enddef;
+let rotatedabout = rotatedaround; % for roundabout people
+
+vardef min(expr u)(text t) = % t is a list of numerics, pairs, or strings
+ save u_; setu_ u; for uu = t: if uu<u_: u_:=uu; fi endfor
+ u_ enddef;
+
+vardef max(expr u)(text t) = % t is a list of numerics, pairs, or strings
+ save u_; setu_ u; for uu = t: if uu>u_: u_:=uu; fi endfor
+ u_ enddef;
+
+def setu_ primary u =
+ if pair u: pair u_ elseif string u: string u_ fi;
+ u_=u enddef;
+
+def flex(text t) = % t is a list of pairs
+ hide(n_:=0; for z=t: z_[incr n_]:=z; endfor
+ dz_:=z_[n_]-z_1)
+ z_1 for k=2 upto n_-1: ...z_[k]{dz_} endfor ...z_[n_] enddef;
+newinternal n_; pair z_[],dz_;
+
+def superellipse(expr r,t,l,b,s)=
+ r{up}...(s[xpart t,xpart r],s[ypart r,ypart t]){t-r}...
+ t{left}...(s[xpart t,xpart l],s[ypart l,ypart t]){l-t}...
+ l{down}...(s[xpart b,xpart l],s[ypart l,ypart b]){b-l}...
+ b{right}...(s[xpart b,xpart r],s[ypart r,ypart b]){r-b}...cycle enddef;
+
+vardef interpath(expr a,p,q) =
+ for t=0 upto length p-1: a[point t of p, point t of q]
+ ..controls a[postcontrol t of p, postcontrol t of q]
+ and a[precontrol t+1 of p, precontrol t+1 of q] .. endfor
+ if cycle p: cycle
+ else: a[point infinity of p, point infinity of q] fi enddef;
+
+vardef solve@#(expr true_x,false_x)= % @#(true_x)=true, @#(false_x)=false
+ tx_:=true_x; fx_:=false_x;
+ forever: x_:=.5[tx_,fx_]; exitif abs(tx_-fx_)<=tolerance;
+ if @#(x_): tx_ else: fx_ fi :=x_; endfor
+ x_ enddef; % now x_ is near where @# changes from true to false
+newinternal tolerance, tx_,fx_,x_; tolerance:=.01;
+
+vardef buildcycle(text ll) =
+ save ta_, tb_, k_, i_, pp_; path pp_[];
+ k_=0;
+ for q=ll: pp_[incr k_]=q; endfor
+ i_=k_;
+ for i=1 upto k_:
+ (ta_[i], length pp_[i_]-tb_[i_]) =
+ pp_[i] intersectiontimes reverse pp_[i_];
+ if ta_[i]<0:
+ errmessage("Paths "& decimal i &" and "& decimal i_ &" don't intersect");
+ fi
+ i_ := i;
+ endfor
+ for i=1 upto k_: subpath (ta_[i],tb_[i]) of pp_[i] .. endfor
+ cycle
+enddef;
+
+
+
+%% units of measure
+
+mm=2.83464; pt=0.99626; dd=1.06601; bp:=1;
+cm=28.34645; pc=11.95517; cc=12.79213; in:=72;
+
+vardef magstep primary m = mexp(46.67432m) enddef;
+
+
+
+%% macros for drawing and filling
+
+def drawoptions(text t) =
+ def _op_ = t enddef
+enddef;
+
+linejoin:=rounded; % parameters that effect drawing
+linecap:=rounded;
+miterlimit:=10;
+
+drawoptions();
+
+pen currentpen;
+picture currentpicture;
+
+def fill expr c = addto currentpicture contour c _op_ enddef;
+def draw expr p =
+ addto currentpicture
+ if picture p:
+ also p
+ else:
+ doublepath p withpen currentpen
+ fi
+ _op_
+enddef;
+def filldraw expr c =
+ addto currentpicture contour c withpen currentpen
+ _op_ enddef;
+def drawdot expr z =
+ addto currentpicture contour makepath currentpen shifted z
+ _op_ enddef;
+
+def unfill expr c = fill c withcolor background enddef;
+def undraw expr p = draw p withcolor background enddef;
+def unfilldraw expr c = filldraw c withcolor background enddef;
+def undrawdot expr z = drawdot z withcolor background enddef;
+def erase text t =
+ def _e_ = withcolor background hide(def _e_=enddef;) enddef;
+ t _e_
+enddef;
+def _e_= enddef;
+
+def cutdraw text t =
+ begingroup interim linecap:=butt; draw t _e_; endgroup enddef;
+
+vardef image(text t) =
+ save currentpicture;
+ picture currentpicture;
+ currentpicture := nullpicture;
+ t;
+ currentpicture
+enddef;
+
+def pickup secondary q =
+ if numeric q: numeric_pickup_ else: pen_pickup_ fi q enddef;
+def numeric_pickup_ primary q =
+ if unknown pen_[q]: errmessage "Unknown pen"; clearpen
+ else: currentpen:=pen_[q];
+ pen_lft:=pen_lft_[q];
+ pen_rt:=pen_rt_[q];
+ pen_top:=pen_top_[q];
+ pen_bot:=pen_bot_[q];
+ currentpen_path:=pen_path_[q] fi; enddef;
+def pen_pickup_ primary q =
+ currentpen:=q;
+ pen_lft:=xpart penoffset down of currentpen;
+ pen_rt:=xpart penoffset up of currentpen;
+ pen_top:=ypart penoffset left of currentpen;
+ pen_bot:=ypart penoffset right of currentpen;
+ path currentpen_path; enddef;
+newinternal pen_lft,pen_rt,pen_top,pen_bot,pen_count_;
+
+vardef savepen = pen_[incr pen_count_]=currentpen;
+ pen_lft_[pen_count_]=pen_lft;
+ pen_rt_[pen_count_]=pen_rt;
+ pen_top_[pen_count_]=pen_top;
+ pen_bot_[pen_count_]=pen_bot;
+ pen_path_[pen_count_]=currentpen_path;
+ pen_count_ enddef;
+
+def clearpen = currentpen:=nullpen;
+ pen_lft:=pen_rt:=pen_top:=pen_bot:=0;
+ path currentpen_path;
+ enddef;
+def clear_pen_memory =
+ pen_count_:=0;
+ numeric pen_lft_[],pen_rt_[],pen_top_[],pen_bot_[];
+ pen currentpen,pen_[];
+ path currentpen_path, pen_path_[];
+ enddef;
+
+vardef lft primary x = x + if pair x: (pen_lft,0) else: pen_lft fi enddef;
+vardef rt primary x = x + if pair x: (pen_rt,0) else: pen_rt fi enddef;
+vardef top primary y = y + if pair y: (0,pen_top) else: pen_top fi enddef;
+vardef bot primary y = y + if pair y: (0,pen_bot) else: pen_bot fi enddef;
+
+vardef penpos@#(expr b,d) =
+ (x@#r-x@#l,y@#r-y@#l)=(b,0) rotated d;
+ x@#=.5(x@#l+x@#r); y@#=.5(y@#l+y@#r) enddef;
+
+def penstroke text t =
+ forsuffixes e = l,r: path_.e:=t; endfor
+ fill path_.l -- reverse path_.r -- cycle enddef;
+path path_.l,path_.r;
+
+
+
+%% High level drawing commands
+
+newinternal ahlength, ahangle;
+ahlength := 4; % default arrowhead length 4bp
+ahangle := 45; % default head angle 45 degrees
+
+vardef arrowhead expr p =
+ save q,e; path q; pair e;
+ e = point length p of p;
+ q = gobble(p shifted -e cutafter makepath(pencircle scaled 2ahlength))
+ cuttings;
+ (q rotated .5ahangle & reverse q rotated -.5ahangle -- cycle) shifted e
+enddef;
+
+path _apth;
+def drawarrow expr p = _apth:=p; _finarr enddef;
+def drawdblarrow expr p = _apth:=p; _findarr enddef;
+
+def _finarr text t =
+ draw _apth t;
+ filldraw arrowhead _apth t
+enddef;
+
+def _findarr text t =
+ draw _apth t;
+ filldraw arrowhead _apth withpen currentpen t;
+ filldraw arrowhead reverse _apth withpen currentpen t
+enddef;
+
+
+
+%% macros for labels
+
+newinternal bboxmargin; bboxmargin:=2bp;
+
+vardef bbox primary p =
+ llcorner p-(bboxmargin,bboxmargin) -- lrcorner p+(bboxmargin,-bboxmargin)
+ -- urcorner p+(bboxmargin,bboxmargin) -- ulcorner p+(-bboxmargin,bboxmargin)
+ -- cycle
+enddef;
+
+string defaultfont;
+newinternal defaultscale, labeloffset;
+defaultfont = "cmr10";
+defaultscale := 1;
+labeloffset := 3bp;
+
+vardef thelabel@#(expr s,z) = % Position s near z
+ save p; picture p;
+ if picture s: p=s
+ else: p = s infont defaultfont scaled defaultscale
+ fi;
+ p shifted (z + labeloffset*laboff@# -
+ (labxf@#*lrcorner p + labyf@#*ulcorner p
+ + (1-labxf@#-labyf@#)*llcorner p
+ )
+ )
+enddef;
+
+def label = draw thelabel enddef;
+newinternal dotlabeldiam; dotlabeldiam:=3bp;
+vardef dotlabel@#(expr s,z) =
+ label@#(s,z);
+ interim linecap:=rounded;
+ draw z withpen pencircle scaled dotlabeldiam;
+enddef;
+def makelabel = dotlabel enddef;
+
+pair laboff, laboff.lft, laboff.rt, laboff.top, laboff.bot;
+pair laboff.ulft, laboff.llft, laboff.urt, laboff.lrt;
+laboff =(0,0); labxf =.5; labyf =.5;
+laboff.lft=(-1,0); labxf.lft=1; labyf.lft=.5;
+laboff.rt =(1,0); labxf.rt =0; labyf.rt =.5;
+laboff.bot=(0,-1); labxf.bot=.5; labyf.bot=1;
+laboff.top=(0,1); labxf.top=.5; labyf.top=0;
+laboff.ulft=(-.7,.7);labxf.ulft=1; labyf.ulft=0;
+laboff.urt=(.7,.7); labxf.urt=0; labyf.urt=0;
+laboff.llft=-(.7,.7);labxf.llft=1; labyf.llft=1;
+laboff.lrt=(.7,-.7); labxf.lrt=0; labyf.lrt=1;
+
+vardef labels@#(text t) =
+ forsuffixes $=t:
+ label@#(str$,z$); endfor
+ enddef;
+vardef dotlabels@#(text t) =
+ forsuffixes $=t:
+ dotlabel@#(str$,z$); endfor
+ enddef;
+vardef penlabels@#(text t) =
+ forsuffixes $$=l,,r: forsuffixes $=t:
+ makelabel@#(str$.$$,z$.$$); endfor endfor
+ enddef;
+
+
+def range expr x = numtok[x] enddef;
+def numtok suffix x=x enddef;
+tertiarydef m thru n =
+ m for x=m+1 step 1 until n: , numtok[x] endfor enddef;
+
+
+
+%% Overall adminstration
+
+string extra_beginfig, extra_endfig;
+extra_beginfig = extra_endfig = "";
+
+def beginfig(expr c) =
+ begingroup
+ charcode:=c;
+ clearxy; clearit; clearpen;
+ pickup defaultpen;
+ drawoptions();
+ scantokens extra_beginfig;
+enddef;
+
+def endfig =
+ scantokens extra_endfig;
+ shipit;
+ endgroup
+enddef;
+
+
+%% last-minute items
+
+vardef z@#=(x@#,y@#) enddef;
+
+def clearxy = save x,y enddef;
+def clearit = currentpicture:=nullpicture enddef;
+def shipit = shipout currentpicture enddef;
+
+let bye = end; outer end,bye;
+
+clear_pen_memory; % initialize the `savepen' mechanism
+clearit;
+
+newinternal defaultpen;
+pickup pencircle scaled .5bp; % set default line width
+defaultpen := savepen;