% output=pdftex engine=pdftex language=uk
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% copyright=pragma-ade readme=readme.pdf licence=cc-by-nc-sa

\startcomponent mfun-012

\environment mfun-000

\chapter {A few applications}

\startintro

For those who need to be inspired, we will demonstrate how
\METAPOST\ can be used to enhance your document with simple
graphics. In these examples we will try to be not too clever,
simply because we lack the experience to be that clever. The
real tricks can be found in the files that come with
\METAPOST.

\stopintro

\section [sec:coils, sec:springs] {Simple drawings}

In the words of John Hobby, the creator of \METAPOST,
\quotation {\METAPOST\ is particularly well||suited for
generating figures for technical documents where some
aspects of a picture may be controlled by mathematical or
geometrical constraints that are best expressed
symbolically. In other words, \METAPOST\ is not meant to
take the place of a freehand drawing tool or even an
interactive graphics editor}.

An example of such a picture is the following one, which is
dedicated to David Arnold, who asked me once how to draw a
spring. So, imagine that we want to draw a schematic view of
a system of four springs.

\startbuffer[a]
def spring (expr a, b, w, h, n) =
  ( ( (0,0) -- (0,h) --
      for i=1 upto n-1: (if odd(i) : - fi w/2,i+h) -- endfor
      (0,n+h) -- (0,n+2h) )
    yscaled ((xpart (b-a) ++ ypart (b-a))/(n+2h))
    rotatedaround(origin,-90+angle(b-a))
    shifted a )
enddef ;
\stopbuffer

\startbuffer[b]
vardef spring (expr a, b, w, h, n) =
  pair vec ; path pat ; numeric len ; numeric ang ;
  vec := (b-a) ;
  pat := for i=1 upto n-1: (if odd(i):-fi w/2,i)--endfor (0,n) ;
  pat := (0,0)--(0,h)-- pat shifted (0,h)--(0,n+h)--(0,n+2h) ;
  len := (xpart vec ++ ypart vec)/(n+2h) ;
  ang := -90+angle(vec) ;
  ( pat yscaled len rotatedaround(origin,ang) shifted a )
enddef ;
\stopbuffer

\startbuffer[c]
path p ; p :=
  (0,0)--spring((.5cm,0),(2.5cm,0),.5cm,0,10)--(3cm,0) ;

draw p withpen pencircle scaled 2pt ;
draw p withpen pencircle scaled 1pt withcolor .8white;
\stopbuffer

\startbuffer[d]
z1 = (+2cm,0) ; z2 = (0,+2cm) ;
z3 = (-2cm,0) ; z4 = (0,-2cm) ;

pickup pencircle scaled 1.5pt ;

drawoptions (withcolor .625red) ;

draw spring (z1, z2, .75cm, 2, 10) ; draw z1 -- 1.5 z1 ;
draw spring (z2, z3, .75cm, 2,  9) ; draw z2 -- 1.1 z2 ;
draw spring (z3, z4, .75cm, 2,  8) ; draw z3 -- 1.5 z3 ;
draw spring (z4, z1, .75cm, 2,  7) ; draw z4 -- 1.1 z4 ;
\stopbuffer

\startbuffer[e]
drawarrow
  (0,0)--spring((.5cm,0),(2.5cm,0),.5cm,0,10)--(3cm,0)
  withpen pencircle scaled 2pt withcolor .625red ;
\stopbuffer

\startbuffer[f]
numeric u ; u := 1mm ; pickup pencircle scaled (u/2) ;
drawoptions (withcolor .625red) ;
draw (0,0)--spring((5u,0),(25u,0),5u,0,10)--(30u,0) ;
drawoptions (dashed evenly withcolor .5white) ;
draw (0,0)--spring((5u,0),(35u,0),(25/35)*5u,0,10)--(40u,0) ;
\stopbuffer

\startlinecorrection[blank]
\processMPbuffer[a,d]
\stoplinecorrection

A rather natural way to define such a system is:

\typebuffer[d]

Here, the macro \type {spring} takes 5~arguments: two points,
the width of the winding, the length of the connecting
pieces, and the number of elements (half windings). The
definition of \type {spring} is less complicated than
readable.

\typebuffer[a]

First we build a path starting in the origin, going left or
right depending on the counter being an odd number.

\starttyping
pat := (0,0) ;
for i=1 upto n-1:
  if odd(i) :
    pat := pat -- (-w/2,i) ;
  else :
    pat := pat -- (+w/2,i) ;
  fi ;
endfor ;
pat := pat -- (0,n) ;
\stoptyping

Once you are accustomed to the way \METAPOST\ interprets
(specialists may say expand) the source code, you will start
using \type {if} and \type {for} statements in assignments.
The previous code can be converted in a one liner, using the
pattern:

\starttyping
pat := for i=1 upto n-1: (x,y)-- endfor (0,n) ;
\stoptyping

The loop splits out a series of \type {(x,y)--} but the last
point is added outside the loop. Otherwise \type {pat} would
have ended with a dangling \type {--}. Of course we need to
replace \type {(x,y)} by something meaningful, so we get:

\starttyping
pat := for i=1 upto n-1: (if odd(i):-fi w/2,i)--endfor (0,n) ;
\stoptyping

We scale this path to the length needed. The expression
$b-a$ calculates a vector, starting at $a$ and ending at
$b$. In \METAPOST, the expression \type {a++b} is identical
to $\sqrt{a^2+b^2}$. Thus, the expression \typ {xpart (b-a)
++ ypart (b-a)} calculates the length of the vector $b-a$.
Because the unscaled spring has length $n+2h$, scaling by
the expression \typ {((xpart (b-a) ++ ypart (b-a)) / (n+2h))}
gives the spring the same length as the vector $b-a$.

Because we have drawn our spring in the vertical position,
we first rotate it 90 degrees clockwise to a horizontal
position, and then rotate it through an angle equal to the
angle in which the vector $b-a$ is pointing. After that, we
shift it to the first point. The main complications are that
we also want to draw connecting lines at the beginning and end,
as well as support springs that connect arbitrary points.
Since no check is done on the parameters, you should be
careful in using this macro.

When we want to improve the readability, we have to use
intermediate variables. Since the macro is expected to return
a path, we must make sure that the content matches this
expectation.

\typebuffer[b]

If you use \type {vardef}, then the last statement is the
return value. Here, when \typ {p := spring (z1, z2, .75cm,
2, 10)} is being parsed, the macro is expanded, the
variables are kept invisible for the assignment, and the path
at the end is considered to be the return value. In a \type
{def} the whole body of the macro is \quote {pasted} in the
text, while in a \type {vardef} only the last line is
visible. We will demonstrate this with a simple example.

\starttyping
def one = (n,n) ; n := n+1 ; enddef ;
def two = n := n + 1 ; (n,n) enddef ;
\stoptyping

Now, when we say:

\starttyping
pair a, b ; numeric n ; n= 10 ; a := one ; b := two ;
\stoptyping

we definitely get an error message. This is because, when
macro \type {two} is expanded, \METAPOST\ sees something:

\starttyping
b := n := n + 1 ;
\stoptyping

By changing the second definition in

\starttyping
vardef two = n := n + 1 ; (n,n) enddef ;
\stoptyping

the increment is expanded out of sight for \type {b :=} and
the pair \type {(n,n)} is returned.

We can draw a slightly better looking spring by drawing twice
with a different pen. The following commands use the spring
macro implemented by the \type {vardef}.

\typebuffer[c]

This time we get:

\startlinecorrection[blank]
\processMPbuffer[a,c]
\stoplinecorrection

Since the \type {spring} macro returns a path, you can do
whatever is possible with a path, like drawing an arrow:

\startlinecorrection[blank]
\processMPbuffer[a,e]
\stoplinecorrection

Or even (watch how we use the neutral unit \type {u} to
specify the dimensions):

\startlinecorrection[blank]
\processMPbuffer[a,f]
\stoplinecorrection

This was keyed in as:

\typebuffer[e]

and:

\typebuffer[f]

\section [sec:free labels] {Free labels}

The \METAPOST\ label macro enables you to position text at
certain points. This macro is kind of special, since it also
enables you to influence the positioning. For that purpose it
uses a special kind of syntax which we will not discuss here
in detail.

\startbuffer[a]
pickup pencircle scaled 1mm ;
path p ; p := fullcircle scaled 3cm ;
draw p withcolor .625yellow ;
dotlabel.rt   ("right"       , point 0 of p) ;
dotlabel.urt  ("upper right" , point 1 of p) ;
dotlabel.top  ("top"         , point 2 of p) ;
dotlabel.ulft ("upper left"  , point 3 of p) ;
dotlabel.lft  ("left"        , point 4 of p) ;
dotlabel.llft ("lower left"  , point 5 of p) ;
dotlabel.bot  ("bottom"      , point 6 of p) ;
dotlabel.lrt  ("lower right" , point 7 of p) ;
\stopbuffer

\typebuffer[a]

The \type {label} command just typesets a text, while \type
{dotlabel} also draws a dot at the position of the label.
The \type {thelabel} command returns a picture.

\startlinecorrection[blank]
\processMPbuffer[a]
\stoplinecorrection

There is a numeric constant \type {labeloffset} that can be
set to influence the distance between the point given and
the content of the label. When we set the offset to zero, we
get the following output.

\startbuffer[x]
interim labeloffset := 0pt ; % local assignment
\stopbuffer

\startlinecorrection[blank]
\processMPbuffer[x,a]
\stoplinecorrection

This kind of positioning works well as long as we know where
we want the label to be placed. However, when we place
labels automatically, for instance in a macro, we have to
apply a few clever tricks. There are fore sure many ways to
accomplish this goal, but here we will follow the mathless
method.

\startbuffer[a]
pickup pencircle scaled 1mm ;
path p ; p := fullcircle scaled 3cm ;
draw p withcolor .625yellow ;
vardef do (expr str) =
  save currentpicture ; picture currentpicture ;
  currentpicture := thelabel(str,origin) ;
  draw boundingbox currentpicture withpen pencircle scaled .5pt ;
  currentpicture
enddef ;
\stopbuffer

\startbuffer[b]
dotlabel.rt   (do("right")       , point 0 of p) ;
dotlabel.urt  (do("upper right") , point 1 of p) ;
dotlabel.top  (do("top")         , point 2 of p) ;
dotlabel.ulft (do("upper left")  , point 3 of p) ;
dotlabel.lft  (do("left")        , point 4 of p) ;
dotlabel.llft (do("lower left")  , point 5 of p) ;
dotlabel.bot  (do("bottom")      , point 6 of p) ;
dotlabel.lrt  (do("lower right") , point 7 of p) ;
\stopbuffer

\startlinecorrection[blank]
\processMPbuffer[x,a,b]
\stoplinecorrection

The previous graphic visualizes the bounding box of the
labels. This bounding box is rather tight and therefore the
placement of labels will always be suboptimal. Compare the
alignment of the left- and rightmost labels. The \type
{btex}||\type {etex} method is better, since then we can add
struts, like:

\starttyping
btex \strut right etex
\stoptyping

to force labels with uniform depths and heights. The next
graphic demonstrates that this looks better indeed. Spending
some time on such details pays back in better graphics.

\startbuffer[b]
dotlabel.rt   (do(btex \strut right       etex) , point 0 of p) ;
dotlabel.urt  (do(btex \strut upper right etex) , point 1 of p) ;
dotlabel.top  (do(btex \strut top         etex) , point 2 of p) ;
dotlabel.ulft (do(btex \strut upper left  etex) , point 3 of p) ;
dotlabel.lft  (do(btex \strut left        etex) , point 4 of p) ;
dotlabel.llft (do(btex \strut lower left  etex) , point 5 of p) ;
dotlabel.bot  (do(btex \strut bottom      etex) , point 6 of p) ;
dotlabel.lrt  (do(btex \strut lower right etex) , point 7 of p) ;
\stopbuffer

\startlinecorrection[blank]
\processMPbuffer[x,a,b]
\stoplinecorrection

Now, what happens when we want to place labels in other
positions? In the worst case, given that we place the labels
manually, we end up in vague arguments in favour for one or
the other placement.

\startbuffer[y]
p := p rotatedaround(center p, 22.5) ;
\stopbuffer

\startlinecorrection[blank]
\processMPbuffer[x,a,y,b]
\stoplinecorrection

Although any automatic mechanism will be sub||optimal, we can
give it a try to write a macro that deals with arbitrary
locations. This macro will accept three arguments and return
a picture.

\starttyping
thefreelabel("some string or picture",a position,the origin)
\stoptyping

Our testcase is just a simple \type {for} loop that places a
series of labels. The \type {freedotlabel} macro is derived
from \type {thefreelabel}.

\startbuffer[c]
pickup pencircle scaled 1mm ;
path p ; p := fullcircle scaled 3cm ;
draw p withcolor .625yellow ;
for i=0 step .5 until 7.5 :
  freedotlabel ("text" , point i of p, center p) ;
endfor ;
\stopbuffer

\typebuffer[c]

As a first step we will simply place the labels without any
correction. We also visualize the bounding box.

\startbuffer[b]
vardef freedotlabel (expr str, loc, ori) =
  drawdot loc ; draw thefreelabel(str,loc,ori) ;
enddef ;

vardef freelabel (expr str, loc, ori) =
  draw thefreelabel(str,loc,ori) ;
enddef ;
\stopbuffer

\startbuffer[a]
vardef thefreelabel (expr str, loc, ori) =
  save s ; picture s ; s := thelabel(str,loc) ;
  draw boundingbox s withpen pencircle scaled .5pt ;
  s
enddef ;
\stopbuffer

\typebuffer[a]

To make our lives more easy, we also define a macro that draws
the dot as well as a macro that draws the label.

\typebuffer[b]

Now we get:

\startlinecorrection[blank]
\processMPbuffer[x,a,b,c]
\stoplinecorrection

The original label macros permits us to align the label at
positions, 4~corners and 4~points halfway the sides. It
happens that circles are also composed of 8~points. Because
in most cases the label is to be positioned in the direction
of the center of a curve and the point at hand, it makes
sense to take circles as the starting points for positioning
the labels.

To help us in positioning, we define a special square path,
\type {freesquare}. This path is constructed out of
8~points that match the positions that are used to align
labels.

\startbuffer[d]
path freesquare ;

freesquare := ((-1,0)--(-1,-1)--(0,-1)--(+1,-1)--
               (+1,0)--(+1,+1)--(0,+1)--(-1,+1)--cycle) scaled .5 ;
\stopbuffer

\typebuffer[d]

We now show this free path together with a circle, using the
following definitions:

\startbuffer[e]
drawpath        fullcircle scaled 3cm ;
drawpoints      fullcircle scaled 3cm ;
drawpointlabels fullcircle scaled 3cm ;
currentpicture := currentpicture shifted (5cm,0) ;
drawpath        freesquare scaled 3cm ;
drawpoints      freesquare scaled 3cm ;
drawpointlabels freesquare scaled 3cm ;
\stopbuffer

\typebuffer[e]

We use two drawing macros that are part of the suite of
visual debugging macros.

\startlinecorrection[blank]
\processMPbuffer[x,d,e]
\stoplinecorrection

As you can see, point~1 is the corner point that suits best
for alignment when a label is put at point~1 of the circle.
We will now rewrite \type {thefreelabel} in such a way that
the appropriate point of the associated \type {freesquare}
is found.

\startbuffer[a]
vardef thefreelabel (expr str, loc, ori) =
  save s, p, q, l ; picture s ; path p, q ; pair l ;
  s := thelabel(str,loc) ;
  p := fullcircle scaled (2*length(loc-ori)) shifted ori ;
  q := freesquare xyscaled (urcorner s - llcorner s) ;
  l := point (xpart (p intersectiontimes (ori--loc))) of q ;
  draw q shifted loc withpen pencircle scaled .5pt ;
  draw l shifted loc withcolor .625yellow ;
  draw loc withcolor .625red ;
  s
enddef ;
\stopbuffer

\typebuffer[a]

The macro xyscaled is part of \METAFUN\ and scales in two
directions at once. The \METAPOST\ primitive \type
{intersectiontimes} returns a pair of time values of the
point where two paths intersect. The first part of the pair
concerns the first path.

\startlinecorrection[blank]
\processMPbuffer[x,a,b,c]
\stoplinecorrection

We are now a small step from the exact placement. If we
change the last line of the macro into:

\starttyping
(s shifted -l)
\stoptyping

we get the displacement we want. Although the final look and
feel is also determined by the text itself, the average
result is quite acceptable.

\startbuffer[a]
vardef thefreelabel (expr str, loc, ori) =
  save s, p, q, l ; picture s ; path p, q ; pair l ;
  s := thelabel(str,loc) ;
  p := fullcircle scaled (2*length(loc-ori)) shifted ori ;
  q := freesquare xyscaled (urcorner s - llcorner s) ;
  l := point (xpart (p intersectiontimes (ori--loc))) of q ;
  draw q shifted loc withpen pencircle scaled .5pt ;
  draw l shifted loc withcolor .625yellow ;
  draw loc withcolor .625red ;
  (s shifted -l)
enddef ;
\stopbuffer

\startlinecorrection[blank]
\processMPbuffer[x,a,b,c]
\stoplinecorrection

Because we also want to pass pictures, and add a bit of
offset too, the final implementation is slightly more
complicated. The picture is handled with an additional
condition, and the offset with the \METAFUN\ macro \type
{enlarged}.

\startbuffer[a]
newinternal freelabeloffset ; freelabeloffset := 3pt ;

vardef thefreelabel (expr str, loc, ori) =
  save s, p, q, l ; picture s ; path p, q ; pair l ;
  interim labeloffset := freelabeloffset ;
  s := if string str : thelabel(str,loc)
       else          : str shifted -center str shifted loc fi ;
  setbounds s to boundingbox s enlarged freelabeloffset ;
  p := fullcircle scaled (2*length(loc-ori)) shifted ori ;
  q := freesquare xyscaled (urcorner s - llcorner s) ;
  l := point (xpart (p intersectiontimes (ori--loc))) of q ;
  setbounds s to boundingbox s enlarged -freelabeloffset ;
  (s shifted -l)
enddef ;
\stopbuffer

\typebuffer[a]

Watch how we temporarily enlarge the bounding box of the
typeset label text. We will now test this macro on a
slightly rotated circle, using labels typeset by \TEX. The
\type {reverse} is there purely for cosmetic reasons, to
suit the label texts.

\startbuffer[b]
pickup pencircle scaled 1mm ;
path p ; p := reverse fullcircle rotated -25 scaled 3cm ;
draw p withcolor .625yellow ; pair cp ; cp := center p ;
freedotlabel (btex \strut We can    etex, point 0 of p, cp) ;
freedotlabel (btex \strut go on     etex, point 1 of p, cp) ;
freedotlabel (btex \strut and on    etex, point 2 of p, cp) ;
freedotlabel (btex \strut in        etex, point 3 of p, cp) ;
freedotlabel (btex \strut defining  etex, point 4 of p, cp) ;
freedotlabel (btex \strut funny     etex, point 5 of p, cp) ;
freedotlabel (btex \strut macros.   etex, point 6 of p, cp) ;
freedotlabel (btex \strut Can't we? etex, point 7 of p, cp) ;
\stopbuffer

\startlinecorrection[blank]
\processMPbuffer[a,b]
\stoplinecorrection

\typebuffer[b]

Unfortunately we can run into problems due to rounding
errors. Therefore we use a less readable but more safe
expression for calculating the intersection points. Instead
of using point \type {loc} as endpoint we use \type {loc}
shifted over a very small distance into the direction \type
{loc} from \type{ori}. In the assignment to~\type {l} we
replace \type {loc} by:

\starttyping
 ( (1+eps) * arclength(ori--loc) * unitvector(loc-ori) )
\stoptyping

\section {Marking angles}

A convenient \METAPOST\ macro is \type {unitvector}. When we
draw a line segment from the origin to the point returned by
this macro, the segment has a length of 1~base point. This
macro has a wide range of applications, but some basic
knowlegde of vector algebra is handy. The following lines of
\METAPOST\ code demonstrate the basics behind unitvectors.

\startbuffer
pair uv ; pickup pencircle scaled 1mm ; autoarrows := true ;
draw fullcircle scaled 2cm withcolor .625red    ;
for i=(10,35), (-40,-20), (85,-15) :
  draw origin--i dashed evenly withcolor .625white ;
  drawarrow origin--unitvector(i) scaled 1cm withcolor .625yellow ;
endfor ;
draw origin withcolor .625red ;
\stopbuffer

\typebuffer

The circle has a radius of 1cm, and the three line segments
are drawn from the origin in the direction of the points that
are passed as arguments. Because the vector has length of~1,
we scale it to the radius to let it touch the circle. By
setting \type {autoarrows} we make sure that the arrowheads
are scaled proportionally to the linewidth of 1~mm.

\startlinecorrection[blank]
\processMPbuffer
\stoplinecorrection

An application of this macro is drawing the angle between
two lines. In the \METAPOST\ manual you can find two macros
for drawing angles: \type {mark_angle} and \type
{mark_rt_angle}. You may want to take a look at their
definitions before we start developing our own alternatives.

\startbuffer[x]
pickup pencircle scaled 1mm ; autoarrows := true ;
drawoptions(withcolor .625white) ;
\stopbuffer

\startbuffer[a]
def anglebetween (expr a, b) =
  (unitvector(a){a rotated 90} .. unitvector(b))
enddef ;
\stopbuffer

\startbuffer[b]
pair a, b ; a := (2cm,-1cm) ; b := (3cm,1cm) ;
drawarrow origin--a ; drawarrow origin--b ;
drawarrow anglebetween(a,b) scaled 1cm withcolor .625red ;
\stopbuffer

\startlinecorrection[blank]
\processMPbuffer[x,a,b]
\stoplinecorrection

The previous graphic demonstrates what we want to
accomplish: a circular curve indicating the angle between
two straight lines. The lines and curve are drawn with the
code:

\typebuffer[b]

where \type {anglebetween} is defined as:

\typebuffer[a]

Both unitvectors return just a point on the line positioned
1~unit (later scaled to 1cm) from the origin. We connect
these points by a curve that starts in the direction at
the first point. If we omit the \type {a rotated 90}
direction specifier, we get:

\startbuffer[a]
def anglebetween (expr a, b) =
  (unitvector(a) .. unitvector(b))
enddef ;
\stopbuffer

\startlinecorrection[blank]
\processMPbuffer[x,a,b]
\stoplinecorrection

These definitions of \type {anglebetween} are far from
perfect. If we don't start in the origin, we get the curve
in the wrong place and when we swap both points, we get the
wrong curve.

\startbuffer[a]
def anglebetween (expr endofa, endofb, common, length) =
  (unitvector (endofa-common){(endofa-common) rotated 90} ..
   unitvector (endofb-common)) scaled length shifted common
enddef ;
\stopbuffer

\startbuffer[b]
pair a, b, c ; a := (2cm,-1cm) ; b := (3cm,1cm) ; c := (-1cm,.5cm) ;
drawarrow c--a ; drawarrow c--b ;
drawarrow anglebetween(a,b,c,1cm) withcolor .625red ;
\stopbuffer

The solution for the displacement is given in the \METAPOST\
manual and looks like this (we package the macro a bit
different):

\typebuffer[a]

As you can see, we compensate for the origin of both
vectors. This macro is called with a few more parameters. We
need to pass the length, since we want to add the shift to
the macro and the shift takes place after the scaling.

\typebuffer[b]

That the results are indeed correct, is demonstrated by the
output of following example:

\startlinecorrection[blank]
\processMPbuffer[x,a,b]
\stoplinecorrection

However, when we swap the points, we get:

\startbuffer[a]
def anglebetween (expr endofb, endofa, common, length) =
  (unitvector (endofa-common){(endofa-common) rotated 90} ..
   unitvector (endofb-common)) scaled length shifted common
enddef ;
\stopbuffer

\startlinecorrection[blank]
\processMPbuffer[x,a,b]
\stoplinecorrection

This means that instead of rotating over $90$ degrees, we have
to rotate over $-90$ or $270$ degrees. That way the arrow will
also point in the other direction. There are undoubtedly more
ways to determine the direction, but the following method
also demonstrates the use of \type {turningnumber}, which
reports the direction of a path. For this purpose we compose
a dummy cyclic path.

\startbuffer[a]
vardef anglebetween (expr endofa, endofb, common, length) =
  save tn ; tn := turningnumber(common--endofa--endofb--cycle) ;
show tn ;
  (unitvector(endofa-common){(endofa-common) rotated (tn*90)} ..
   unitvector(endofb-common)) scaled length shifted common
enddef ;
\stopbuffer

\typebuffer[a]

Because we use an intermediate variable, just to keep things
readable, we have to use \type {vardef} to hide the
assignment for the outside world. We demonstrate this macro
using the following code:

\startbuffer[b]
pair a, b, c ; a := (2cm,-1cm) ; b := (3cm,1cm) ; c := (-1cm,.5cm) ;
drawarrow c--a ; drawarrow c--b ;
drawarrow anglebetween(a,b,c,0.75cm) withcolor .625red ;
drawarrow anglebetween(b,a,c,1.50cm) withcolor .625red ;
\stopbuffer

\typebuffer[b]

Watch how both arrows point in the direction of the line that
is determined by the second point.

\startlinecorrection[blank]
\processMPbuffer[x,a,b]
\stoplinecorrection

We now have the framework of an angle drawing macro ready
and can start working placing the label.

\startbuffer[a]
vardef anglebetween (expr endofa, endofb, common, length, str) =
  save curve, where ; path curve ; numeric where ;
  where := turningnumber (common--endofa--endofb--cycle) ;
  curve := (unitvector(endofa-common){(endofa-common) rotated (where*90)}
         .. unitvector(endofb-common)) scaled length shifted common ;
  draw thefreelabel(str,point .5 of curve,common) withcolor black ;
  curve
enddef ;
\stopbuffer

\typebuffer[a]

The macro \type {thefreelabel} is part of \METAFUN\ and is
explained in detail in \in {section} [sec:free labels]. This
macro tries to place the label as good as possible without
user interference.

\startbuffer[b]
pair a ; a := (2cm,-1cm) ; drawarrow origin--a ;
pair b ; b := (3cm, 1cm) ; drawarrow origin--b ;
drawarrow
  anglebetween(a,b,origin,1cm,btex $\alpha$ etex)
  withcolor .625red ;
\stopbuffer

\typebuffer[b]

Instead of a picture we may also pass a string, but using
\TEX\ by means of \type {btex}||\type {etex} often leads to
better results.

\startlinecorrection[blank]
\processMPbuffer[x,a,b]
\stoplinecorrection

Because in most cases we want the length to be consistent
between figures and because passing two paths is more
convenient than passing three points, the final definition
looks slightly different.

\startbuffer[a]
numeric anglelength ; anglelength := 20pt ;

vardef anglebetween (expr a, b, str) = % path path string
  save endofa, endofb, common, curve, where ;
  pair endofa, endofb, common ; path curve ; numeric where ;
  endofa := point length(a) of a ;
  endofb := point length(b) of b ;
  if round point 0 of a = round point 0 of b :
    common := point 0 of a ;
  else :
    common := a intersectionpoint b ;
  fi ;
  where := turningnumber (common--endofa--endofb--cycle) ;
  curve := (unitvector (endofa-common){(endofa-common) rotated (where*90)} ..
            unitvector (endofb-common)) scaled anglelength shifted common ;
  draw thefreelabel(str,point .5 of curve,common) withcolor black ;
  curve
enddef ;
\stopbuffer

\typebuffer[a]

This macro has a few more \type {if}'s than its predecessor.
First we test if the label is a string, and if so, we
calculate the picture ourselves, otherwise we leave this to
the user.

\startbuffer[b]
path a, b, c, d, e, f ;
a := origin--( 2cm, 1cm) ; b := origin--( 1cm, 2cm) ;
c := origin--(-2cm, 2cm) ; d := origin--(-2cm,-1cm) ;
e := origin--(-1cm,-2cm) ; f := origin--( 1cm,-2cm) ;
for i=a, b, c, d, e, f : drawarrow i ; endfor ;
anglelength := 1.0cm ; drawoptions(withcolor .625red) ;
drawarrow         anglebetween(a,b,btex $\alpha  $ etex) ;
drawarrow         anglebetween(c,d,btex $\gamma  $ etex) ;
drawarrow         anglebetween(e,f,btex $\epsilon$ etex) ;
anglelength := 1.5cm ; drawoptions(withcolor .625yellow) ;
drawdblarrow      anglebetween(b,c,btex $\beta   $ etex) ;
drawarrow reverse anglebetween(d,e,btex $\delta  $ etex) ;
drawarrow         anglebetween(a,f,btex $\zeta   $ etex) ;
\stopbuffer

\typebuffer[b]

Because \type {anglebetween} returns a path, you can apply
transformations to it, like reversing. Close reading of the
previous code learns that the macro handles both directions.

\startlinecorrection[blank]
\processMPbuffer[x,a,b]
\stoplinecorrection

Multiples of 90 degrees are often identified by a
rectangular symbol. We will now extend the previously
defined macro in such a way that more types can be drawn.

\startbuffer[a]
numeric anglelength ; anglelength := 20pt ;
numeric anglemethod ; anglemethod :=    1 ;

vardef anglebetween (expr a, b, str) = % path path string
  save pointa, pointb, common, middle, offset ;
  pair pointa, pointb, common, middle, offset ;
  save curve ; path curve ;
  save where ; numeric where ;
  if round point 0 of a = round point 0 of b :
    common := point 0 of a ;
  else :
    common := a intersectionpoint b ;
  fi ;
  pointa := point anglelength on a ;
  pointb := point anglelength on b ;
  where  := turningnumber (common--pointa--pointb--cycle) ;
  middle := ((common--pointa) rotatedaround (pointa,-where*90))
                            intersectionpoint
            ((common--pointb) rotatedaround (pointb, where*90)) ;
  if     anglemethod = 1 :
    curve  := pointa{unitvector(middle-pointa)}.. pointb;
    middle := point .5 along curve ;
  elseif anglemethod = 2 :
    middle := common rotatedaround(.5[pointa,pointb],180) ;
    curve  := pointa--middle--pointb ;
  elseif anglemethod = 3 :
    curve  := pointa--middle--pointb ;
  elseif anglemethod = 4 :
    curve  := pointa..controls middle..pointb ;
    middle := point .5 along curve ;
  fi ;
  draw thefreelabel(str, middle, common) withcolor black ;
  curve
enddef ;
\stopbuffer

\typebuffer[a]

\startbuffer[p]
anglemethod := 1 ;
\stopbuffer

\startbuffer[q]
anglemethod := 2 ;
\stopbuffer

\startbuffer[r]
anglemethod := 3 ;
\stopbuffer

\startbuffer
\startcombination[3*1]
  {\processMPbuffer[x,a,p,b]} {method 1}
  {\processMPbuffer[x,a,q,b]} {method 2}
  {\processMPbuffer[x,a,r,b]} {method 3}
\stopcombination
\stopbuffer

\placefigure
  [here][fig:three methods]
  {Three ways of marking angles.}
  {\getbuffer}

\in {Figure} [fig:three methods] shows the first three
alternative methods implemented here. Instead of using
\type {unitvectors}, we now calculate the points using the
\type {arctime} and \type {arclength} primitives. Instead
of complicated expressions, we use the \METAFUN\ operators
\type {along} and \type {on}. The following expressions are
equivalent.

\starttyping
pointa := point anglelength on a ;
middle := point .5 along curve ;

pointa := point (arctime anglelength of a) of a ;
middle := arctime (.5(arclength curve)) of curve) of curve ;
\stoptyping

The third method can be implemented in different, more math
intensive ways, but the current implementation suits rather
well and is understood by the author.

\section [sec:color circles]{Color circles}

In \in {chapter} [sec:embedding] we showed a few color
circles. Drawing such a graphic can be done in several ways,
and here we will show a few methods. First we will
demonstrate how you can apply \type {cutafter} and \type
{curbefore}, next we will show how the \METAPOST\ macro
\type {buildpath} can be used, and finally we will present a
clean solution using \type {subpath}. We will assume that
the circle is called with the macro:

\starttyping
colorcircle (4cm, red, green, blue) ;
\stoptyping

\startbuffer[circle]
vardef colorcircle (expr size, red, green, blue) =
  save r, g, b, rr, gg, bb, cc, mm, yy ;
  save b_r, b_g, g_r, g_b ;
  save radius ;

  path r, g, b, rr, bb, gg, cc, mm, yy ;
  pair b_r, b_g, g_r, g_b ;

  numeric radius ; radius := 3cm ;

  pickup pencircle scaled (radius/20) ;

  r := g := b := fullcircle scaled radius shifted (0,radius/4);

  r := r rotatedaround(origin, 15) ; % drawarrow r withcolor red   ;
  g := g rotatedaround(origin,135) ; % drawarrow g withcolor green ;
  b := b rotatedaround(origin,255) ; % drawarrow b withcolor blue  ;

  b_r :=         b intersectionpoint r ; % draw b_r ;
  b_g :=         b intersectionpoint g ; % draw b_g ;
  g_r := reverse g intersectionpoint r ; % draw g_r ;
  g_b := reverse g intersectionpoint b ; % draw g_b ;

  bb := b cutafter  b_r ; bb := bb cutbefore b_g ; % drawarrow bb ;
  gg := g cutbefore b_g ; gg := gg cutafter  g_r ; % drawarrow gg ;
  rr := r cutbefore g_r &        r cutafter  b_r ; % drawarrow rr ;

  cc := b cutbefore b_r ; cc := cc cutafter  g_b ; % drawarrow br ;
  yy := g cutbefore g_r ; yy := yy cutafter  g_b ; % drawarrow rg ;
  mm := r cutbefore g_r &        r cutafter  b_r ; % drawarrow gb ;

  bb := gg -- rr -- reverse bb -- cycle ;
  gg := bb rotatedaround(origin,120) ;
  rr := bb rotatedaround(origin,240) ;

  cc := mm -- cc -- reverse yy -- cycle ;
  yy := cc rotatedaround(origin,120) ;
  mm := cc rotatedaround(origin,240) ;

  fill fullcircle scaled radius withcolor white ;

  fill rr withcolor red   ; fill cc withcolor white-red   ;
  fill gg withcolor green ; fill mm withcolor white-green ;
  fill bb withcolor blue  ; fill yy withcolor white-blue  ;

  for i = rr,gg,bb,cc,mm,yy : draw i withcolor .5white ; endfor ;

  currentpicture := currentpicture xsized size ;
enddef ;
\stopbuffer

We need to calculate seven paths. The first implementation
does all the handywork itself and thereby is rather long,
complicated and unreadable. It does not really use the
strenghth of \METAPOST\ yet.

\typebuffer[circle]

In determining the right intersection points, you need to
know where the path starts and in what direction it moves.
In case of doubt, drawing the path as an arrow helps. If you
want to see the small paths used, you need to comment the
lines with the \type {fill}'s and uncomment the lines with
\type {draw}'s. Due to the symmetry and the fact that we
keep the figure centered around the origin, we only need to
calculate two paths since we can rotate them.

There are for sure more (efficient) ways to draw such a
figure, but this one demonstrates a few new tricks, like
grouping. We use grouping here because we want to use \type
{mm} to indicate the magenta path, and \type {mm} normally
means millimeter. Within a group, you can save variables.
These get their old values when the group is left.

With \type {for} we process multiple paths after each other.
In this case it hardly saves tokens, but it looks more
clever.

One of the more efficient methods is using the \type
{buildcycle} macro. This macro takes two or more paths and
calculates the combined path. Although this is a rather
clever macro, you should be prepared to help it a bit when
paths have multiple intersection points. Again, we could
follow a more secure mathematical method, but the next one
took only a few minutes of trial and error. To save some
memory, we redefine the \type {colors} graphic.

\startbuffer[demo]
colorcircle(4cm, red, green, blue) ;
\stopbuffer

When we call this macro as:

\typebuffer[demo]

we get:

\startlinecorrection[blank]
\processMPbuffer[circle,demo]
\stoplinecorrection

\startbuffer[circle]
vardef colorcircle (expr size, red, green, blue) =
  save r, g, b, rr, gg, bb, cc, mm, yy ; save radius ;
  path r, g, b, rr, bb, gg, cc, mm, yy ; numeric radius ;

  radius := 5cm ; pickup pencircle scaled (radius/25) ;

  r := g := b := fullcircle scaled radius shifted (0,radius/4) ;

  r := r rotatedaround (origin, 15) ;
  g := g rotatedaround (origin,135) ;
  b := b rotatedaround (origin,255) ;

  r := r rotatedaround(center r,-90) ;
  g := g rotatedaround(center g, 90) ;

  gg := buildcycle(buildcycle(reverse r,b),g) ;
  cc := buildcycle(buildcycle(b,reverse g),r) ;

  rr := gg rotatedaround(origin,120) ;
  bb := gg rotatedaround(origin,240) ;

  yy := cc rotatedaround(origin,120) ;
  mm := cc rotatedaround(origin,240) ;

  fill fullcircle scaled radius withcolor white ;

  fill rr withcolor red   ; fill cc withcolor white-red   ;
  fill gg withcolor green ; fill mm withcolor white-green ;
  fill bb withcolor blue  ; fill yy withcolor white-blue  ;

  for i = rr,gg,bb,cc,mm,yy : draw i withcolor .5white ; endfor ;

  currentpicture := currentpicture xsized size ;
enddef ;
\stopbuffer

\typebuffer [circle]

Since we don't want to duplicate a graphic, this time we
show the dark alternatives.

\startbuffer[demo]
colorcircle(4cm, .5red, .5green, .5blue) ;
\stopbuffer

\typebuffer[demo]

This kind of unsafe path calculations are very sensitive to
breaking. Changing the \type {radius/4} into something else
demonstrates this but we will not challenge this macro that
much. Therefore, the 50\% color circle shows up as:

\startlinecorrection[blank]
\processMPbuffer[circle,demo]
\stoplinecorrection

This command is part of \METAFUN\ and can be used to
determine nice color combinations by also looking at their
complementary colors.

\startbuffer[demo]
colorcircle (4cm, .7red, .5green, .3blue) ;
\stopbuffer

\typebuffer[demo]

\startlinecorrection[blank]
\processMPbuffer[circle,demo]
\stoplinecorrection

The next circle that we draw shows the three main colors
used in this document. This circle is not that beautiful.

\startbuffer[demo]
colorcircle(4cm,.625red,.625yellow,.625white) ;
\stopbuffer

\typebuffer[demo]

\startlinecorrection[blank]
\processMPbuffer[circle,demo]
\stoplinecorrection

This definition can be cleaned up a bit by using \type
{transform}, but the fuzzy \type {buildcycle}'s remain.

\startbuffer[circle]
vardef colorcircle (expr size, red, green, blue) =
  save r, g, b, rr, gg, bb, cc, mm, yy ; save radius ;
  path r, g, b, rr, bb, gg, cc, mm, yy ; numeric radius ;

  radius := 5cm ; pickup pencircle scaled (radius/25) ;

  transform t ; t := identity rotatedaround(origin,120) ;

  r := fullcircle scaled radius
   shifted (0,radius/4) rotatedaround(origin,15) ;

  g := r transformed t ; b := g transformed t ;

  r := r rotatedaround(center r,-90) ;
  g := g rotatedaround(center g, 90) ;

  gg := buildcycle(buildcycle(reverse r,b),g) ;
  cc := buildcycle(buildcycle(b,reverse g),r) ;

  rr := gg transformed t ; bb := rr transformed t ;
  yy := cc transformed t ; mm := yy transformed t ;

  fill fullcircle scaled radius withcolor white ;

  fill rr withcolor red   ; fill cc withcolor white-red   ;
  fill gg withcolor green ; fill mm withcolor white-green ;
  fill bb withcolor blue  ; fill yy withcolor white-blue  ;

  for i = rr,gg,bb,cc,mm,yy : draw i withcolor .5white ; endfor ;

  currentpicture := currentpicture xsized size ;
enddef ;
\stopbuffer

\typebuffer [circle]

\startbuffer[demo]
colorcircle(4cm,(.4,.6,.8),(.8,.4,.6),(.6,.8,.4));
\stopbuffer

\startlinecorrection[blank]
\processMPbuffer[circle,demo]
\stoplinecorrection

This rather nice circle is defined as:

\typebuffer[demo]

The final implementation, which is part of \METAFUN, is
slightly more efficient.

\startbuffer[circle]
vardef colorcircle (expr size, red, green, blue) =
  save r, g, b, c, m, y, w ; save radius ;
  path r, g, b, c, m, y, w ; numeric radius ;

  radius := 5cm ; pickup pencircle scaled (radius/25) ;

  transform t ; t := identity rotatedaround(origin,120) ;

  r := fullcircle rotated 90 scaled radius
         shifted (0,radius/4) rotatedaround(origin,135) ;

  b := r transformed t ; g := b transformed t ;

  c := buildcycle(subpath(1,7) of g, subpath(1,7) of b) ;
  y := c transformed t ; m := y transformed t ;

  w := buildcycle(subpath(3,5) of r,
         subpath(3,5) of g, subpath(3,5) of b) ;

  pushcurrentpicture ;

  fill r withcolor red   ;
  fill g withcolor green ;
  fill b withcolor blue  ;
  fill c withcolor white-red   ;
  fill m withcolor white-green ;
  fill y withcolor white-blue  ;
  fill w withcolor white ;

  for i = r,g,b,c,m,y : draw i withcolor .5white ; endfor ;

  currentpicture := currentpicture xsized size ;

  popcurrentpicture ;
enddef ;
\stopbuffer

\typebuffer [circle]

Here, we first fill the primary circles, next we fill the
secondary ones. These also cover the center, which is why
finally we fill the center with white.

\startbuffer[demo]
colorcircle(4cm,(.2,.5,.8),(.8,.2,.5),(.5,.8,.2));
\stopbuffer

\startlinecorrection[blank]
\processMPbuffer[circle,demo]
\stoplinecorrection

The circle uses the following colors:

\typebuffer[demo]

The next graphic demonstrates how the subpaths look that
build the shapes.

\startbuffer[circle]
vardef colorcircle (expr size, red, green, blue) =
  save r, g, b, c, m, y, w ; save radius ;
  path r, g, b, c, m, y, w ; numeric radius ;

  radius := 5cm ; pickup pencircle scaled (radius/25) ;

  transform t ; t := identity rotatedaround(origin,120) ;

  r := fullcircle rotated 90 scaled radius
         shifted (0,radius/4) rotatedaround(origin,135) ;

  b := r transformed t ; g := b transformed t ;

  c := buildcycle(subpath(1,7) of g,subpath(1,7) of b) ;
  y := c transformed t ; m := y transformed t ;

  w := buildcycle(subpath(3,5) of r,
         subpath(3,5) of g, subpath(3,5) of b) ;

  pushcurrentpicture ;

  def do_it =
    fill r withcolor red   ;
    fill g withcolor green ;
    fill b withcolor blue  ;
    fill c withcolor white-red   ;
    fill m withcolor white-green ;
    fill y withcolor white-blue  ;
    fill w withcolor white ;
    for i = r,g,b,c,m,y : draw i withcolor .5white ; endfor ;
  enddef ;

  autoarrows := true ;

  do_it ;
  for i=r,g,b : drawarrow i withcolor black ; endfor ;
  currentpicture := currentpicture shifted (-2radius,0) ;

  do_it ;
  for i=r,g,b : drawarrow subpath(1,7) of i withcolor black ; endfor ;
  currentpicture := currentpicture shifted (-2radius,0) ;

  do_it ;
  for i=r,g,b : drawarrow subpath(3,5) of i withcolor black ; endfor ;
  currentpicture := currentpicture shifted (+4radius,2radius) ;

  drawarrow r withpen pencircle scaled (radius/10) withcolor red   ;
  drawarrow g withpen pencircle scaled (radius/20) withcolor green ;
  drawarrow b withpen pencircle scaled (radius/40) withcolor blue  ;
  currentpicture := currentpicture shifted (-2radius,0) ;

  drawarrow c withpen pencircle scaled (radius/10) withcolor white-red   ;
  drawarrow m withpen pencircle scaled (radius/20) withcolor white-green ;
  drawarrow y withpen pencircle scaled (radius/40) withcolor white-blue  ;
  currentpicture := currentpicture shifted (-2radius,0) ;

  drawarrow w withcolor black ;

  currentpicture := currentpicture xsized 3size ;

  popcurrentpicture ;
enddef ;
\stopbuffer

\startlinecorrection[blank]
\processMPbuffer[circle,demo]
\stoplinecorrection

We did not mention what the push and pop commands are
responsible for. Scaling the current picture is well defined
as long as we deal with one graphic. However, if the current
picture already has some content, this content is also
scaled. The push and pop commands let us add content to the
current picture as well as manipulating the picture as a
whole without any side effects. The final result is put on
top of the already drawn content. Instead of the sequence:

\starttyping
pushcurrentpicture ;
  ...
  currentpicture := currentpicture ... transformations ... ;
popcurrentpicture ;
\stoptyping

you can say:

\starttyping
pushcurrentpicture ;
  ...
popcurrentpicture ... transformations ... ;
\stoptyping

Both are equivalent to:

\starttyping
draw image ( ... ) ... transformations ... ;
\stoptyping

For larger sequences of commands, the push||pop
alternative gives a bit more more readable code.

\section {Fool yourself}

When doing a literature search on the human perception of
black||white edges, I ran into several articles with
graphics that I remember having seen before in books on
psychology, physiology and|/|or ergonomics. One of the
articles was by Edward H.~Adelson of MIT and we will use a
few of his example graphics in our exploration to what
extend \METAPOST\ can be of help in those disciplines. Since
such graphics normally occur in typeset documents, we will
define them in the document source.

\startbuffer[a]
\startbuffer
interim linecap := butt ; numeric u ; u := 1cm ;
pickup pencircle scaled .5u ;
for i=1u step u until 5u :
  draw (0,i) -- (5u,i) ;
endfor ;
for i=2u step u until 4u :
  draw   (u,i) -- (2u,i)                   withcolor .5white ;
  draw ((3u,i) -- (4u,i)) shifted (0,-.5u) withcolor .5white ;
endfor ;
\stopbuffer
\stopbuffer

\startbuffer[b]
\placefigure
  [here][fig:tricked 1]
  {White's illusion.}
  {\processMPbuffer}
\stopbuffer

\getbuffer[a,b]

Unless you belong to the happy few whose visual
capabilities are not distorted by neural optimizations, in
\in {figure} [fig:tricked 1] the gray rectangles at the
left look lighter than those on the right. Alas, you can
fool yourself, but \METAPOST\ does not cheat. This graphic,
referred to as White's illusion, is defined as follows.

\typebuffer[a]

Watch how we include the code directly. We have packaged this
graphic in a buffer which we include as a floating figure.

\typebuffer[b]

When passed to \METAPOST, this code is encapsulated in its
\type {beginfig} and \type {endfig} macros and thereby
grouped. But any change to a variable that is not explicitly
saved, migrates to the outer level. In order to prevent all
successive graphics to have butt'd linecaps, we have to
change this line characteristic locally. Because \type
{linecap} is defined as an internal variable, we have to use
\type {interim} to overload its value. Because \type {u} is
a rather commonly used scratch variable, we don't save its
value.

Watch how we use \type {u} as the loop step. In spite of
what your eyes tell you, this graphic only has two explicit
color directives, both being 50\% black. In the next example
we will use some real colors.

\startbuffer[a]
\startuseMPgraphic{first}
  numeric size, delta ;
  size := 2.5cm ; delta := size/3 ;
  color mainshade, topshade, bottomshade, centershade ;
  mainshade := \MPcolor{funcolor} ;
  topshade := .9mainshade ; bottomshade := .5mainshade ;
  centershade := .5[topshade,bottomshade] ;
\stopuseMPgraphic
\stopbuffer

\startbuffer[b]
\startuseMPgraphic{second}
  \includeMPgraphic{first}
  fill fullsquare scaled size  withcolor topshade ;
  fill fullsquare scaled delta withcolor centershade ;
\stopuseMPgraphic
\stopbuffer

\startbuffer[c]
\startuseMPgraphic{third}
  \includeMPgraphic{first}
  fill fullsquare scaled size  withcolor bottomshade ;
  fill fullsquare scaled delta withcolor centershade ;
\stopuseMPgraphic
\stopbuffer

\getbuffer[a,b,c]

\startbuffer[d]
\startcombination[5*2]
  {\definecolor[funcolor][red]   \useMPgraphic{second}} {}
  {\definecolor[funcolor][green] \useMPgraphic{second}} {}
  {\definecolor[funcolor][blue]  \useMPgraphic{second}} {}
  {\definecolor[funcolor][yellow]\useMPgraphic{second}} {}
  {\definecolor[funcolor][white] \useMPgraphic{second}} {}
  {\definecolor[funcolor][red]   \useMPgraphic{third}}  {}
  {\definecolor[funcolor][green] \useMPgraphic{third}}  {}
  {\definecolor[funcolor][blue]  \useMPgraphic{third}}  {}
  {\definecolor[funcolor][yellow]\useMPgraphic{third}}  {}
  {\definecolor[funcolor][white] \useMPgraphic{third}}  {}
\stopcombination
\stopbuffer

\placefigure
  [here][fig:tricked 2]
  {The simultaneous contrast effect.}
  {\getbuffer[d]}

In \in {figure} [fig:tricked 2] the small squares in the
center of each colored pair of big squares have the same
shade, but the way we perceive them is influenced by their
surroundings. Both sets of squares are defined using usable
graphics. The top squares are defined as:

\typebuffer[b]

and the bottom squares are coded as:

\typebuffer[c]

Because both graphics share code, we have defined that code
as a separate graphic, that we include. The only point of
interest in this definition is the fact that we let
\METAPOST\ interpolate between the two colors using \type
{.5[ ]}.

\typebuffer[a]

The color \type {funcolor} is provided by \CONTEXT, and
since we want to use this graphic with different colors,
this kind of mapping is quite convenient. The bunch of
graphics is packaged in a combination with empty captions.
Note how we set the color before we include the graphic.

\typebuffer[d]

We use a similar arrangement for the following graphic, where
we have replaced the definitions of \type {first}, \type
{second} and \type {third} by new definitions.

\startbuffer[a]
\startuseMPgraphic{first}
  numeric height, width, radius, gap ; gap := 1mm ;
  height = 2.5cm ; width := height/2 ; radius := height/2.5 ;
  color mainshade, leftshade, rightshade, centershade ;
  mainshade := \MPcolor{funcolor} ;
  leftshade := .9mainshade ; rightshade := .5mainshade ;
  centershade := .5[leftshade,rightshade] ;
  fill unitsquare xyscaled ( width,height) withcolor leftshade ;
  fill unitsquare xyscaled (-width,height) withcolor rightshade ;
  draw (fullcircle scaled radius) shifted (0,height/2)
    withpen pencircle scaled (radius/2) withcolor centershade ;
\stopuseMPgraphic
\stopbuffer

\startbuffer[b]
\startuseMPgraphic{second}
  \includeMPgraphic{first}
  interim linecap := butt ; pickup pencircle scaled gap ;
  draw (0,0) -- (0,height) withcolor white ;
\stopuseMPgraphic
\stopbuffer

\startbuffer[c]
\startuseMPgraphic{third}
  \includeMPgraphic{first}
  picture p, q ; p := q := currentpicture ;
  clip p to unitsquare xscaled  width yscaled height ;
  clip q to unitsquare xscaled -width yscaled height ;
  currentpicture := p ;
  addto currentpicture also q shifted (0,radius/2) ;
\stopuseMPgraphic
\stopbuffer

\getbuffer[a,b,c]

\startbuffer[d]
\startcombination[5*3]
  {\definecolor[funcolor][red]   \useMPgraphic{first}}  {}
  {\definecolor[funcolor][green] \useMPgraphic{first}}  {}
  {\definecolor[funcolor][blue]  \useMPgraphic{first}}  {}
  {\definecolor[funcolor][yellow]\useMPgraphic{first}}  {}
  {\definecolor[funcolor][white] \useMPgraphic{first}}  {}
  {\definecolor[funcolor][red]   \useMPgraphic{second}} {}
  {\definecolor[funcolor][green] \useMPgraphic{second}} {}
  {\definecolor[funcolor][blue]  \useMPgraphic{second}} {}
  {\definecolor[funcolor][yellow]\useMPgraphic{second}} {}
  {\definecolor[funcolor][white] \useMPgraphic{second}} {}
  {\definecolor[funcolor][red]   \useMPgraphic{third}}  {}
  {\definecolor[funcolor][green] \useMPgraphic{third}}  {}
  {\definecolor[funcolor][blue]  \useMPgraphic{third}}  {}
  {\definecolor[funcolor][yellow]\useMPgraphic{third}}  {}
  {\definecolor[funcolor][white] \useMPgraphic{third}}  {}
\stopcombination
\stopbuffer

\placefigure
  [here][fig:tricked 3]
  {Koffka's examples of manipulating contrast by changing
   the spatial configuration.}
  {\getbuffer[d]}

The definition of the first row of \in {figure} [fig:tricked
3] is used in the second and third and therefore is the most
complicated. We use quite some scratch variables to reach a
high level of abstraction. The \type {xyscaled} operator is
a \METAFUN\ macro.

\typebuffer[a]

The graphics of the second row extend those of the first by
drawing a white line through the middle. In this example
setting the linecap is not really needed, because rounded
top and bottoms in white are invisible and the part that
extends beyond the points does not count in calculating the
bounding box.

\typebuffer[b]

The third row graphics again extend the first graphic. First
we copy the picture constructed so far. Watch the double
assignment. Next we clip the pictures in half, and shift the
right half down over the width of the circle.

\typebuffer[c]

% \section {Puzzles}
%
% {\em Maybe.}
%
% \section {Flow charts}
%
% {\em Instead of starting anew every time, you can use
% predefined macros, like those in the flow chart module. Let's
% see how we can influence the \METAPOST\ code. Maybe not here.}
%
% \section {Chemistry}
%
% {\em \METAPOST\ can do it's work unseen, as in the chemistry
% module that comes with \CONTEXT. There, \METAPOST\ is used
% as one of the graphical plug||ins. It demonstrates that we
% can put \METAPOST\ to work without seeing any code.}

\section {Growing graphics}

Although \METAPOST\ is not really suited as a simulation
engine, it is possible to build graphics that are built and
displayed incrementally with a sequence of mouse clicks. The
following example is the result of an email discussion David
Arnold and the author had while \METAFUN\ evolved.

Instead of defining the graphics in a separate \METAPOST\
file, we will incorporate them in the document source in
which they are used. We can use several methods.

\startitemize[n]
\item  Define macros and figures in a separate file and
       include the graphics as external graphics.
\item  Define everything in the document source as usable
       graphics and include the graphics using \type
       {\useMPgraphic}.
\item  Package the graphic components in buffers and paste
       those together as graphics that can be processed at
       run time.
\stopitemize

The first method is the most independent one, which has its
advantages if we want to use the graphics in other
applications too. The second method works well in graphics
where parts of the definitions change between invocations of
the graphic. This method follows the template:

\starttyping
\startuseMPgraphic{whatever}
 ...
\stopuseMPgraphic

\startuseMPgraphic{result}
  ...
  \includeMPgraphic{whatever}
  ...
\stopuseMPgraphic

\useMPgraphic{result}
\stoptyping

The disadvantage of this method is that it cannot be
combined with \type {btex}||\type {etex} since it is nearly
impossible to determine when, how, and to what extent the
content of a graphic should be expanded before writing it
to the temporary \METAPOST\ file.

Therefore, we will demonstrate how buffers can be used. This
third method closely parallels the first way of defining
graphics. A nice side effect is that we can easily typeset
these buffers verbatim, which we did to typeset this
document.

We are going to do a classic compass and straightedge
construction, the bisection of a line segment joining two
arbitrary points. We will construct five graphics, where each
one displays one step of the construction. We will embed
each graphic in a start||stop command. Later we will see the
advantage of this strategy.

\startbuffer
\startbuffer[a]
def start_everything = enddef ;
def stop_everything  = enddef ;
\stopbuffer
\stopbuffer

\typebuffer \getbuffer

\startbuffer
\startbuffer[b]
numeric u, w ; u := .5cm ; w := 1pt ;

pickup pencircle scaled w ;

def draw_dot expr p =
  draw p withpen pencircle scaled 3w ;
enddef ;

def stand_out =
  drawoptions(withcolor .625red) ;
enddef ;
\stopbuffer
\stopbuffer

We are going to draw a few dots, and to force consistency we
first define a macro \type {draw_dot}. The current step will
be highlighted in red using \type {stand_out}.

\typebuffer \getbuffer

\startbuffer
\startbuffer[c]
def draw_basics =
  pair pointA, pointB ; path lineAB ;
  pointA := origin ; pointB := pointA shifted (5u,0) ;
  lineAB := pointA -- pointB ;
  draw lineAB ;
  draw_dot pointA ; label.lft(btex A etex, pointA) ;
  draw_dot pointB ; label.rt (btex B etex, pointB) ;
enddef ;
\stopbuffer
\stopbuffer

First, we construct the macro that will plot two points $A$
and $B$ and connect them with a line segment.

\typebuffer \getbuffer

\startbuffer
\startbuffer[1]
start_everything ;
  stand_out ; draw_basics ;
stop_everything ;
\stopbuffer
\stopbuffer

The code in this buffer executes the preceding macros. The
\type {..._everything} commands are still undefined, but
later we can use these hooks for special purposes.

\typebuffer \getbuffer

This graphic can now be embedded by the \CONTEXT\ command
\type {\processMPbuffer}. This command, like the ordinary
buffer inclusion commands, accepts a list of buffers.

\startbuffer
\startlinecorrection[blank]
\ruledhbox{\processMPbuffer[a,b,c,1]}
\stoplinecorrection
\stopbuffer

\typebuffer

We use \type {\ruledhbox} to show the tight bounding box of
the graphic. The line correction takes care of proper
spacing around non textual content, like graphics. \footnote
{These spacing commands try to get the spacing around the
content visually compatible, and take the height and depth
of the preceding and following text into account.} This is
only needed when the graphic is part of the text flow!

\getbuffer

Next, we draw two circles of equal radius, one centered
at point $A$, the other at point $B$.

\startbuffer
\startbuffer[d]
def draw_circles =
  path circleA, circleB ; numeric radius, distance ;
  distance := (xpart pointB) - (xpart pointA) ;
  radius := 2/3 * distance ;
  circleA := fullcircle scaled (2*radius) ;
  circleB := circleA shifted pointB ;
  draw circleA ;
  draw circleB ;
enddef ;
\stopbuffer
\stopbuffer

\typebuffer \getbuffer

\startbuffer
\startbuffer[2]
start_everything ;
  draw_basics ; stand_out ; draw_circles ;
stop_everything ;
\stopbuffer
\stopbuffer

As you can see, we move down the \type {stand_out} macro so
that only the additions are colored red.

\typebuffer \getbuffer

We now use \type{\processMPbuffer[a,b,c,d,2]} to include the
latest step.

\startlinecorrection[blank]
\ruledhbox{\processMPbuffer[a,b,c,d,2]}
\stoplinecorrection

The next step in the construction of the perpendicular
bisector requires that we find and label the points of
intersection of the two circles centered at points $A$ and
$B$. The intersection points are calculated as follows.
Watch the \type {reverse} operation, which makes sure that
we get the second intersection point.

\startbuffer
\startbuffer[e]
def draw_intersection =
  pair pointC, pointD ;
  pointC := circleA intersectionpoint circleB ;
  pointD := (reverse circleA) intersectionpoint (reverse circleB) ;
  draw_dot pointC ; label.lft(btex C etex, pointC shifted (-2w,0)) ;
  draw_dot pointD ; label.lft(btex D etex, pointD shifted (-2w,0)) ;
enddef ;
\stopbuffer
\stopbuffer

\typebuffer \getbuffer

\startbuffer
\startbuffer[3]
start_everything ;
  draw_basics ; draw_circles ; stand_out ; draw_intersection ;
stop_everything ;
\stopbuffer
\stopbuffer

In placing the label, we must make sure that the text runs
free of the lines and curves. Again, move the \type
{stand_out} macro just prior to \type {draw_intersection}
macro, so that this step is highlighted in the drawing
color, while prior steps are drawn in the default color (in
this case black).

\typebuffer \getbuffer

\startlinecorrection[blank]
\ruledhbox{\processMPbuffer[a,b,c,d,e,3]}
\stoplinecorrection

The line drawn through points $C$ and $D$ will be the
perpendicular bisector of the line segment connecting points
$A$ and $B$. In the next step we will draw a line using the
plain \METAPOST\ \type {drawdblarrow} macro that draws
arrowheads at each end of a path.

\startbuffer
\startbuffer[f]
def draw_bisector =
  path lineCD ;
  lineCD := origin -- origin shifted (2*distance,0) ;
  lineCD := lineCD rotated 90 shifted 0.5[pointA,pointB] ;
  lineCD := lineCD shifted (0,-distance) ;
  drawdblarrow lineCD ;
enddef ;
\stopbuffer

\startbuffer[4]
start_everything ;
  draw_basics ; draw_circles ; draw_intersection ; stand_out ;
  draw_bisector ;
stop_everything ;
\stopbuffer
\stopbuffer

\typebuffer \getbuffer

\startlinecorrection[blank]
\ruledhbox{\processMPbuffer[a,b,c,d,e,f,4]}
\stoplinecorrection

The following code draws the intersection of line $C-D$ and
line segment $A-B$, which can be shown to be the midpoint of
segment $A-B$.

\startbuffer
\startbuffer[g]
def draw_midpoint =
  pair pointM ;
  pointM := lineCD intersectionpoint lineAB ;
  draw_dot pointM ; label.llft(btex M etex, pointM) ;
enddef ;
\stopbuffer

\startbuffer[5]
start_everything ;
  draw_basics ; draw_circles ; draw_intersection ; draw_bisector ;
  stand_out ; draw_midpoint ;
stop_everything ;
\stopbuffer
\stopbuffer

\typebuffer \getbuffer

\startlinecorrection[blank]
\ruledhbox{\processMPbuffer[a,b,c,d,e,f,g,5]}
\stoplinecorrection

As long as we place the graphics as individual insertions in
our document, everything is fine. However, if we wish to
place them all at once, or as we shall see later, place them
on top of one another in a fieldstack, it makes sense to
give them all the same bounding box. We can do this by
completing the \type {start_everything} and \type
{stop_everything} commands.

\startbuffer
\startbuffer[a]
def start_everything =
  path bb ;
  draw_basics ;
  draw_circles ;
  draw_intersection ;
  draw_bisector ;
  draw_midpoint ;
  bb := boundingbox currentpicture ;
  currentpicture := nullpicture ;
enddef ;

def stop_everything =
  setbounds currentpicture to bb ;
enddef ;
\stopbuffer
\stopbuffer

\typebuffer \getbuffer

In \in {figure} [fig:1 till 5] we demonstrate the effect of
this redefinition. For this purpose we scale down the
graphic to a comformatable 40\%, of course by using an
additional buffer. We also visualize the bounding box.

\startbuffer
\startbuffer[h]
def stop_everything =
  setbounds currentpicture to bb ;
  draw bb withpen pencircle scaled .5pt withcolor .625yellow ;
  currentpicture := currentpicture scaled .4 ;
enddef ;
\stopbuffer
\stopbuffer

\typebuffer \getbuffer

The graphic itself is defined as follows. Watch how we use
the default buffer to keep the definitions readable.

\startbuffer
\startbuffer
\startcombination[5*1]
  {\processMPbuffer[a,b,c,h,d,e,f,g,1]} {step 1}
  {\processMPbuffer[a,b,c,h,d,e,f,g,2]} {step 2}
  {\processMPbuffer[a,b,c,h,d,e,f,g,3]} {step 3}
  {\processMPbuffer[a,b,c,h,d,e,f,g,4]} {step 4}
  {\processMPbuffer[a,b,c,h,d,e,f,g,5]} {step 5}
\stopcombination
\stopbuffer

\placefigure
  [here][fig:1 till 5]
  {The five graphics, each with the same bounding box.}
  {\getbuffer}
\stopbuffer

\typebuffer \getbuffer

As the original purpose of these graphics was not to show them
side by side, but to present them as field stack in a
document to be viewed at the computer screen. For this
purpose we have to define the graphics as symbols.

\startbuffer
\definesymbol[step 1][{\processMPbuffer[a,b,c,d,e,f,g,1]}]
\definesymbol[step 2][{\processMPbuffer[a,b,c,d,e,f,g,2]}]
\definesymbol[step 3][{\processMPbuffer[a,b,c,d,e,f,g,3]}]
\definesymbol[step 4][{\processMPbuffer[a,b,c,d,e,f,g,4]}]
\definesymbol[step 5][{\processMPbuffer[a,b,c,d,e,f,g,5]}]
\stopbuffer

\typebuffer \getbuffer

A field stack is a sequence of overlayed graphics. We will
arrange these to cycle manually, with clicks of the mouse,
through the sequence of graphs depicting the construction of
the midpoint of segment $A-B$. So, in fact we are dealing
with a manual simulation. The definition of such a stack is
as follows:

\startbuffer
\definefieldstack
  [midpoint construction]
  [step 1, step 2, step 3, step 4, step 5]
  [frame=on,offset=3pt,framecolor=darkyellow,rulethickness=1pt]
\stopbuffer

\typebuffer \getbuffer

The first argument is to be a unique identifier, the second
argument takes a list of symbols, while the third argument
accepts settings. More on this command can be found in the
\CONTEXT\ manuals.

The stack is shown as \in {figure} [fig:steps]. Its caption
provides a button, which enables the reader to cycle through
the stack. We call this a stack because the graphics are
positioned on top of each other. Only one of them is visible
at any time.

\startbuffer
\placefigure
  [here][fig:steps]
  {Bisecting a line segment with compass and straightedge? Just
   click \goto {here} [JS(Walk_Field{midpoint construction})] to
   walk through the construction! (This stack is only visible
   in a \PDF\ viewer that supports widgets.)}
  {\fieldstack[midpoint construction]}
\stopbuffer

\typebuffer

{\setupinteraction[color=darkred,contrastcolor=darkred]\getbuffer}

At the start of this section, we mentioned three methods.
When we use the first method of putting all the graphics in
an external \METAPOST\ file, the following framework suits.
We assume that the file is called \type {step.mp} and that
it is kept by the user along with his document source. We
start with the definitions of the graphic steps. These are
the same as the ones shown previously.

\starttyping
def draw_basics       = ... enddef ;
def draw_circles      = ... enddef ;
def draw_intersection = ... enddef ;
def draw_bisector     = ... enddef ;
def draw_midpoint     = ... enddef ;
def stand_out         = ... enddef ;
\stoptyping

We can safe some code by letting the \type {..._everything}
take care of the \type {beginfig} and \type {endfig} macros.

\starttyping
def start_everything (expr n) = beginfig(n) ; ... enddef ;
def stop_everything           =    ... ; endfig ; enddef ;
\stoptyping

The five graphics now become:

\starttyping
start_everything (1) ;
  stand_out ; draw_basics ;
stop_everything ;

start_everything (2) ;
  draw_basics ; stand_out ; draw_circles ;
stop_everything ;

start_everything (3) ;
  draw_basics ; draw_circles ; stand_out ; draw_intersection ;
stop_everything ;

start_everything (4) ;
  draw_basics ; draw_circles ; draw_intersection ; stand_out ;
  draw_bisector ;
stop_everything ;

start_everything (5) ;
  draw_basics ; draw_circles ; draw_intersection ; draw_bisector ;
  stand_out ; draw_midpoint ;
stop_everything ;
\stoptyping

The definitions of the symbols now refer to an external
figure.

\starttyping
\definesymbol[step 1][{\externalfigure[step.1]}]
\definesymbol[step 2][{\externalfigure[step.2]}]
\definesymbol[step 3][{\externalfigure[step.3]}]
\definesymbol[step 4][{\externalfigure[step.4]}]
\definesymbol[step 5][{\externalfigure[step.5]}]
\stoptyping

Which method is used, depends on the way the graphics are
used. In this example we wanted to change the definition of
\type {..._everything}, so here the third method was quite
useful.

\section {Simple Logos}

\startbuffer[ns]
numeric width, height, line, delta ;
width = 5cm ; height = width/2 ; line = height/4 ; delta = line ;

linejoin := mitered ; pickup pencircle scaled line ;

color nsblue   ; nsblue   := (0,0,1) ;
color nsyellow ; nsyellow := (1,1,0) ;

z1 = (0, height/2) ;
z2 = (width/2-height/4, y1) ;
z3 = (width/2+height/4, y4) ;
z4 = (width, 0) ;

z5 = (x4+height/2, y1) ;
z6 = (x4, 2y1) ;
z7 = 1.5[z5,z6] ;

path p ; p := z1--z2--z3--z4 ; path q ; q := z3--z4--z5--z7 ;

numeric d, lx, ly, ux, uy ; d = line/2 ;

lx = -3d - d/3  ; ly = -d ; ux = rt x5 + d/3 ; uy = top y6 ;

path r ; r := (lx,ly)--(ux,ly)--(ux,uy)--(lx,uy)--cycle;

lx := lx-delta ; ly := ly-delta ; ux := ux+delta ; uy := uy+delta ;

path s ; s := (lx,ly)--(ux,ly)--(ux,uy)--(lx,uy)--cycle;

draw p withcolor nsblue ; draw q withcolor nsblue ;

addto currentpicture also currentpicture
  rotatedaround (.5[z2,z3],180) shifted (height/4,height/2) ;

picture savedpicture ; savedpicture := currentpicture ;

clip currentpicture to r ;
setbounds currentpicture to r ;

savedpicture := currentpicture ; currentpicture := nullpicture ;

fill s withcolor nsyellow ;
addto currentpicture also savedpicture ;
\stopbuffer

Many company logos earn their beauty from their simplicity.
One of the logos that most Dutch people have imprinted in
their mind is that of the Dutch Railway Company (NS). An
interesting feature of this logo is that, although it is
widely known, drawing it on a piece of paper from mind is a
task that many people fail.

\startlinecorrection[blank]
\processMPbuffer[ns]
\stoplinecorrection

This logo makes a good candidate for demonstrating
a few fine points of drawing graphics, like using
linear equations, setting line drawing characteristics,
clipping and manipulating bounding boxes.

The implementation below is quite certainly not according to
the official specifications, but it can nevertheless serve
as an example of defining such logos.

\startbuffer[a]
numeric width  ; width  = 3cm ;
numeric height ; height = width/2 ;
numeric line   ; line   = height/4 ;
\stopbuffer

As always, we need to determine the dimensions first. Here,
both the height and line width depend on the width of the
graphic.

Instead of calculating the blue shape such that it will be a
filled outline, we will draw the logo shape using line
segments. This is why we need the \type {line} parameter.

\typebuffer[a]

We want sharp corners which can be achieved by setting \type
{linejoin} to \type {mitered}.

\startbuffer[b]
linejoin := mitered ; pickup pencircle scaled line ;
\stopbuffer

\typebuffer[b]

The colors are rather primary blue and yellow. At the time
of writing this manual, Dutch trains are still painted
yellow, so we will use that shade as background color.

\startbuffer[c]
color nsblue   ; nsblue   := (0,0,1) ;
color nsyellow ; nsyellow := (1,1,0) ;
\stopbuffer

\typebuffer[c]

We will now describe the main curves. Although these
expressions are not that advanced, they demonstrate that we
can express relationships instead of using assignments.

\startbuffer[d]
z1 = (0, height/2) ;
z2 = (width/2-height/4, y1) ;
z3 = (width/2+height/4, y4) ;
z4 = (width, 0) ;

path p ; p := z1--z2--z3--z4 ;
\stopbuffer

\typebuffer[d]

Although it is accepted to consider \type {z} to be a
variable, it is in fact a \type {vardef} macro, that expands
into a pair \type {(x,y)}. This means that the previous
definitions internally become:

\starttyping
(x1,y1) = (0, height/2) ;
(x2,y2) = (width/2-height/4, y1) ;
(x3,y3) = (width/2+height/4, y4) ;
(x4,y4) = (width, 0) ;
\stoptyping

These 8 relations can be solved by \METAPOST, since all
dependencies are known.

\starttyping
x1 = 0                ; y1 = height/2 ;
x2 = width/2-height/4 ; y2 = y1       ;
x3 = width/2+height/4 ; y3 = y4       ;
x4 = width            ; y4 = 0        ;
\stoptyping

Since we express the variables \type {x} and \type {y} in
terms of relations, we cannot reuse them, since that would
mean that inconsistent relations occur. So, the following
lines will lead to an error message:

\starttyping
z1 = (10,20) ; z1 = (30,50) ;
\stoptyping

For similar reasons, we may not assign a value (using \type
{:=}) to such a \type {z} variable. Within a \METAPOST\
figure, \type {z} variables are automatically saved, which
means that they can be reused for each figure.

\startbuffer[x]
drawpath p ; drawpoints p ; drawpointlabels p ;
\stopbuffer

So far, we have defined the following segment of the logo.

\startlinecorrection[blank]
\processMPbuffer[a,b,c,d,x]
\stoplinecorrection

\startbuffer[e]
z5 = (x4+height/2, y1) ;
z6 = (x4, 2y1) ;
z7 = 1.5[z5,z6] ;

path q ; q := z3--z4--z5--z7 ;
\stopbuffer

The next expressions are used to define the second segment.
The third expression determines \type {z7} to be positioned
on the line \type {z5--z6}, where we extend this line by
50\%.

\typebuffer[e]

\startbuffer[x]
drawpath q ; drawpoints q ; drawpointlabels q ;
\stopbuffer

\startlinecorrection[blank]
\processMPbuffer[a,b,c,d,e,x]
\stoplinecorrection

If we combine these two segments, we get:

\startbuffer[x]
drawpath p ; drawpoints p ; drawpointlabels p ;
swappointlabels := true ;
drawpath q ; drawpoints q ; drawpointlabels q ;
\stopbuffer

\startlinecorrection[blank]
\processMPbuffer[a,b,c,d,e,x]
\stoplinecorrection

However, when we draw them using the right linewidth and
color, you will notice that we're not yet done:

\startbuffer[f]
draw p withcolor nsblue ; draw q withcolor nsblue ;
\stopbuffer

\startlinecorrection[blank]
\processMPbuffer[a,b,c,d,e,f]
\stoplinecorrection

The second curve is similar to the first one, but rotated
over 180 degrees.

\startbuffer[g]
addto currentpicture also currentpicture
  rotatedaround (.5[z2,z3],180) shifted (height/4,height/2) ;
\stopbuffer

\typebuffer[g]

\startlinecorrection[blank]
\processMPbuffer[a,b,c,d,e,f,g]
\stoplinecorrection

In order to get the sharp edges, we need to clip off part of
the curves and at first sight, we may consider using a
scaled bounding box. However, when we show the natural
bounding box, you will notice that a more complicated bit of
calculations is needed.

\startbuffer[x]
draw boundingbox currentpicture
  withpen pencircle scaled .5mm withcolor .625white ;
\stopbuffer

\startlinecorrection[blank]
\processMPbuffer[a,b,c,d,e,f,g,x]
\stoplinecorrection

The right clip path is calculated using the following
expressions. Watch how we use \type {rt} and \type {top} to
correct for the linewidth.

\startbuffer[h]
numeric d, lx, ly, ux, uy ; d = line/2 ;

lx = -3d - d/3  ; ly = -d ; ux = rt x5 + d/3 ; uy = top y6 ;

path r ; r := (lx,ly)--(ux,ly)--(ux,uy)--(lx,uy)--cycle;
\stopbuffer

\typebuffer[h]

The clipping path is applied by saying:

\startbuffer[i]
clip currentpicture to r ;
\stopbuffer

\typebuffer[i]

The result is quite acceptable:

\startlinecorrection[blank]
\processMPbuffer[a,b,c,d,e,f,g,h,i]
\stoplinecorrection

But, if you watch closely to how this graphic extends into
to left margin of this document, you will see that the
bounding box is not yet right.

\startlinecorrection[blank]
\processMPbuffer[a,b,c,d,e,f,g,h,i,x]
\stoplinecorrection

\startbuffer[j]
setbounds currentpicture to r ;
\stopbuffer

\typebuffer[j]

We use the same path \type {r} to correct the bounding box.

\startlinecorrection[blank]
\processMPbuffer[a,b,c,d,e,f,g,h,i,j,x]
\stoplinecorrection

There are a few subtle points involved, like setting the
\type {linejoin} variable. If we had not set it to \type
{mitered}, we would have got round corners. We don't set the
\type {linecap}, since a flat cap would not extend far
enough into the touching curve and would have left a small
hole. The next example shows what happens if we set these
variables to the wrong values:

\startbuffer[bb]
linejoin := rounded ; linecap := mitered ;
\stopbuffer

\startlinecorrection[blank]
\processMPbuffer[a,b,bb,c,d,e,f,g,h,i,j]
\stoplinecorrection

In fact we misuse the fact that both curves overlay each
other.

\startbuffer[f]
draw p withcolor nsblue ; draw q withcolor .625white ;
\stopbuffer

\startlinecorrection[blank]
\processMPbuffer[a,b,c,d,e,f,g,h,i,j]
\stoplinecorrection

The complete logo definition is a bit more extensive since
we also want to add a background. Because we need to clip
the blue foreground graphic, we must temporarily store it
when we fill the background.

\typebuffer[ns]

For practical use it makes sense to package this definition
in a macro to which we pass the dimensions.

\section {Music sheets}

The next example demonstrates quite some features. Imagine
that we want to make us a couple of sheets so that we can
write a musical masterpiece. Let's also forget that \TEX\
can draw lines, which means that somehow we need to use
\METAPOST.

Drawing a bar is not that complicated as the following code
demonstrates.

\startbuffer
\startusableMPgraphic{bar}
  vardef MusicBar (expr width, gap, linewidth, barwidth) =
    image
      ( interim linecap := butt ;
        for i=1 upto 5 :
          draw ((0,0)--(width,0)) shifted (0,(i-1)*gap)
            withpen pencircle scaled linewidth ;
        endfor ;
        for i=llcorner currentpicture -- ulcorner currentpicture ,
              lrcorner currentpicture -- urcorner currentpicture :
          draw i withpen pencircle scaled barwidth ;
        endfor ; )
  enddef ;
\stopusableMPgraphic
\stopbuffer

\typebuffer \getbuffer

We can define the sidebars a bit more efficient using two
predefined subpaths:

\starttyping
for i=leftboundary currentpicture, rightboundary currentpicture :
\stoptyping

We define a macro \type {MusicBar} that takes four
arguments. The first two determine the dimensions, the last
two concern the line widths. Now watch how we can use this
macro:

\startbuffer
\includeMPgraphic{bar} ;
draw MusicBar (200pt, 6pt, 1pt, 2pt) ;
draw MusicBar (300pt, 6pt, 1pt, 2pt) shifted (0,-30pt) ;
\stopbuffer

\typebuffer

\startlinecorrection[blank]
\processMPbuffer
\stoplinecorrection

As you can see in this example, the bar is a picture that
can be transformed (shifted in our case). However, a close
look at the macro teaches us that it does a couple of draws
too. This is possible because we wrap the whole in an image
using the \type {image} macro. This macro temporary saves
the current picture, and at the end puts the old
\type {currentpicture} under the new one.

We wrap the whole in a \type {vardef}. This means that the
image is returned as if it was a variable. Actually, the
last thing in a \type {vardef} should be a proper return
value, in our case a picture. This also means that we may
not end the \type {vardef} with a semi colon. So, when the
content of the \type {vardef} is expanded, we get something

\starttyping
draw some_picture ... ;
\stoptyping

Because we are still drawing something, we can add transform
directives and set attributes, like the color.

The second \type {for} loop demonstrates two nice features. Instead
of repeating the draw operation by copying code, we apply it
to a list, in our case a list of paths. This list contains
two simple line paths. Because an \type {image} starts
with a fresh \type {currentpicture}, we can safely use the
bounding box data to determine the height of the line.

The next step in producing the sheets of paper is to put
several bars on a page, preferable with the width of the
current text. This time we will use a reusable graphic,
because each bar is the same.

\startbuffer
\startreusableMPgraphic{bars}
  \includeMPgraphic{bar} ;
  draw MusicBar (TextWidth, 6pt, 1pt, 2pt) withcolor .625yellow ;
\stopreusableMPgraphic
\stopbuffer

\typebuffer \getbuffer

\startlinecorrection[blank]
\reuseMPgraphic{bars}
\stoplinecorrection

Instead of going trough the trouble of letting \METAPOST\
calculate the positions of the bars, we will use \TEX. We
put 12 bars on a page and let \TEX\ take care of the
inter||bar spacing. Because we only want stretchable space
between bars, called glue in \TEX, we need to remove the
last added glue.

\startnotmode[screen]

\startbuffer[music]
\startstandardmakeup[doublesided=no,page=]
  \dorecurse{15}{\reuseMPgraphic{bars}\vfill}\removelastskip
\stopstandardmakeup
\stopbuffer

\stopnotmode

\startmode[screen]

\startbuffer[music]
\startstandardmakeup[doublesided=no,page=]
  \dorecurse{10}{\reuseMPgraphic{bars}\vfill}\removelastskip
\stopstandardmakeup
\stopbuffer

\stopmode

\typebuffer[music]

\startusableMPgraphic{bar}
  vardef MusicBar (expr width, gap, linewidth, barwidth) =
    image
      ( interim linecap := butt ;
        for i=1 upto 5 :
          draw ((0,0)--(width,0)) randomized (1pt,1.5pt) shifted (0,(i-1)*gap)
            withpen pencircle scaled linewidth ;
        endfor ;
        for i=llcorner currentpicture -- ulcorner currentpicture ,
              lrcorner currentpicture -- urcorner currentpicture :
          draw i randomized 2pt withpen pencircle scaled barwidth ;
        endfor ; )
  enddef ;
\stopusableMPgraphic

\startreusableMPgraphic{bars} % trigger a new one
  \includeMPgraphic{bar} ;
  draw MusicBar (TextWidth, 6pt, 1pt, 2pt) withcolor .625yellow ;
\stopreusableMPgraphic

It may add to the atmosphere of handy||work if you slightly
randomize the lines. We leave it up to the reader to figure
out how the code should be changed to accomplish this.

\startlinecorrection[blank]
\reuseMPgraphic{bars}
\stoplinecorrection

The complete result is shown on the next page.

\startpostponing
\getbuffer[music]
\stoppostponing

\section {The euro symbol}

When Patrick Gundlach posted a nice \METAPOST\ version of
the euro symbol to the \CONTEXT\ discussion list, he added
the comment \quotation {The official construction is
ambiguous: how thick are the horizontal bars? How much do
they stick out to the left? Is this thing a circle or what?
Are the angles on the left side of the bars the same as the
one on the right side? \unknown} The alternative below is
probably not as official as his, but permits a finetuning.
You are warned: whatever you try, the euro {\em is} and
{\em will remain} an ugly symbol.

We use a couple of global variables to control the euro
shape within reasonable bounds. Next we define two circles.
Next we define a vertical line that we use in a couple of
cut and paste operations. Watch how the top left point of
the outer circle determines the slant of the line that we
use to slice the vertical bars.

\startbuffer[euro]
boolean trace_euro ; trace_euro := false ;

vardef euro_symbol = image ( % begin_of_euro

if unknown euro_radius   : euro_radius   := 2cm             ; fi ;
if unknown euro_width    : euro_width    := 3euro_radius/16 ; fi ;
if unknown euro_r_offset : euro_r_offset := euro_width      ; fi ;
if unknown euro_l_offset : euro_l_offset := euro_radius/32  ; fi ;
if unknown euro_l_shift  : euro_l_shift  := euro_r_offset   ; fi ;
if unknown euro_v_delta  : euro_v_delta  := euro_width/4    ; fi ;

save
  outer_circle, inner_circle, hor_bar,
  right_line, right_slant, top_slant,  bot_slant,
  euro_circle, euro_topbar, euro_botbar ;

path
  outer_circle, inner_circle, hor_bar,
  right_line, right_slant, top_slant, bot_slant,
  euro_circle, euro_topbar, euro_botbar ;

outer_circle := fullcircle scaled euro_radius ;
inner_circle := fullcircle scaled (euro_radius-euro_width) ;

if trace_euro : for i = outer_circle, inner_circle :
  draw i withpen pencircle scaled 1pt withcolor .5white ;
endfor ; fi ;

right_line :=
  (lrcorner outer_circle -- urcorner outer_circle)
  shifted (-euro_r_offset,0) ;

outer_circle := outer_circle cutbefore right_line ;

right_slant :=
  point 0 of outer_circle
  -- origin shifted (0,ypart lrcorner outer_circle) ;

euro_circle := buildcycle(outer_circle, right_line,
  reverse inner_circle, reverse right_slant) ;

hor_bar := (-euro_radius,0) -- (euro_radius,0) ;

top_slant :=
  right_slant shifted (-euro_radius+euro_r_offset-euro_l_offset,0) ;

bot_slant :=
  top_slant shifted (0,-euro_l_shift) ;

if trace_euro : for i = right_line, right_slant, top_slant, bot_slant :
  draw i withpen pencircle scaled 1pt withcolor .5white ;
endfor ; fi ;

euro_topbar := buildcycle
  (top_slant, hor_bar shifted (0, euro_v_delta),
   right_slant, hor_bar shifted (0, euro_v_delta+euro_width/2)) ;

euro_botbar := buildcycle
  (bot_slant, hor_bar shifted (0,-euro_v_delta),
   right_slant, hor_bar shifted (0,-euro_v_delta-euro_width/2)) ;

for i = euro_circle, euro_topbar, euro_botbar :
  draw i withpen pencircle scaled 0 ;
endfor ;
for i = euro_circle, euro_topbar, euro_botbar :
  fill i withpen pencircle scaled 0 ;
endfor ;

if trace_euro :
  drawpoints euro_circle withcolor red   ;
  drawpoints euro_topbar withcolor green ;
  drawpoints euro_botbar withcolor blue  ;
fi ;

) enddef ; % end_of_euro
\stopbuffer

\typebuffer[euro]

We only set a parameter when it is not yet set. This has
the advantage that we don't have to set them when we change
one. This way of manipulating paths (cutting and building)
does not always work well because of rounding errors, but
here it does work.

\startbuffer[demo]
euro_radius := 4cm ; trace_euro := true ; draw euro_symbol ;
\stopbuffer

\typebuffer[demo]

For educational purposes, we have added as a bit of
tracing. When enables, the euro shows up as:

\startlinecorrection[blank]
\processMPbuffer[euro,demo]
\stoplinecorrection

Of course it would be best to define the euro as one shape,
but we won't go though that process right now. By packaging
the combined paths in an image, we can conveniently color
the euro symbol:

\startbuffer[demo]
draw euro_symbol withcolor .625red ;
\stopbuffer

\typebuffer[demo]

\startlinecorrection[blank]
\processMPbuffer[euro,demo]
\stoplinecorrection

You may wonder why we both draw and fill the euro, using a
pen with zero width. We've done this in order to demonstrate
the \type {redraw} and \type {refill} macros.

\startbuffer[extra]
redraw currentpicture withpen pencircle scaled 4pt withcolor .625yellow ;
refill currentpicture withcolor .625white ;
setbounds currentpicture to boundingbox currentpicture enlarged 2pt ;
\stopbuffer

\typebuffer[extra]

\startlinecorrection[blank]
\processMPbuffer[euro,demo,extra]
\stoplinecorrection

\section {Killing time}

Not seldom \TEX\ users want to use this program and its
meta||relatives as general purpose tools, even at the cost
of quite some effort or suboptimal results. Imagine that
you are under way from our planet to Mars. After a long
period of sleep you wake up and start wondering on what
track you are. You even start questioning the experts that
send you on your way, so you pop open your laptop, launch
your editor and start metaposting.

First you need to determine the begin and end points of your
journey. For this it is enough to know the relative angle of
the paths that both planets follow as well as the path
themselves. We assume circular paths.

\startbuffer
path a ; a := fullcircle scaled 3cm ;
path b ; b := fullcircle scaled 2cm rotated 120 ;

draw a withpen pencircle scaled 1mm withcolor .625red ;
draw b withpen pencircle scaled 1mm withcolor .625yellow ;

draw point 0 of a withpen pencircle scaled 2mm ;
draw point 0 of b withpen pencircle scaled 2mm ;
\stopbuffer

\typebuffer

The rotation 120 can be calculated from the relative
starting points and time the journey will take.
Alternatively we can use the time along the path, but this
would be a bit more fuzzy later on. \footnote {In case you
wonder why \METAPOST\ talks about the time on a path, you
now have a cue.}

\startlinecorrection[blank]
\processMPbuffer
\stoplinecorrection

After a bit of playing with drawing paths between the two
points, you decide to make a macro. We want to feed the
angle between the paths but also the connecting path. So,
we have to pass a path, but unfortunately we don't have
direct access to the points. By splitting the argument
definition we can pass an expression first, and a wildcard
argument next.

\startbuffer
\startuseMPgraphic{gamble}
def Gamble (expr rot) (text track) =
  path a ; a := fullcircle scaled 3cm ;
  path b ; b := fullcircle scaled 2cm rotated rot ;

  pair aa ; aa := point 0 of a ;
  pair bb ; bb := point 0 of b ;
  path ab ; ab := track ;

  draw a  withpen pencircle scaled 1mm withcolor .625red ;
  draw b  withpen pencircle scaled 1mm withcolor .625yellow ;

  draw aa withpen pencircle scaled 2mm ;
  draw bb withpen pencircle scaled 2mm ;

  drawarrow ab withpen pencircle scaled 1mm withcolor .625white ;

  setbounds currentpicture to boundingbox a enlarged 2mm ;
  draw boundingbox currentpicture withpen pencircle scaled .25mm ;
enddef ;
\stopuseMPgraphic
\stopbuffer

\typebuffer \getbuffer

Because at this distance nobody will bother us with the
thickness of the pen and colors, we code them the hard way.
We create our own universe by setting a fixed boundingbox.

We leave the Earth in the most popular way, straight
upwards and after a few cycles, we leave in parallel to the
surface. The path drawn reminds much of the trajectories
shown in popular magazines.

\startbuffer
\startMPcode
\includeMPgraphic{gamble} ;
Gamble(120, aa {(0,1)} .. bb) ;
\stopMPcode
\stopbuffer

\typebuffer

\startlinecorrection[blank] \getbuffer \stoplinecorrection

According to \METAPOST, when we leave the Earth straight
upwards and want a smooth trajectory, we have to pass
through outer space.

\startbuffer
\startMPcode
\includeMPgraphic{gamble} ;
Gamble(120,aa {(1,0)} .. bb) ;
\stopMPcode
\stopbuffer

\typebuffer

\startlinecorrection[blank] \getbuffer \stoplinecorrection

Given that we want a smooth path as well as a short journey,
we can best follow Mars' path. Here we face the risk that
when we travel slower than Mars does, we have a problem.

\startbuffer
\startMPcode
\includeMPgraphic{gamble} ;
Gamble(120,aa {dir 90} .. {precontrol 0 of b rotated 90} bb) ;
\stopMPcode
\stopbuffer

\typebuffer

\startlinecorrection[blank] \getbuffer \stoplinecorrection

We can even travel a shorter path when we leave Earth at the
surface that faces the point of arrival.

\startbuffer
\startMPcode
\includeMPgraphic{gamble} ;
Gamble(120,aa .. {precontrol 0 of b rotated 90} bb) ;
\stopMPcode
\stopbuffer

\typebuffer

\startlinecorrection[blank] \getbuffer \stoplinecorrection

In the end we decide that although the trajectories look
impressive, we will not trust our lives to \METAPOST. A
beautiful path is not neccessarily a good path. But even
then, this macro provides a nice way to experiment with
directions, controls and tensions.

% \section {Animations}
%
% {\em Although \METAPOST\ is not that well suited for free
% hand drawings, you can use it to make stylistics animations.}

\stopcomponent
