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<h2>10. Describing locations
<a name="10. Describing locations"></a>
</h2>
<p style="margin-top: 1em"><font color="#000000">The
location of points can be described in many different ways.
All these forms are interchangeable as for as the <b>pic</b>
language syntax is concerned; where you can use one, any of
the others that would make semantic sense are
allowed.</font></p>
<p style="margin-top: 1em"><font color="#000000">The
special label <b>Here</b> always refers to the current
position.</font></p>
<h3>10.1. Absolute Coordinates
<a name="10.1. Absolute Coordinates"></a>
</h3>
<p style="margin-top: 1em"><font color="#000000">The
simplest is absolute coordinates in inches; <b>pic</b> uses
a Cartesian system with (0,0) at the lower left corner of
the virtual drawing surface for each picture (that is,
X increases to the right and Y increases upwards).
An absolute location may always be written in the
conventional form as two comma-separated numbers surrounded
by parentheses (and this is recommended for clarity). In
contexts where it creates no ambiguity, the pair of X and
Y coordinates suffices without parentheses.</font></p>
<p style="margin-top: 1em"><font color="#000000">It is a
good idea to avoid absolute coordinates, however. They tend
to make picture descriptions difficult to understand and
modify. Instead, there are quite a number of ways to specify
locations relative to <b>pic</b> objects and previous
locations.</font></p>
<p style="margin-top: 1em"><font color="#000000">Another
possibility of surprise is the fact that <b>pic</b> crops
the picture to the smallest bounding box before writing it
out. For example, if you have a picture consisting of a
small box with its lower left corner at (2,2) and another
small box with its upper right corner at (5,5), the width
and height of the image are both 3 units and
not 5. To get the origin at (0,0) included, simply add
an invisible object to the picture, positioning the
object’s left corner at (0,0).</font></p>
<h3>10.2. Locations Relative to Objects
<a name="10.2. Locations Relative to Objects"></a>
</h3>
<p style="margin-top: 1em"><font color="#000000">The symbol
<b>Here</b> always refers to the position of the last object
drawn or the destination of the last <b>move</b>.</font></p>
<p style="margin-top: 1em"><font color="#000000">Alone and
unqualified, a <b>last circle</b> or any other way of
specifying a closed-object or arc location refers as a
position to the geometric center of the object. Unqualified,
the name of a line or spline object refers to the position
of the object start.</font></p>
<p style="margin-top: 1em"><font color="#000000">Also,
<b>pic</b> objects have quite a few named locations
associated with them. One of these is the object center,
which can be indicated (redundantly) with the suffix
<b>.center</b> (or just <b>.c</b>). Thus, <b>last circle
.center</b> is equivalent to <b>last circle</b>.</font></p>
<h4>10.2.1. Locations Relative to Closed Objects
<a name="10.2.1. Locations Relative to Closed Objects"></a>
</h4>
<p style="margin-top: 1em"><font color="#000000">Every
closed object (box, circle, ellipse, or block composite)
also has eight compass points associated with it;</font></p>
<p align="center" style="margin-top: 1em"><font color="#000000"><img src="img/pic31.png" alt="Image img/pic31.png"></font></p>
<p align="center" style="margin-top: 1em"><font color="#000000">Figure
10-1: Compass points</font></p>
<p style="margin-top: 1em"><font color="#000000">these are
the locations where eight compass rays from the geometric
center would intersect the figure. So when we say <b>last
circle .s</b> we are referring to the south compass point of
the last circle drawn. The explanation of Figure 8-3’s
program is now complete.</font></p>
<p style="margin-top: 1em"><font color="#000000">(In case
you dislike compass points, the names <b>.top</b>,
<b>.bottom</b>, <b>.left</b> and <b>.right</b> are synonyms
for <b>.n</b>, <b>.s</b>, <b>.e</b>, and <b>.w</b>
respectively; they can even be abbreviated to <b>.t</b>,
<b>.b</b>, <b>.l</b> and <b>.r</b>).</font></p>
<p style="margin-top: 1em"><font color="#000000">The names
<b>center</b>, <b>top</b>, <b>bottom</b>, <b>left</b>,
<b>right</b>, <b>north</b>, <b>south</b>, <b>east</b>, and
<b>west</b> can also be used (without the leading dot) in a
prefix form marked by <b>of</b>; thus, <b>center of last
circle</b> and <b>top of 2nd last ellipse</b> are both valid
object references. Finally, the names <b>left</b> and
<b>right</b> can be prefixed with <b>upper</b> and
<b>lower</b> which both have the obvious meaning.</font></p>
<p style="margin-top: 1em"><font color="#000000">Arc
objects also have compass points; they are the compass
points of the implied circle.</font></p>
<p style="margin-top: 1em"><font color="#000000">Non-closed
objects (line, arrow, or spline) have compass points too,
but the locations of them are completely arbitrary. In
particular, different <b>pic</b> implementations return
different locations.</font></p>
<h4>10.2.2. Locations Relative to Open Objects
<a name="10.2.2. Locations Relative to Open Objects"></a>
</h4>
<p style="margin-top: 1em"><font color="#000000">Every open
object (line, arrow, arc, or spline) has three named points:
<b>.start</b>, <b>.center</b> (or <b>.c</b>), and
<b>.end</b>. They can also be used without leading dots in
the <b>of</b> prefix form. The center of an arc is the
center of its circle, but the center of a line, path, or
spline is halfway between its endpoints.</font></p>
<p align="center" style="margin-top: 1em"><font color="#000000"><img src="img/pic32.png" alt="Image img/pic32.png"></font></p>
<p align="center" style="margin-top: 1em"><font color="#000000">Figure
10-2: Special points on open objects</font></p>
<h3>10.3. Ways of Composing Positions
<a name="10.3. Ways of Composing Positions"></a>
</h3>
<p style="margin-top: 1em"><font color="#000000">Once you
have two positions to work with, there are several ways to
combine them to specify new positions.</font></p>
<h4>10.3.1. Vector Sums and Displacements
<a name="10.3.1. Vector Sums and Displacements"></a>
</h4>
<p style="margin-top: 1em"><font color="#000000">Positions
may be added or subtracted to yield a new position (to be
more precise, you can only add a position and an expression
pair; the latter must be on the right side of the addition
or subtraction sign). The result is the conventional vector
sum or difference of coordinates. For example, <b>last box
.ne + (0.1, 0)</b> is a valid position. This example
illustrates a common use, to define a position slightly
offset from a named one (say, for captioning
purposes).</font></p>
<h4>10.3.2. Interpolation Between Positions
<a name="10.3.2. Interpolation Between Positions"></a>
</h4>
<p style="margin-top: 1em"><font color="#000000">A position
may be interpolated between any two positions. The syntax is
‘<i>fraction</i> <b>of the way between</b>
<i>position1</i> <b>and</b> <i>position2</i>’. For
example, you can say <b>1/3 of the way between Here and last
ellipse .ne</b>. The fraction may be in
numerator/denominator form or may be an ordinary number
(values are <i>not</i> restricted to [0,1]). As an
alternative to this verbose syntax, you can say
‘<i>fraction</i> <b><</b><i>position1</i> <b>,</b>
<i>position2</i><b>></b>’; thus, the example could
also be written as <b>1/3 <Here, last
ellipse></b>.</font></p>
<p align="center" style="margin-top: 1em"><font color="#000000"><img src="img/pic33.png" alt="Image img/pic33.png"></font></p>
<p align="center" style="margin-top: 1em"><font color="#000000">Figure
10-3: <b>P: 1/3 of the way between last arrow .start and
last arrow .end</b></font></p>
<p style="margin-top: 1em"><font color="#000000">This
facility can be used, for example, to draw double
connections.</font></p>
<p align="center" style="margin-top: 1em"><font color="#000000"><img src="img/pic34.png" alt="Image img/pic34.png"></font></p>
<p align="center" style="margin-top: 1em"><font color="#000000">Figure
10-4: Doubled arrows</font></p>
<p style="margin-top: 1em"><font color="#000000">You can
get Figure 10-4 from the following program:</font></p>
<p style="margin-left:10%; margin-top: 1em"><font color="#000000">.PS
<br>
A: box "yin"; move; <br>
B: box "yang"; <br>
arrow right at 1/4 <A.e,A.ne>; <br>
arrow left at 1/4 <B.w,B.sw>; <br>
.PE</font></p>
<p style="margin-top: 1em"><font color="#000000">Note the
use of the short form for interpolating points.</font></p>
<h4>10.3.3. Projections of Points
<a name="10.3.3. Projections of Points"></a>
</h4>
<p style="margin-top: 1em"><font color="#000000">Given two
positions <i>p</i> and <i>q</i>, the position
<b>(</b><i>p</i><b>,</b> <i>q</i><b>)</b> has the
X coordinate of <i>p</i> and the Y coordinate of
<i>q</i>. This can be helpful in placing an object at one of
the corners of the virtual box defined by two other
objects.</font></p>
<p align="center" style="margin-top: 1em"><font color="#000000"><img src="img/pic35.png" alt="Image img/pic35.png"></font></p>
<p align="center" style="margin-top: 1em"><font color="#000000">Figure
10-5: Using (<i>x</i>, <i>y</i>) composition</font></p>
<h3>10.4. Using Locations
<a name="10.4. Using Locations"></a>
</h3>
<p style="margin-top: 1em"><font color="#000000">There are
four ways to use locations; <b>at</b>, <b>from</b>,
<b>to</b>, and <b>with</b>. All four are object modifiers;
that is, you use them as suffixes to a drawing
command.</font></p>
<p style="margin-top: 1em"><font color="#000000">The
<b>at</b> modifier says to draw a closed object or arc with
its center at the following location, or to draw a
line/spline/arrow starting at the following
location.</font></p>
<p style="margin-top: 1em"><font color="#000000">The
<b>to</b> modifier can be used alone to specify a move
destination. The <b>from</b> modifier can be used alone in
the same way as <b>at</b>.</font></p>
<p style="margin-top: 1em"><font color="#000000">The
<b>from</b> and <b>to</b> modifiers can be used with a
<b>line</b> or <b>arc</b> command to specify start and end
points of the object. In conjunction with named locations,
this offers a very flexible mechanism for connecting
objects. For example, the following program</font></p>
<p style="margin-left:10%; margin-top: 1em"><font color="#000000">.PS
<br>
box "from" <br>
move 0.75; <br>
ellipse "to" <br>
arc cw from 1/3 of the way \ <br>
between last box .n and last box .ne to last ellipse .n;
<br>
.PE</font></p>
<p style="margin-top: 1em"><font color="#000000">yields:</font></p>
<p align="center" style="margin-top: 1em"><font color="#000000"><img src="img/pic36.png" alt="Image img/pic36.png"></font></p>
<p align="center" style="margin-top: 1em"><font color="#000000">Figure
10-6: A tricky connection specified with English-like
syntax</font></p>
<p style="margin-top: 1em"><font color="#000000">The
<b>with</b> modifier allows you to identify a named
attachment point of an object (or a position within the
object) with another point. This is very useful for
connecting objects in a natural way. For an example,
consider these two programs:</font></p>
<p align="center" style="margin-top: 1em"><font color="#000000"><img src="img/pic37.png" alt="Image img/pic37.png"></font></p>
<p align="center" style="margin-top: 1em"><font color="#000000">Figure
10-7: Using the <b>with</b> modifier for
attachments</font></p>
<h3>10.5. The ‘chop’ Modifier
<a name="10.5. The ‘chop’ Modifier"></a>
</h3>
<p style="margin-top: 1em"><font color="#000000">When
drawing lines between circles that don’t intersect
them at a compass point, it is useful to be able to shorten
a line by the radius of the circle at either or both ends.
Consider the following program:</font></p>
<p style="margin-left:10%; margin-top: 1em"><font color="#000000">.PS
<br>
circle "x" <br>
circle "y" at 1st circle - (0.4, 0.6) <br>
circle "z" at 1st circle + (0.4, -0.6) <br>
arrow from 1st circle to 2nd circle chop <br>
arrow from 2nd circle to 3rd circle chop <br>
arrow from 3rd circle to 1st circle chop <br>
.PE</font></p>
<p style="margin-top: 1em"><font color="#000000">It yields
the following:</font></p>
<p align="center" style="margin-top: 1em"><font color="#000000"><img src="img/pic38.png" alt="Image img/pic38.png"></font></p>
<p align="center" style="margin-top: 1em"><font color="#000000">Figure
10-8: The <b>chop</b> modifier</font></p>
<p style="margin-top: 1em"><font color="#000000">Notice
that the <b>chop</b> attribute moves arrowheads rather than
stepping on them. By default, the <b>chop</b> modifier
shortens both ends of the line by <b>circlerad</b>. By
suffixing it with a number you can change the amount of
chopping.</font></p>
<p style="margin-top: 1em"><font color="#000000">If you say
<b>line ... chop</b> <i>r1</i> <b>chop</b> <i>r2</i> with
<i>r1</i> and <i>r2</i> both numbers, you can vary the
amount of chopping at both ends. You can use this in
combination with trigonometric functions to write code that
deals with more complex intersections.</font></p>
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