<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE language SYSTEM "klayout_doc.dtd">

<doc>

  <title>The Geometry API</title>

  <h2-index/>

  <h2>The Geometry API</h2>

  <p>
  The central class of the layout database is the <class_doc href="Layout"/> class, which provides
  a concept for layers, shape containers, cells and hierarchy. Separate from that, a set of 
  classes exist, which represent basic shapes and geometrical objects. While shapes embedded in 
  a Layout object are not independent and care has to be taken when manipulating them inside the 
  database API, the free objects are easy to manage, manipulate and in general to work with as they
  are normal objects in the script interpreter's context.
  </p>

  <p>
  In addition, the gometry API provides basic objects such as points, edges and transformations for
  general use throughout the API. Higher objects such as regions and edge collections provide implementations
  for geometrical algorithms like boolean operations and DRC checks.
  </p>

  <p>
  Most classes of the geometry API provide a hash value so they can be used as keys in Ruby 
  hashes.
  </p>

  <h2>The Point class</h2>

  <keyword name="Point"/>
  <p>
  The <class_doc href="Point"/> object represents an integer coordinate in the 2-dimensional layout space,
  expressed in database units. 
  That object provides the x and y coordinates through the <class_doc href="Point#x"/> and <class_doc href="Point#y"/> attributes. Points can be added, 
  subtracted, the euclidian distance and its square can be computed using the <class_doc href="Point#distance"/> and <class_doc href="Point#sq_distance"/>
  methods. The <b>*</b> operator used with a factor as the second operand will scale both the x and y coordinates.
  </p>

  <h2>Points and vectors</h2>

  <keyword name="Point"/>
  <p>
  Next to points there is a corresponding vector class (<class_doc href="Vector"/>). 
  A vector is basically the difference between two points.
  It is meant to describe the distance and direction between two points.
  The following rules therefore apply:
  </p>

  <ul>
    <li>Subtracting a point from a point renders a vector</li>
    <li>Adding a vector to a point renders a point</li>
  </ul>

  <p>
  As points, vectors have an x and a y component which can be accessed with <class_doc href="Vector#x"/> and <class_doc href="Vector#y"/>.
  Vectors offer two functions to compute the vector product (<class_doc href="Vector#vprod"/>) and the scalar product
  (<class_doc href="Vector#sprod"/>). For some applications it's sufficient to know the sign of the product. You can 
  get that with <class_doc href="Vector#vprod_sign"/> and <class_doc href="Vector#sprod_sign"/> respectively.
  </p>

  <p>
  Vectors don't transform the same way than points. On transformation, only rotation, mirror and scaling (if applicable) 
  is applied. Displacement is not applied. This way, the following two forms are equivalent:
  </p>

  <pre>(p1 - p2).transformed(t) == p1.transformed(t) - p2.transformed(t)</pre>

  <h2>The Box class</h2>

  <keyword name="Box"/>
  <p>
  <img src="/programming/box.png"/>
  </p>

  <p>
  The <class_doc href="Box"/> object represents a rectangle whose sides
  are parallel to the axes. The coordinates are integer values and express the rectangle's dimensions in database units.
  The basic specification consists of two points, giving the lower left
  and upper right corner. The constructor takes either two points (which are ordered internally) or four coordinates
  representing the left, bottom, right and top coordinates.
  </p>

  <p>
  A box has a couple of attributes which are shown in the figure above. The <class_doc href="Box#area"/> method delivers the area of 
  the rectangle. A box can be "empty" (non-existing). Such a rectangle can be created by the default constructor
  without parameters. An empty box basically behaves as a region without a point, i.e. testing the intersection with
  an empty box always returns false. An empty box returns true on <class_doc href="Box#empty?"/>. A box whose lower left point is
  identical to the upper right one contains just a single point. Such a box returns true on <class_doc href="Box#is_point?"/>.
  </p>

  <p>
  A box supports a couple of operations:
  </p>

  <ul>
    <li><b>Box &amp; Box</b>: this operator computes the intersection of two boxes. If the boxes do not intersect, an
    empty box is returned. 
    </li>
    <li><b>Box * Box</b>: computes the "convolution" of two boxes. This operation 
    can basically be viewed as the box which results when the first box is painted with a pen that has the size 
    and displacement of the second box. 
    </li>
    <li><b>Box * factor</b>: scales the box, i.e. multiplies all coordinates with the 
    given factor.
    </li>
    <li><b>Box + Box</b>: computes the box that encloses both boxes given in the
    operation.
    </li>
    <li><b>Box + Point</b>: computes the box that encloses the box and the point given
    as the second operand. It basically enlarges the first box so it includes the second point as well.
    </li>
    <li><b>contains?(Point)</b>: returns true, if the point is inside the box or on its edges.
    </li>
    <li><b>enlarge(Point)</b>: will enlarge the box by the x and y coordinates of the Point. Basically the x value
    of the point is subtracted from the left side and added to the right. The same way, the y coordinate is added
    to the top and subtracted from the bottom coordinate.
    </li>
    <li><b>enlarged(Point)</b>: returns the enlarged box without modifying the box this method is called on (out-of-place operation).</li>
    <li><b>move(Point)</b>: moves the box by the displacement given by the point. Basically the x value of the point
    is added to the left and right coordinate while the y coordinate is added to the top and bottom coordinates.
    </li>
    <li><b>moved(Point)</b>: returns the moved box without modifying the box this method is called on (out-of-place operation).</li>
    <li><b>inside?(Box)</b>: returns true, if the box the method is called on is inside the box given by the method's argument.</li>
    <li><b>overlaps?(Box)</b>: returns true, if the given box overlaps with the box the method is called on.</li>
    <li><b>touches?(Box)</b>: returns true, if the given box touches the box the method is called on.</li>
    <li><b>transformed(Trans)</b>: returns the box transformed with the given transformation.</li>

    <li>
    <p>
    <b>transformed(ICplxTrans)</b>: returns the box transformed with the given complex, integer-based transformation (see <class_doc href="ICplxTrans"/>). 
    Note, that if the complex transformation includes a rotation by a non-90-degree angle (for example 45 degree), this operation
    does not return a rotated box, because by definition a box has edges which are parallel to the axes. Hence the general 
    solution is to convert the box to a polygon:
    </p>

    <pre># Wrong result
box = RBA::Box::new(0, 0, 100, 200)
transformed_box = box.transformed(RBA::ICplxTrans::new(1, 45, false, RBA::Vector::new))
# -&gt; (-141,0;71,212)

# Correct result
transformed_box_as_polygon = RBA::Polygon::new(box).transformed(RBA::ICplxTrans::new(1, 45, false, RBA::Vector::new))
# -&gt; (0,0;-141,141;-71,212;71,71)</pre>
    </li>

    <li><b>transformed(CplxTrans)</b>: behaves like the previous "transformed" method but returns a 
    floating-point coordinate object which is the target coordinate type of the CplxTrans object (see <class_doc href="CplxTrans"/>).
    </li>

  </ul>

  <p>A box object can be constructed from a floating-point object (for floating-point objects see below). 
  Lacking a database unit, no conversion from micrometer units
  to database units is performed. Instead, the floating-point coordinates are rounded to the nearest integer coordinates:
  </p>

  <pre>dbox = RBA::DBox::new(2.1, 3.1, 10.7, 11.8)
box = RBA::Box::new(dbox)
# -&gt; (2,3;11,12)</pre>

  <p>An integer box can be turned into a floating-point unit box using <class_doc href="Box#to_dtype"/></p>

  <p>Floating-point boxes can be transformed using the <class_doc href="DTrans"/>, the <class_doc href="DCplxTrans"/> or the <class_doc href="VCplxTrans"/>
  transformations. The latter delivers an integer-type box and provides the reverse flavour transformation to "CplxTrans".
  </p>
  
  <h2>The SimplePolygon class</h2>
  
  <keyword name="SimplePolygon"/>
  <p>
  <img src="/programming/simple_polygon.png"/>
  </p>

  <p>
  A <class_doc href="SimplePolygon"/> object is a polygon that does not have holes. It consists of
  a single, closed contour. Such polygons are compatible with the GDS2 format for example.
  A SimplePolygon object can be created from a Box object or from an array of Point objects.
  Internally, the points will be ordered into a clockwise orientation. 
  </p>

  <p>The class method <class_doc href="SimplePolygon#from_dpoly"/> creates a integer-coordinate type SimplePolygon object from a floating-point coordinate type DSimplePolygon
  object (see below). The floating-point coordinates are rounded to the nearest integer coordinates.
  </p>
  
  <p>
  The <class_doc href="SimplePolygon#bbox"/> method returns the bounding box of the polyon. <class_doc href="SimplePolygon#area"/> will return the area of the polygon.
  <class_doc href="SimplePolygon#num_points"/> returns the number of points, while <class_doc href="SimplePolygon#point"/> returns the point for a given index. <class_doc href="SimplePolygon#points="/>
  replaces the polygon by a new polygon with the given array of points.
  </p>

  <p>The <class_doc href="SimplePolygon#each_point"/> iterator will deliver each point in clockwise orienation, starting form the bottom/leftmost one.
  <class_doc href="SimplePolygon#each_edge"/> will deliver all edges of the polygon (connecting every point with the next one). 
  </p>

  <p>
  <class_doc href="SimplePolygon#compress"/> will remove points that connect two collinear edges. It has a parameter that controls whether to remove
  reflecting edges (spikes) as well. <class_doc href="SimplePolygon#inside?"/> returns true, if a given point is inside the polygon. <class_doc href="SimplePolygon#minkowski_sum"/>
  computes the Minkowski sum between a polygon and another object in various flavors. <class_doc href="SimplePolygon#move"/> will displace the polygon
  by the distance given by the Point argument. <class_doc href="SimplePolygon#moved"/> will return the moved polygon without modifying the 
  polygon it is called on (out-of-place operation). <class_doc href="SimplePolygon#transformed"/> will return the 
  transformed polygon, either with a simple or a complex transformation (see the description of the Box object and the
  section about transformations below for a discussion of transformations). Finally, <class_doc href="SimplePolygon#round_corners"/> will apply a
  corner rounding to a copy of the polygon and return that copy without modifying the polygon.
  </p>

  <p>
  Please note that using the Point-array constructors it is possible to create polygons with self-intersecting or 
  twisted contours. Such polygons may not behave as expected.
  </p>

  <h2>The Polygon class</h2>
  
  <keyword name="Polygon"/>
  <p>
  <img src="/programming/polygon.png"/>
  </p>

  <p>
  The <class_doc href="Polygon"/> object is basically an extension of the SimplePolygon object and 
  in contrast to that object, supports holes. Polygon objects are not compatible with the GDS2 or OASIS format
  and will be converted to SimplePolygon objects by introducing cutlines when writing them.
  </p>

  <p>
  A polygon consists of an outer contour (the hull) and zero to many hole contours. The outer contour like for the SimplePolygon
  is oriented clockwise while the hole contours are oriented counterclockwise. This orientation is established
  internally. For the API, contours are represented by arrays of Point objects.
  </p>

  <p>
  A Polygon differs from a SimplyPolygon by providing methods to modify holes and the hull. Holes can be inserted by 
  using the <class_doc href="Polygon#insert_hole"/> method. A hole is identified by an index. The <class_doc href="Polygon#holes"/> method returns the number of holes. The 
  hole index runs from 0 to the number of holes minus one.
  </p>

  <p>The class method <class_doc href="Polygon#from_dpoly"/> creates a integer-coordinate type Polygon object from a floating-point coordinate type DPolygon
  object (see below). The floating-point coordinates are rounded to the nearest integer coordinates. A constructor is provided
  that creates a Polygon object from a SimplePolygon object.
  </p>
  
  <p>
  <class_doc href="Polygon#each_edge"/> will deliver all edges. The orientation of the edges determines whether they belong to a hole or the hull contour.
  <class_doc href="Polygon#each_point_hull"/> will iterated over all points of the hull and <class_doc href="Polygon#each_point_hole"/> over the points of the hole with the 
  given index. <class_doc href="Polygon#num_points_hull"/> will return the number of points for the hull and <class_doc href="Polygon#num_points_hole"/> the number of 
  points for the given hole. <class_doc href="Polygon#point_hull"/> will return a specific point from the hull and <class_doc href="Polygon#point_hole"/> a specific point
  for the given hole. <class_doc href="Polygon#num_points"/> returns the total number of points.
  </p>

  <p>
  <class_doc href="Polygon#resolve_holes"/> will remove all holes and connect them with the hull by introducing cutlines that connect the hole with
  the hull. This operation introduces new vertexes and hence may apply some distortion due to grid snapping. <class_doc href="Polygon#resolved_holes"/>
  returns the polygon with the holes removed without modifying the object it is called on (out-of-place operation). <class_doc href="Polygon#to_simple_polygon"/>
  basically does the same but returns a SimplePolygon object.
  </p>

  <p>
  Using <class_doc href="Polygon#assign_hull"/> (or <class_doc href="Polygon#hull="/>) and <class_doc href="Polygon#assign_hole"/> the hull contour or a hole contour can be replaced with the given array of Point objects.
  Please note that it is possible to create invalid polygons where the holes are not completely contained in the hull. Such polygons
  may not behave as expected. The same is true for polygons with self-intersecting or twisted contours.
  </p>

  <h2>The Path class</h2>
  
  <keyword name="Path"/>
  <p>
  <img src="/programming/path.png"/>
  </p>

  <p>
  A <class_doc href="Path"/> object represents a line with a certain width. The figure above depicts the basic properties
  of a path. The basic geometry of a path is defined by its spine - a sequence of points that the path follows.
  By default, a path has rectangular end caps with variable length. The end caps can be round, in which case the extension 
  should be half the path width to avoid paths which are not compatible with the GDS2 format. Round-ended paths return
  true on <class_doc href="Path#is_round?"/>. An example for such a path is depicted in the following figure:
  </p>

  <p>
  <img src="/programming/round_ended_path.png"/>
  </p>

  <p>
  A path with is given in database units. The hull contour of paths with an odd width cannot be represented on-grid and should
  be avoided. Such path objects are allowed in the GDS2 format, but not in OASIS.
  </p>

  <p>
  The spine points of a path can be iterated with <class_doc href="Path#each_point"/>. <class_doc href="Path#num_points"/> returns the number of points.
  <class_doc href="Path#points="/> allows replacing of the spine with the given array of Point objects.
  <class_doc href="Path#round="/> sets the "round ended" flag. <class_doc href="Path#bbox"/> and <class_doc href="Path#area"/> deliver the bounding box and the area, where the area
  is only approximate and is computed from the spine's length including the extensions times the width for efficiency. For
  certain acute-angle configurations that value may not be the exact area.
  </p>

  <p>
  <class_doc href="Path#polygon"/> returns the polygon representing the path's hull. <class_doc href="Path#simple_polygon"/> returns a SimplePolygon
  object that represents the hull. <class_doc href="Path#move"/>, <class_doc href="Path#moved"/> and <class_doc href="Path#transformed"/> basically work
  like for the other objects.
  </p>

  <p>The class method <class_doc href="Path#from_dpath"/> creates a integer-coordinate type Path object from a floating-point coordinate type DPath
  object (see below). The floating-point coordinates are rounded to the nearest integer coordinates.
  </p>
  
  <h2>The Text class</h2>
  
  <keyword name="Text"/>
  <p>
  A <class_doc href="Text"/> object is basically a point with a label attached. The extension of a text object
  only includes the point, not the text itself. For display purposes, a text orientation, font and alignment options can 
  be specified.
  </p>

  <p>
  The location and the orientation of the text are combined in a <class_doc href="Trans"/> object. That transformation
  can be read and written with the <class_doc href="Text#trans"/> attribute. The text orientation is not always shown in the drawing. Whether the
  text is shown in the orientation specified depends on the application settings.
  </p>

  <p>
  Font number and alignment flags can be read and written using the <class_doc href="Text#font"/>, <class_doc href="Text#halign"/> and <class_doc href="Text#valign"/> attributes.
  If one of these attributes is not set, the application-provided default is used. The font is currently just a number and 
  is provided to support the respective property of the GDS2 format. For the documentation of the alignment flags see the 
  class documentation for the <class_doc href="Text"/> class.
  </p>

  <p>
  The text string is accessible with the <class_doc href="Text#string"/> attribute. Not all characters are supported. Depending on 
  the format, only a subset of the ASCII character set are supported. Sometimes, line-feed control characters are found
  in text strings. Following the strict OASIS specification for example, such characters are not allowed. The 
  lower and upper case letters and most of the special printing characters of the ASCII character set are usually safe
  for use in text strings.
  </p>

  <p>
  <class_doc href="Text#move"/>, <class_doc href="Text#moved"/> and <class_doc href="Text#transformed"/> basically work
  like for the other objects.
  </p>

  <p>
  Passing a DText object to the Text constructor creates an integer-coordinate type Text object from a floating-point coordinate type DText
  object (see below). The floating-point coordinates are rounded to the nearest integer coordinates.
  </p>

  <h2>The Edge class</h2>

  <keyword name="Edge"/>
  <p>
  <class_doc href="Edge"/> objects are basically connections between two points. Edge objects are provided to support
  special applications and are mapped to zero-width, two-point paths in the GDS2 format. 
  Edge objects are not supported as editable objects in KLayout currently. Edge objects may be created by script and 
  are useful sometimes to represent the output of a design rule check tool for example.
  </p>

  <p>
  Edge objects however a frequently used as raw objects in various applications, i.e. as one output option of a boolean operation.
  Therefore, the Edge class provides a couple of operations on edges, mainly tests for the geometrical relationship of two
  edges and points in relation to edges. An edge also may represent a straight line through the two endpoints of the edge, i.e.
  for the <class_doc href="Edge#crossed_by?"/> check.
  </p>

  <p>An edge is defined by two points: the start and end point. <class_doc href="Edge#p1"/> is the attribute for the start point, <class_doc href="Edge#p2"/> the 
  attribute for the end point. Various properties deliver extensions of the edge (<class_doc href="Edge#dx"/> and <class_doc href="Edge#dy"/> for the horizontal
  and vertical extension, <class_doc href="Edge#length"/> and <class_doc href="Edge#sq_length"/> for the length and the square of the length, <class_doc href="Edge#ortho_length"/>
  for the Manhattan distance between the points). <class_doc href="Edge#swap_points"/> swaps p1 and p2 and effectively inverts the orientation 
  of the edge. <class_doc href="Edge#bbox"/> is the bounding box of the edge.
  </p>

  <p>Various methods test the relationship between two edges or an edge and a point. See the following figure for a 
  summary of these methods:
  </p>

  <p>
  <img src="/programming/edge_relations.png"/>
  </p>

  <p>
  <class_doc href="Edge#move"/>, <class_doc href="Edge#moved"/> and <class_doc href="Edge#transformed"/> basically work
  like for the other objects.
  The class method <class_doc href="Edge#from_dedge"/> creates a integer-coordinate type Edge object from a floating-point coordinate type DEdge
  object (see below). The floating-point coordinates are rounded to the nearest integer coordinates.
  </p>

  <h2>The floating-point geometrical primitives</h2>

  <keyword name="DPoint"/>
  <keyword name="DBox"/>
  <keyword name="DText"/>
  <keyword name="DPath"/>
  <keyword name="DPolygon"/>
  <keyword name="DSimplePolygon"/>
  <keyword name="DEdge"/>
  <p>
  Beside the integer-coordinate primitives like Box, Edge, Polygon etc., KLayout provides floating-point coordinate variants
  as well. These objects require more memory and are subject to floating-point rounding issues and are therefore not employed
  inside the database. They are provided however, to allow a temporary representation of micron-unit objects or for use
  int the <class_doc href="Marker"/> class for example.
  </p>

  <p>
  The class names for the floating-point variants is the same the integer-coordinate type prefixed with a "D" (for example, DBox
  is the floating-point variant of Box).
  Since floating-point variants support fractional coordinates, scaling with an arbitrary value is not connected with a 
  loss of accuracy due to rounding. That is why floating-point coordinates are the target type of general transformations including
  an arbitrary scaling. For example:
  </p>

  <pre>RBA::Box.new(1,1,2,2)*2.5
# -&gt; (3,3;5,5)
# but:
RBA::DBox.new(1,1,2,2)*2.5
# -&gt; (2.5,2.5;5,5)</pre>

  <p>
  However, the higher precision of floating-point coordinates comes with subtle issues originating from the 
  finite-precision representation. For example, the value of 0.1 cannot be precisely represented by a 
  floating-point value. The value of 0.1 is approximated and therefore depending on the way it is approximated,
  0.1 is not necessarily equal to 0.1. For example (try this in the Ruby console):
  </p>

  <pre>a = 0.1
b = 1e-6*1e5
a
# -&gt; 0.1
b
# -&gt; 0.1
# but:
a == b
# -&gt; false!
# the reason is:
"%.21g"%a
# -&gt; 0.100000000000000005551
"%.21g"%b
# -&gt; 0.0999999999999999916733</pre>
 
  <p>
  Because of that, direct comparison of coordinate values should be avoided.
  The internal precision used is 0.00001. By convention, floating-point type objects are supposed to 
  be used for micrometer-unit objects, hence this precision corresponds to 0.01 nm which corresponds to 
  the tenth of an atom and should be well below physical tolerances. 
  </p>

  <p>
  Some operations which are implemented on integer coordinates (like the removal of holes in polygons)
  are not available for the floating-point type objects.
  </p>

  <p>
  All floating-point type objects have a "to_itype" method (i.e. <class_doc href="DBox#to_itype"/>) which convert the floating-point 
  type object to the integer-coordinate type object. This method can be given a database unit for scaling from micrometer units. 
  Integer-coordinate type objects can be converted to floating-point type objects the same way through the "to_dtype" method (i.e.
  <class_doc href="Box#to_dtype"/>. A database unit can be passed to this method too for conversion to micrometer units.
  These methods however require some rounding and are therefore responsible for a potential distortion of the geometry of the object.
  </p>

  <p>
  For the point and vector object there is also a floating-point equivalent (<class_doc href="DPoint"/> and <class_doc href="DVector"/>). 
  Both behaves like their integer-coordinate equivalent.
  </p>

  <h2>Transformations</h2>

  <keyword name="Trans"/>
  <keyword name="DTrans"/>
  <keyword name="DCplxTrans"/>
  <keyword name="CplxTrans"/>
  <keyword name="ICplxTrans"/>
  <keyword name="VCplxTrans"/>

  <p>
  Transformations in KLayout are restricted affine transformations. Shear transformations and anisotropic scaling is not
  supported.
  All transformation in KLayout follows the conventions implied by the GDS2 and OASIS formats and include in that order:
  </p>

  <ul>
    <li>Mirroring around the x axis (optional)</li>
    <li>Rotation</li>
    <li>Scaling</li>
    <li>Displacement</li>
  </ul>

  <p>
  For memory performance, a restricted version of that transformation is used if possible. In that restricted version, the
  rotation angles are confined to a multiple of 90 degree and scaling is not supported. This restricted affine transformation
  is provided through the <class_doc href="Trans"/> class. The 8 possible rotation/mirror variants can be coded in 
  a single rotation/mirror code which is used frequently throughout KLayout (see <link href="/about/transformations.xml"/>).
  </p>

  <p>
  For transforming floating-point coordinates, the <class_doc href="DTrans"/> object is provided. The basic difference 
  is that the displacement uses floating-point coordinates (by employing a DVector object).
  </p>

  <p>
  To support more complex affine transformations include arbitrary-angle rotations and scaling, the complex transformation objects
  are provided. In addition, the complex transformation object can translate between integer and floating-point coordinate types.
  </p>

  <ul>
    <li><class_doc href="CplxTrans"/>: takes integer coordinates and delivers floating-point coordinates. 
    Therefore it uses a DVector for the displacement. The inverse of this transformation is a VCplxTrans
    class (see below).
    </li>
    <li><class_doc href="DCplxTrans"/>: takes floating-point coordinates and delivers floating-point coordinates. 
    Therefore it also uses a DVector for the displacement.
    </li>
    <li><class_doc href="ICplxTrans"/>: takes integer coordinates and delivers integer coordinates. 
    Therefore it uses a Vector for the displacement. This transformation is convenient to provide complex transformations
    for database operations but implies rounding errors due to rounding to integer coordinates on output. It is safe to use however
    for integer-factor scaling operations for example.
    </li>
    <li><class_doc href="VCplxTrans"/>: takes floating-point coordinates and integer coordinates. 
    Therefore it also uses a Vector for the displacement. The inverse of this transformation is a 
    CplxTrans object.
    </li>
  </ul>

  <p>
  Multiplication of a transformation with an object renders the transformed object:
  </p>

  <pre>t = ... # a transformation
p = ... # some point

# compute the transformed point:
q = t * p</pre>

  <p>
  Multiplication of two transformations corresponds to concatenation of two transformations:
  </p>

  <pre>t1 = ... # a transformation
t2 = ... # another transformation

# Multiplication of t2 and t1 renders an equivalent transformation
# that corresponds to "apply t1 first and then t2":
(t2 * t1) * p == t2 * (t1 * p)</pre>

  <p>
  When multiplying two complex transformations, the resulting transformation type will 
  have the corresponding input and output types. For example when multiplying a VCplxTrans with 
  a CplxTrans, a ICplxTrans object will be created. This is because the first transformation
  takes integers and the second one delivers integers. Hence the resulting transformation
  is ICplxTrans.
  </p>

  <h2>The Region class</h2>

  <keyword name="Region"/>

  <p>
  Regions are basically collections of polygons. Regions provide higher functions such as boolean operations and
  sizing which cannot be implemented on pure polygons because their output may be a number of polygons.
  A Region is a general representation of a set of points in a two-dimensional area while a polygon is 
  a coherence set of points.
  </p>

  <p>
  Regions can be created by starting with an empty region and filling the latter with polygons. Regions
  can also be created from a <class_doc href="RecursiveShapeIterator"/> which allows feeding layout data
  into the region in a very flexible way. In the latter case, boxes and paths will be converted to polygons when feeding them 
  into the region.
  </p>

  <p>
  Regions feature some important concepts:
  </p>

  <ul>
    <li><b>Merged semantics:</b> A region can be created from a series of polygons which
      potentially overlap or touch. In "merged semantics" the region will merge
      the polygons forming big polygons from touching ones and removing overlaps.
    </li>
    <li><b>Minimum coherence:</b> In certain cases, the output of merging polygons is 
      ambiguous. In the "kissing corner" case, the touching polygons may be either 
      considered separate ("minimum coherence") or belonging together.
    </li>
    <li><b>strict mode:</b> In strict mode, some operations are performed even if 
      they are not necessary. For example, an XOR between a region and an empty
      region will render the first input. Hence the implementation can simply copy
      the first input in that case. With strict mode, the operation will be performed
      in every case which is less efficient but renders merged polygons.
    </li>
  </ul>

  <p>
  The region object is very mighty and easy to use. Here is an example which
  computes the difference of two boxes (rendering a ring) and sizes the latter:
  </p>

  <pre>r1 = RBA::Region::new
r1.insert(RBA::Box::new(-2000, -3000, 2000, 3000))

r2 = RBA::Region::new
r2.insert(RBA::Box::new(-1500, -2000, 1500, 2000))

r = (r1 - r2).sized(100)

puts r
</pre>

  <p>
  Using a Region object with input from a cell is almost as simple as this.
  The following example will take the input from layer 11 and 21 from the top 
  cell and the hierarchy below (as is flat) and compute the intersection 
  in layer 100:</p>

  <pre>ly = RBA::CellView::active.layout

l11 = ly.layer(11, 0)
l21 = ly.layer(21, 0)

r11 = RBA::Region::new(ly.top_cell.begin_shapes_rec(l11))
r21 = RBA::Region::new(ly.top_cell.begin_shapes_rec(l21))

ly.top_cell.shapes(ly.layer(100, 0)).insert(r11 &amp; r21)
  </pre>

  <p>
  A variety of operations is implemented on regions:
  </p>

  <p><class_doc href="Region#&amp;"/>, <class_doc href="Region#|"/>, <class_doc href="Region#-"/> and
     <class_doc href="Region#^"/> implement boolean operations (AND, OR, NOT and XOR).
     The operations can be combined with assignment (in-place) using <class_doc href="Region#&amp;="/> etc.
     <class_doc href="Region#+"/> adds the polygons from another region to self which is basically
     the same then a boolean OR.
  </p>
  <p><class_doc href="Region#area"/> and <class_doc href="Region#bbox"/> deliver the area and bounding
    box of the region.
  </p>
  <p><class_doc href="Region#each"/> will deliver all original polygons (unless the region was already merged).</p>
  <p><class_doc href="Region#each_merged"/> will deliver the merged polygons.</p>
  <p><class_doc href="Region#edges"/> will translate the polygons to edges covering their boundaries.
    If merged semantics is specified, the edges will only cover the outer edges, not inner ones.</p>
  <p><class_doc href="Region#enclosing_check"/>, <class_doc href="Region#inside_check"/>, 
    <class_doc href="Region#isolated_check"/>, <class_doc href="Region#notch_check"/>, 
    <class_doc href="Region#separation_check"/>, <class_doc href="Region#space_check"/>
    and <class_doc href="Region#width_check"/> implement DRC functions. DRC functions deliver 
    <class_doc href="EdgePairs"/> edge pair collections, not regions. Each edge pair marks a
    violation of the given check.
  </p>
  <p><class_doc href="Region#extents"/> replaces each polygon with its bounding box.</p>
  <p><class_doc href="Region#grid_check"/> performs an on-grid check returning edge pairs for off-grid markers.</p>
  <p><class_doc href="Region#holes"/> identifies holes and delivers a new region with the holes as filled polygons.</p>
  <p><class_doc href="Region#hulls"/> identifies holes and delivers a new region without the holes.</p>
  <p><class_doc href="Region#insert"/> adds polygons in various flavors.</p>
  <p><class_doc href="Region#interacting"/> and <class_doc href="Region#not_interacting"/> select polygons 
    interacting (overlapping or touching) or not interacting with polygons from another region.</p>
  <p><class_doc href="Region#members_of"/> selects polygons which are contained in the same 
    way in another region.</p>
  <p><class_doc href="Region#merge"/> and <class_doc href="Region#merged"/> merge the polygons. This feature allows selecting overlapping polygons
    (minimum wrap count parameter).</p>
  <p><class_doc href="Region#minkowski_sum"/> computes the Minkowski sum of the region with other objects (edges, single polygons and other regions).</p>
  <p><class_doc href="Region#move"/>, <class_doc href="Region#moved"/>, <class_doc href="Region#transform"/> and
    <class_doc href="Region#transformed"/> apply geometrical transformations.</p>
  <p><class_doc href="Region#rectangles"/>, <class_doc href="Region#rectilinear"/>, <class_doc href="Region#non_rectangles"/> and
    <class_doc href="Region#non_rectilinear"/> filter out polygons by their appearance.</p>
  <p><class_doc href="Region#inside"/>, <class_doc href="Region#outside"/>, <class_doc href="Region#not_inside"/> and
    <class_doc href="Region#not_outside"/> filter out polygons by their relation to the polygons in the other region.</p>
  <p><class_doc href="Region#round_corners"/> and <class_doc href="Region#rounded_corners"/> apply corner rounding
    to polygons.</p>
  <p><class_doc href="Region#size"/> and <class_doc href="Region#sized"/> will size the polygons
    (shift edges).</p>
  <p><class_doc href="Region#smooth"/> smoothes out coarse steps of the polygons.</p>
  <p><class_doc href="Region#snap"/> and <class_doc href="Region#snapped"/> apply grid snapping.</p>
  <p><class_doc href="Region#with_angle"/> marks polygon vertices which satisfy a certain angle criterion.</p>
  <p><class_doc href="Region#with_perimeter"/>, <class_doc href="Region#with_area"/> and many more "with_..."
    methods select polygons based on their properties.</p>

  <h2>The Edges class</h2>

  <keyword name="Edges"/>

  <p>
  <class_doc href="Edges"/> represents a collection of edges which comprise either full polygons
  (forming closed contours) or parts of polygons. Edges can be derived from Region objects
  using the <class_doc href="Region#edges"/>
  </p>

  <p>
  A variety of operations is implemented on regions:
  </p>

  <p><class_doc href="Edges#&amp;"/>, <class_doc href="Edges#|"/>, <class_doc href="Edges#-"/> and
     <class_doc href="Edges#^"/> implement boolean operations (AND, OR, NOT and XOR).
     The operations can be combined with assignment (in-place) using <class_doc href="Region#&amp;="/> etc.
     <class_doc href="Edges#+"/> adds the polygons from another edge set to self which is basically
     the same then a boolean OR. Boolean AND and NOT are available between edges and regions as well 
     and will deliver the edge parts inside the given region or not inside the given region.
  </p>
  <p><class_doc href="Edges#length"/> and <class_doc href="Edges#bbox"/> deliver the total length and bounding
    box of the edge set.
  </p>
  <p><class_doc href="Edges#each"/> will deliver the edges in the collection.</p>
  <p><class_doc href="Edges#centers"/> will deliver the center parts of the edges.</p>
  <p><class_doc href="Edges#enclosing_check"/>, <class_doc href="Edges#inside_check"/>, 
    <class_doc href="Edges#separation_check"/>, <class_doc href="Edges#space_check"/>
    and <class_doc href="Edges#width_check"/> implement DRC functions. DRC functions deliver 
    <class_doc href="EdgePairs"/> edge pair collections, not edge sets. Each edge pair marks a
    violation of the given check.
  </p>
  <p><class_doc href="Edges#start_segments"/> and <class_doc href="Edges#end_segments"/> replaces
    each edge with a part at the beginning or end.</p>
  <p><class_doc href="Edges#extended"/>, <class_doc href="Edges#extended_in"/> and
    <class_doc href="Edges#extended_out"/> create polygons on that represent the edge with a 
    certain width.</p>
  <p><class_doc href="Edges#extents"/> returns a region with the bounding boxes of the edges.</p>
  <p><class_doc href="Edges#inside_part"/> and <class_doc href="Edges#outside_part"/> 
    return the parts of the edge set which are inside or outside a given region. The
    effect is comparable to the boolean operations but differs at the exact boundary
    of the polygons of the region.</p>
  <p><class_doc href="Edges#interacting"/> and <class_doc href="Edges#not_interacting"/> select edges
    interacting (overlapping or touching) or not interacting with edges of another edge set or
    polygons from another region.</p>
  <p><class_doc href="Edges#members_of"/> selects edges which are contained in the same 
    way in another edge set.</p>
  <p><class_doc href="Edges#merge"/> and <class_doc href="Edges#merged"/> merge (join) the edges.</p>
  <p><class_doc href="Edges#move"/>, <class_doc href="Edges#moved"/>, <class_doc href="Region#transform"/> and
    <class_doc href="Edges#transformed"/> apply geometrical transformations.</p>
  <p><class_doc href="Edges#with_length"/>, <class_doc href="Region#with_angle"/> and many more "with_..."
    methods select edges based on their properties.</p>

  <h2>The EdgePair class</h2>

  <keyword name="EdgePair"/>

  <p>
  Edge pairs are handy objects to describe the output of a DRC violation. A space violation marker
  for example will consist of two edges which mark the opposite parts of the space violation.
  Such markers are represented by objects of <class_doc href="EdgePair"/>.
  </p>

  <p>
  Edge pairs are simple objects consisting of two edges ("first" and "second").
  Polygons can generated from edge pairs covering the marked edges and their connecting
  files using <class_doc href="EdgePair#polygon"/>. Edge pairs can be transformed using
  <class_doc href="EdgePair#transformed"/>.
  </p>

  <p>
  Edge pairs can be "normalized" using <class_doc href="EdgePair#normalized"/>.
  This method returns the normalized version of the edge pair.
  Normalization will bring the edge pairs in a form such that when connecting their start and end
  points a closed loop without intersections is formed. Normalized edge pairs will produce
  nicer polygons later on.
  </p>

  <p>
  A floating-point version of the EdgePair class exists as well: <class_doc href="DEdgePair"/>.
  </p>

  <h2>The EdgePairs class</h2>

  <keyword name="EdgePairs"/>

  <p>
  <class_doc href="EdgePairs"/> provides a collection of edge pairs and is the preferred output
  of DRC functions which deliver one DRC marker for each violation.
  </p>

  <p>
  EdgePair collections can be conveniently split into individual edges using <class_doc href="EdgePairs#edges"/>
  or converted into polygons covering the markers with <class_doc href="EdgePairs#polygons"/>. The
  first and second edges can be extracted as a Edges edge collection with <class_doc href="EdgePairs#first_edges"/>
  and <class_doc href="EdgePairs#second_edges"/>. <class_doc href="EdgePairs#extents"/> will deliver
  a region containing the bounding boxes of the edge pairs.
  </p>

</doc>
