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      <div style="padding: 5px;" align="center"> <img src="images/boost.png" border="0" height="86" width="277"><a title="www.boost.org home page" tabindex="2" style="border: medium none ;" href="http://www.boost.org/"> </a></div>
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      <h3 class="navbar">Contents</h3>
      <ul>
        <li><a href="index.htm">Boost.Polygon Main Page</a></li>
        <li><a href="gtl_design_overview.htm">Design Overview</a></li>
        <li><a href="gtl_isotropy.htm">Isotropy</a></li>
        <li><a href="gtl_coordinate_concept.htm">Coordinate Concept</a></li>
        <li><a href="gtl_interval_concept.htm">Interval Concept</a></li>
        <li><a href="gtl_point_concept.htm">Point Concept</a></li>
        <li><a href="gtl_segment_concept.htm">Segment Concept</a></li>
        <li><a href="gtl_rectangle_concept.htm">Rectangle Concept</a></li>
        <li><a href="gtl_polygon_90_concept.htm">Polygon 90 Concept</a></li>
        <li><a href="gtl_polygon_90_with_holes_concept.htm">Polygon 90
With Holes Concept</a></li>
        <li><a href="gtl_polygon_45_concept.htm">Polygon 45 Concept</a></li>
        <li><a href="gtl_polygon_45_with_holes_concept.htm">Polygon 45
With Holes Concept</a></li>
        <li><a href="gtl_polygon_concept.htm">Polygon Concept</a></li>
        <li><a href="gtl_polygon_with_holes_concept.htm">Polygon With
Holes Concept</a></li>
        <li><a href="gtl_polygon_90_set_concept.htm">Polygon 90 Set
Concept</a></li>
        <li><a href="gtl_polygon_45_set_concept.htm">Polygon 45 Set
Concept</a></li>
        <li><a href="gtl_polygon_set_concept.htm">Polygon Set Concept</a></li>
        <li><a href="gtl_connectivity_extraction_90.htm">Connectivity
Extraction 90</a></li>
        <li><a href="gtl_connectivity_extraction_45.htm">Connectivity
Extraction 45</a></li>
        <li><a href="gtl_connectivity_extraction.htm">Connectivity
Extraction</a></li>
        <li><a href="gtl_property_merge_90.htm">Property Merge 90</a></li>
        <li><a href="gtl_property_merge_45.htm">Property Merge 45</a></li>
        <li><a href="gtl_property_merge.htm">Property Merge</a></li>
        <li><a href="voronoi_main.htm">Voronoi Main Page </a></li>
        <li><a href="voronoi_benchmark.htm">Voronoi Benchmark</a> </li>
        <li>Voronoi Builder</li>
        <li><a href="voronoi_diagram.htm">Voronoi Diagram</a></li>
      </ul>
      <h3 class="navbar">Other Resources</h3>
      <ul>
        <li><a href="GTL_boostcon2009.pdf">GTL Boostcon 2009 Paper</a></li>
        <li><a href="GTL_boostcon_draft03.pdf">GTL Boostcon 2009
Presentation</a></li>
        <li><a href="analysis.htm">Performance Analysis</a></li>
        <li><a href="gtl_tutorial.htm">Layout Versus Schematic Tutorial</a></li>
        <li><a href="gtl_minkowski_tutorial.htm">Minkowski Sum Tutorial</a></li>
        <li><a href="voronoi_basic_tutorial.htm">Voronoi Basic Tutorial</a></li>
        <li><a href="voronoi_advanced_tutorial.htm">Voronoi Advanced
Tutorial</a></li>
      </ul>
      </div>
      <h3 class="navbar">Polygon Sponsor</h3>
      <div style="padding: 5px;" align="center"> <img src="images/intlogo.gif" border="0" height="51" width="127"><a title="www.adobe.com home page" tabindex="2" style="border: medium none ;" href="http://www.adobe.com/"> </a></div>
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      <td style="padding-left: 10px; padding-right: 10px; padding-bottom: 10px;" valign="top" width="100%"><!-- End Header --> <br>
      <h1>Voronoi Builder </h1>
The Voronoi builder is the event generator structure, that implements
the <a href="http://en.wikipedia.org/wiki/Sweep_line_algorithm">sweepline
algorithm</a>,
scanning 2D space and spawning the two types of events: <a href="http://www.ams.org/samplings/feature-column/fcarc-voronoi">site
events and circle events</a>. Each event is reported to the output data
structure builder.
The structure shares Voronoi name, as the events generated by it
provide
enough information to construct the Voronoi diagram of a set of points
and linear segments. The requirements for the coordinate type of
the input/output geometries, configured through the coordinate type
traits template argument, are the following:<br>
      <ul>
        <li>The input coordinate type (for input points and enpoints of
the input segments) is not required to be integral
(while it
still should be an integer type)</li>
        <li>The output coordinate type (for
Voronoi vertices) is required to be IEEE-754 floating point type</li>
      </ul>
      <h2>Important</h2>
The users are encouraged to use the default static methods defined in
the <a href="../../../boost/polygon/voronoi.hpp">voronoi.hpp</a>
header for the Voronoi diagram construction. However it's also possible
to
use Voronoi builder explicitly, especially if the user wants to drop
the external dependencies such as MPL (metaprogramming library). The
following include set doesn't depend on any external headers
(except STL and boost/cstdint.hpp), even
the Boost.Polygon library:<br>
      <br>
      <span style="font-family: Courier New,Courier,monospace;">#include
&lt;voronoi_builder.hpp&gt;</span><br style="font-family: Courier New,Courier,monospace;">
      <span style="font-family: Courier New,Courier,monospace;">#include
&lt;voronoi_diagram.hpp&gt;</span><br>
      <h2>Declaration</h2>
Header: <a href="../../../boost/polygon/voronoi_builder.hpp">boost/polygon/voronoi_builder.hpp</a><br>
      <br>
      <font face="Courier New"> <span style="font-family: 'Courier New',Courier,monospace;">template
&lt;typename T,</span><br style="font-family: 'Courier New',Courier,monospace;">
      <span style="font-family: 'Courier New',Courier,monospace;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
typename CTT = detail::voronoi_ctype_traits&lt;T&gt;,</span><br style="font-family: 'Courier New',Courier,monospace;">
      <span style="font-family: 'Courier New',Courier,monospace;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
typename VP = detail::voronoi_predicates&lt;CTT&gt; &gt;</span><br style="font-family: 'Courier New',Courier,monospace;">
      <span style="font-family: 'Courier New',Courier,monospace;">class
voronoi_builder;</span><br>
      <br>
      <span style="font-family: 'Courier New',Courier,monospace;">T</span></font>
- specifies the coordinate type of the input geometries (points and
segments).<br>
      <font face="Courier New"> <span style="font-family: 'Courier New',Courier,monospace;">CTT</span></font>
- defines the input/output coordinate type traits used by the Voronoi
predicates (VP).<br>
      <font face="Courier New"> <span style="font-family: 'Courier New',Courier,monospace;">VP</span></font>
- the predicate kernel, that contains robust and
efficient predicates and functors.<br>
The Voronoi builder data structure is ready to use from the box with
32-bit signed integer input coordinate type. The user may extend the
input coordinate range to the other integer types (e.g. 64-bit
integer), however this will also require manual setup of the
coordinate type traits. Default predicate kernel provides correct and
efficient predicates as long as types
defined by the coordinate type traits satisfy the requirements
explained at the end of this page. In case
those
requirements are not satisfied,<font face="Courier New"><span style="font-family: 'Courier New',Courier,monospace;"></span></font>
the proper predicate kernel
implementation is required.<span style="font-family: Courier New,Courier,monospace;"></span><br>
      <h2>Member Functions</h2>
      <table style="text-align: left; width: 100%;" border="1" cellpadding="2" cellspacing="2">
        <tbody>
          <tr>
            <td style="font-family: 'Courier New',Courier,monospace;"><span style="font-weight: bold;">voronoi_builder</span>()</td>
            <td width="693">Default
constructor.</td>
          </tr>
          <tr>
            <td><span style="font-family: Courier New,Courier,monospace;">size_t <span style="font-weight: bold;">insert_point</span>(const int_type&amp; x,<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
const int_type&amp; y)</span> </td>
            <td>Inserts a point object with
the specified coordinates into the Voronoi builder.<br>
Returns index of the inserted point. </td>
          </tr>
          <tr>
            <td><span style="font-family: Courier New,Courier,monospace;">size_t <span style="font-weight: bold;">insert_segment</span>(const int_type&amp;
x1,<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
const int_type&amp; y1,<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
const int_type&amp; x2,<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
const int_type&amp; y2)</span> </td>
            <td>Inserts a segment object
with the specified coordinates into the Voronoi builder.<br>
Returns index of the inserted segment. </td>
          </tr>
          <tr>
            <td style="font-family: 'Courier New',Courier,monospace;">template
&lt;typename OUTPUT&gt;<br>
void <span style="font-weight: bold;">construct</span>(OUTPUT* output)
            </td>
            <td width="693">Runs the sweepline
algorithm over the set of inserted geometries and generates site and
circle events to the OUTPUT data structure. It's the responsibility of
the
output data structure to process them.<br>
Complexity: O(N * log N), where N is the total number of input points and segments.<br>
 </td>
          </tr>
          <tr>
            <td style="font-family: 'Courier New',Courier,monospace;">void
            <span style="font-weight: bold;">clear</span>() </td>
            <td width="693">Clears the
list of the inserted geometries. Sets the input geometry index to
zero. </td>
          </tr>
        </tbody>
      </table>
      <h1>Voronoi Coordinate Type Traits</h1>
      <p>The library provides the default coordinate type traits
specialization for the
32-bit signed integer type:</p>
      <font style="font-family: 'Courier New',Courier,monospace;" face="Courier New">
      <p>template &lt;typename T&gt;<br>
struct voronoi_ctype_traits;<br>
      <br>
template &lt;&gt;<br>
struct voronoi_ctype_traits&lt;int32&gt; {<br>
&nbsp;&nbsp;&nbsp; typedef int32 int_type;<br>
&nbsp;&nbsp;&nbsp; typedef int64 int_x2_type;<br>
&nbsp;&nbsp;&nbsp; typedef uint64 uint_x2_type;<br>
&nbsp;&nbsp;&nbsp; typedef extended_int&lt;128&gt; big_int_type;<br>
&nbsp;&nbsp;&nbsp; typedef fpt64 fpt_type;<br>
&nbsp;&nbsp;&nbsp; typedef extended_exponent_fpt&lt;fpt_type&gt;
efpt_type;<br>
&nbsp;&nbsp;&nbsp; typedef ulp_comparison&lt;fpt_type&gt; ulp_cmp_type;<br>
&nbsp;&nbsp;&nbsp; typedef type_converter_fpt to_fpt_converter_type;<br>
&nbsp;&nbsp;&nbsp; typedef type_converter_efpt to_efpt_converter_type;<br>
};</p>
      </font> One
of the most important features of the library is that Voronoi builder
output geometries are constructed within defined relative error (64
machine epsilons). That means, the more mantissa bits the user provided
fpt_type has, the better precision of the output geometries will be. In
order for the user defined traits to be consistent with the default
Voronoi builder predicate kernel user should define the following set
of traits (the assumption is made that input geometries have
X-bit signed integer coordinate type):<br>
      <br>
      <table style="text-align: left; width: 100%;" border="1" cellpadding="2" cellspacing="2">
        <tbody>
          <tr>
            <td style="font-family: 'Courier New',Courier,monospace; font-weight: bold;">int_type
            </td>
            <td style="vertical-align: top;">At least X-bit signed
integer type. </td>
          </tr>
          <tr>
            <td style="vertical-align: top; font-family: 'Courier New',Courier,monospace; font-weight: bold;">int_x2_type
            </td>
            <td style="vertical-align: top;">At least 2X-bit signed
integer type. </td>
          </tr>
          <tr>
            <td style="vertical-align: top; font-family: 'Courier New',Courier,monospace; font-weight: bold;">uint_x2_type
            </td>
            <td style="vertical-align: top;">At least 2X-bit unsigned
integer type. </td>
          </tr>
          <tr>
            <td style="vertical-align: top; font-family: 'Courier New',Courier,monospace; font-weight: bold;">big_int_type
            </td>
            <td style="vertical-align: top;">At least 8X-bit signed
integer type if input dataset contains only points.<br>
At least 64X-bit signed integer type if input dataset contains
segments. </td>
          </tr>
          <tr>
            <td style="vertical-align: top; font-family: 'Courier New',Courier,monospace; font-weight: bold;">fpt_type
            </td>
            <td style="vertical-align: top;">IEEE-754 floating point
type, with mantissa at least (X+16) bits and exponent able to handle
32X-bit unsigned integer type. </td>
          </tr>
          <tr>
            <td style="vertical-align: top; font-family: 'Courier New',Courier,monospace; font-weight: bold;">efpt_type
            </td>
            <td style="vertical-align: top;">IEEE-754 floating point
type, with mantissa at least (X+16) bits and exponent able to handle
64X-bit unsigned integer type. </td>
          </tr>
          <tr>
            <td style="vertical-align: top; font-family: 'Courier New',Courier,monospace; font-weight: bold;">ulp_cmp_type
            </td>
            <td style="vertical-align: top;">Ulp comparison structure,
that checks if two fpt_type values are within the given ulp range
(relative error range). </td>
          </tr>
          <tr>
            <td style="vertical-align: top; font-family: 'Courier New',Courier,monospace; font-weight: bold;">to_fpt_converter_type
            </td>
            <td style="vertical-align: top;">Type converter structure,
that converts any of the integer types above plus efpt_type to the
fpt_type using operator(). </td>
          </tr>
          <tr>
            <td style="vertical-align: top; font-family: 'Courier New',Courier,monospace; font-weight: bold;">to_efpt_converter_type
            </td>
            <td style="vertical-align: top;">Type converter structure,
that converts any of the integer types above to the efpt_type using
operator(). </td>
          </tr>
        </tbody>
      </table>
      <p>Notes:</p>
      <ul>
        <li>Four different integer types are used (instead of a single
big_int_type) to slightly improve algorithm performance and memory
usage.</li>
        <li>As the maximum required size of the big_int_type is known
in advance, it's possible to use fixed, stack allocated, multiprecision
integer type.</li>
        <li>Two separate floating-point types are defined, because for
the input geometries
with
32-bit signed integer coordinates, double type won't be able to handle
2048-bit (64 chunks of 32 bits each) integers, as they will
overflow its exponent. On the
gcc compiler it's possible to use 80-bit long doubles for both fpt
types, however this is not supported by the MSVC compiler.</li>
        <li>efpt_type and to_efpt_converter_type are not used to
construct the Voronoi diagram of a set of points (mock implementation
will work).</li>
        <li>For an example of the user defined coordinate type traits
check the <a href="voronoi_advanced_tutorial.htm">advanced Voronoi
tutorial</a>.</li>
      </ul>
      </td>
    </tr>
    <tr>
      <td style="background-color: rgb(238, 238, 238);" nowrap="1">&nbsp;</td>
      <td style="padding-left: 10px; padding-right: 10px; padding-bottom: 10px;" valign="top" width="100%">
      <table class="docinfo" id="table2" frame="void" rules="none">
        <colgroup> <col class="docinfo-name"><col class="docinfo-content"> </colgroup> <tbody valign="top">
          <tr>
            <th class="docinfo-name">Copyright:</th>
            <td>Copyright © Andrii Sydorchuk 2010-2013.</td>
          </tr>
          <tr class="field">
            <th class="docinfo-name">License:</th>
            <td class="field-body">Distributed under the Boost Software
License, Version 1.0. (See accompanying file <tt class="literal"><span class="pre">LICENSE_1_0.txt</span></tt> or copy at <a class="reference" target="_top" href="http://www.boost.org/LICENSE_1_0.txt">
http://www.boost.org/LICENSE_1_0.txt</a>)</td>
          </tr>
        </tbody>
      </table>
      </td>
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