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  <title>Boost Polygon Library: Isotropy</title>
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      <div style="padding: 5px;" align="center"> <img
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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>Isotropy</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<br />
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        <li><a href="voronoi_diagram.htm">Voronoi Diagram</a></li>
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      <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>
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      <p>
      </p>
      <h1>Isotropy</h1>
      <p> </p>
      <p align="left">What is isotropy?</p>
      <p:colorscheme
 colors="#ffffff,#000000,#808080,#000000,#bbe0e3,#333399,#009999,#99cc00">
      </p:colorscheme>
      <div class="O" style="text-align: center;" v:shape="_x0000_s1026">
      <p style="text-align: left;"> <span style="">Isotropy -
Function: <i>adjective</i> Etymology: International Scientific
Vocabulary<br />
      <b>:</b> exhibiting properties (as velocity of light
transmission) with the same values when measured along axes in all
directions &lt;an <i>isotropic</i> crystal&gt;</span></p>
      </div>
      <p align="left">In computational geometry things are often
symmetric and invariant to direction and orientation.&nbsp; This
invariance to direction is called isotropy.&nbsp; In such situations it
is convenient to parameterize direction or orientation and write code
that is invariant to the direction or orientation in which it is
applied.&nbsp; To do this effectively we provide an internally
consistent set of isotropic data types to represent program data that
describes orientations and directions.&nbsp; These data types are:</p>
      <ul>
        <li>direction_1d - has one of the following 2 states: LOW, HIGH
        </li>
        <li>orientation_2d - has one of the following 2 states:
HORIZONTAL, VERTICAL</li>
        <li>direction_2d - has one of the following 4 states: WEST,
EAST, SOUTH, NORTH</li>
        <li>orientation_3d - has one of the following 3 states:
HORIZONTAL, VERTICAL, PROXIMAL</li>
        <li>direction_3d - has one of the following 6 states: WEST,
EAST, SOUTH, NORTH, DOWN, UP</li>
      </ul>
      <p align="left">The isotropic types create a system and interact
with each other in various ways, such as casting.&nbsp; Together they
create a language for describing isotropic situations
programmatically.&nbsp; For instance, to get the positive direction_2d
from an orientation_2d you would call a member function of
orientation_2d and pass a direction_1d:</p>
      <p align="left"><font face="Courier New">orientation_2d orient =
HORIZONTAL;<br />
direction_2d dir = orient.get_direction(direction_1d(HIGH));<br />
assert(dir == EAST);</font></p>
      <p align="left">The motivation for providing isotropic data types
is to encourage programming at a higher level of abstraction where
program behavior is controlled by program data passed into function
calls rather than flow control syntax.&nbsp; Isotropic programming
style is particularly applicable to working with points, intervals and
rectangles.&nbsp; Often times the implementation of such logic is
identical when implemented for the x or y coordinates, except that the
names of functions and data members are changed in a mechanical way
leading to code duplication and bloat that results in copy-paste
programming errors and maintenance problems where changes made to a
given code block relating to x coordiantes are not duplicated to the
code block that refers to y.&nbsp; Isotropy therefore represents an
opportunity to refactor and improve the quality of low level geometry
code especially in regard to inter-relating coordinates, points,
intervals and rectangles.</p>
      <h2>direction_1d</h2>
      <p> </p>
      <p align="left">The direction_1d data type has two possible
states.&nbsp; These are the positive and negative directions on a
continuum such as the number line.&nbsp;&nbsp; These states can be
described by one of several direction_1d_enum values:&nbsp; We make
clockwise and counterclockwise winding orientation of polygons a
direction 1d value instead of providing a separate winding_orientation
data type.&nbsp; This is because winding orientation can be thought of
as positive and negative directions in a 1d (although cyclic)
space.&nbsp; We assign counterclockwise to be the positive direction of
travel in the 1d cyclic space to conform with the mathematical
convention frequently described as the "right hand rule" which assigns
positive normal value to counterclockwise and negative normal value to
clockwise as well as the common convention that counterclockwise
polygon winding corresponds to positive polygonal regions where as
clockwise polygon winding corresponds to hole (negative) polygonal
regions.</p>
      <p align="left"><font face="Courier New">enum direction_1d_enum
{LOW = 0, HIGH = 1,<br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
LEFT = 0, RIGHT = 1,<br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
CLOCKWISE = 0, COUNTERCLOCKWISE = 1,<br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
REVERSE = 0, FORWARD = 1,<br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
NEGATIVE = 0, POSITIVE = 1 };</font></p>
      <h2>Member Functions</h2>
      <table id="table1" border="1" width="100%">
        <tbody>
          <tr>
            <td width="586"><font face="Courier New"><b>direction_1d</b>(direction_1d_enum
val = LOW)</font></td>
            <td>Constructor defaults to LOW. </td>
          </tr>
          <tr>
            <td width="586"><b><font face="Courier New">direction_1d</font></b><font
 face="Courier New">(const direction_1d&amp; that)</font></td>
            <td>Copy construct.</td>
          </tr>
          <tr>
            <td width="586"><b><font face="Courier New">direction_1d</font></b><font
 face="Courier New">(const direction_2d&amp; that)</font></td>
            <td>Down cast direction_2d, extracting out whether positive
or negative</td>
          </tr>
          <tr>
            <td width="586"><b><font face="Courier New">direction_1d</font></b><font
 face="Courier New">(const direction_3d&amp; that)</font></td>
            <td>Down cast direction_3d, extracting out whether positive
or negative</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">direction_1d&amp; <b>operator=</b>(const
direction_1d dir)</font></td>
            <td>Assignment</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">direction_1d&amp; <b>operator==</b>(const
direction_1d dir) const</font></td>
            <td>Equivalence</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">direction_1d&amp; <b>operator!=</b>(const
direction_1d dir) const</font></td>
            <td>Inequivalence</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">unsigned int <b>to_int</b>()
const</font></td>
            <td>Convert to the integer enum value of current state to
use as index.&nbsp; Auto-cast to int is disallowed for type safety
reasons.</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">direction_1d&amp; <b>backward</b>()</font></td>
            <td>Inverts direction.</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">int <b>get_sign</b>()
const</font></td>
            <td>Returns positive 1 if positive direction and negative
one if negative direction.</td>
          </tr>
        </tbody>
      </table>
      <h2>orientation_2d</h2>
      <p> </p>
      <p align="left">The orientation_2d data type has two possible
states.&nbsp; These are the horizontal and vertical axis of a 2d
Cartesian coordinate system.&nbsp;&nbsp; These states can be described
by one of the two orientation_2d_enum values:</p>
      <p align="left"><font face="Courier New">enum orientation_2d_enum
{ HORIZONTAL = 0, VERTICAL = 1 };</font></p>
      <h2>Member Functions</h2>
      <table id="table2" border="1" width="100%">
        <tbody>
          <tr>
            <td width="586"><b><font face="Courier New">orientation_2</font></b><font
 face="Courier New"><b>d</b>(orientation_2d_enum val = HORIZONTAL)</font></td>
            <td>Constructor defaults to HORIZONTAL. </td>
          </tr>
          <tr>
            <td width="586"><b><font face="Courier New">orientation_2</font></b><font
 face="Courier New"><b>d</b>(const orientation_2d&amp; that)</font></td>
            <td>Copy construct.</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">explicit </font><b>
            <font face="Courier New">orientation_2</font></b><font
 face="Courier New"><b>d</b>(const direction_2d&amp; that)</font></td>
            <td>Down cast direction_2d, extracting out whether
horizontal or vertical direction type</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">orientation_2d&amp;
            <b>operator=</b>(const orientation_2d o)</font></td>
            <td>Assignment</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">orientation_2d&amp;
            <b>operator==</b>(const orientation_2d o) const</font></td>
            <td>Equivalence</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">orientation_2d&amp;
            <b>operator!=</b>(const orientation_2d o) const</font></td>
            <td>Inequivalence</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">unsigned int <b>to_int</b>()
const</font></td>
            <td>Convert to the integer enum value of current state to
use as index.&nbsp; Auto-cast to int is disallowed for type safety
reasons</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">orientation_2d&amp;
            <b>turn_90</b>()</font></td>
            <td>Change to orthogonal orientation</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">int <b>get_perpendicular</b>()
const</font></td>
            <td>Returns orthogonal orientation</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">int <b>get_direction</b>(direction_1d
dir) const</font></td>
            <td>Returns the positive or negative direction_2d depending
on the value of dir</td>
          </tr>
        </tbody>
      </table>
      <h2>direction_2d</h2>
      <p> </p>
      <p align="left">The direction_2d data type has four possible
states.&nbsp; These are the cardinal directions of the 2D Cartesian
coordinate system.&nbsp;&nbsp; These states can be described by one of
several direction_2d_enum values:</p>
      <p align="left"><font face="Courier New">enum direction_2d_enum {
WEST = 0, EAST = 1, SOUTH = 2, NORTH = 3 };</font></p>
      <h2>Member Functions</h2>
      <table id="table3" border="1" width="100%">
        <tbody>
          <tr>
            <td width="586"><font face="Courier New"><b>direction_2d</b>(direction_2d_enum
val = WEST)</font></td>
            <td>Constructor defaults to WEST. </td>
          </tr>
          <tr>
            <td width="586"><b><font face="Courier New">direction_2d</font></b><font
 face="Courier New">(const direction_2d&amp; that)</font></td>
            <td>Copy construct.</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">direction_1d&amp; <b>operator=</b>(const
direction_2d dir)</font></td>
            <td>Assignment</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">direction_1d&amp; <b>operator==</b>(const
direction_2d dir) const</font></td>
            <td>Equivalence</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">direction_1d&amp; <b>operator!=</b>(const
direction_2d dir) const</font></td>
            <td>Inequivalence</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">unsigned int <b>to_int</b>()
const</font></td>
            <td>Convert to the integer enum value of current state to
use as index.&nbsp; Auto-cast to int is disallowed for type safety
reasons.</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">direction_2d&amp; <b>backward</b>()</font></td>
            <td>Inverts direction.</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">direction_2d&amp; <b>turn</b>(direction_1d
dir)</font></td>
            <td>Changes to direction_2d to the left if dir is LOW, to
the right if dir is HIGH</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">direction_2d&amp; <b>left</b>()</font></td>
            <td>Changes to the direction_2d to the left</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">direction_2d&amp; <b>right</b>()</font></td>
            <td>Changes to the direction_2d to the right</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">int <b>is_positive</b>()
const</font></td>
            <td>Returns true if EAST or NORTH</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">int <b>is_negative</b>()
const</font></td>
            <td>Returns true if WEST or SOUTH</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">int <b>get_sign</b>()
const</font></td>
            <td>Returns positive 1 if positive direction and negative
one if negative direction.</td>
          </tr>
        </tbody>
      </table>
      <h2>orientation_3d</h2>
      <p> </p>
      <p align="left">The orientation_3d data type has three possible
states.&nbsp; These are the horizontal, vertical and proximal (x, y, z)
axis of a 3d Cartesian coordinate system.&nbsp;&nbsp; These states can
be described by one of the orientation_2d_enum values or by the
orientation_3d_enum value:</p>
      <p align="left"><font face="Courier New">enum orientation_3d_enum
{ PROXIMAL = 2 };</font></p>
      <h2>Member Functions</h2>
      <table id="table6" border="1" width="100%">
        <tbody>
          <tr>
            <td width="586"><b><font face="Courier New">orientation_3</font></b><font
 face="Courier New"><b>d</b>(orientation_2d_enum val = HORIZONTAL)</font></td>
            <td>Constructor defaults to HORIZONTAL. </td>
          </tr>
          <tr>
            <td width="586"><b><font face="Courier New">orientation_3</font></b><font
 face="Courier New"><b>d</b>(const orientation_3d&amp; that)</font></td>
            <td>Copy construct.</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">explicit </font><b>
            <font face="Courier New">orientation_3</font></b><font
 face="Courier New"><b>d</b>(const direction_2d&amp; that)</font></td>
            <td>Extract out the orientation of the direction</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">explicit </font><b>
            <font face="Courier New">orientation_3</font></b><font
 face="Courier New"><b>d</b>(const direction_3d&amp; that)</font></td>
            <td>Extract out the orientation of the direction</td>
          </tr>
          <tr>
            <td width="586"><b><font face="Courier New">orientation_3</font></b><font
 face="Courier New"><b>d</b>(const orientation_2d&amp; that)</font></td>
            <td>Up cast orientation_2d to orientation_3d.</td>
          </tr>
          <tr>
            <td width="586"><b><font face="Courier New">orientation_3</font></b><font
 face="Courier New"><b>d</b>(const orientation_3d_enum&amp; that)</font></td>
            <td>Construct from constant value</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">orientation_3d&amp;
            <b>operator=</b>(const orientation_3d o)</font></td>
            <td>Assignment</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">orientation_3d&amp;
            <b>operator==</b>(const orientation_3d o) const</font></td>
            <td>Equivalence</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">orientation_3d&amp;
            <b>operator!=</b>(const orientation_3d o) const</font></td>
            <td>Inequivalence</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">unsigned int <b>to_int</b>()
const</font></td>
            <td>Convert to the integer enum value of current state to
use as index.&nbsp; Auto-cast to int is disallowed for type safety
reasons</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">int <b>get_direction</b>(direction_1d
dir) const</font></td>
            <td>Returns the positive or negative direction_2d depending
on the value of dir</td>
          </tr>
        </tbody>
      </table>
      <h2>direction_3d</h2>
      <p> </p>
      <p align="left">The direction_3d data type has six possible
states.&nbsp; These are the cardinal directions of the 3D Cartesian
coordinate system.&nbsp;&nbsp; These states can be described by one of
the direction_2d_enum values or the direction_3d_enum values:</p>
      <p align="left"><font face="Courier New">enum direction_3d_enum {
DOWN = 4, UP = 5 };</font></p>
      <h2>Member Functions</h2>
      <table id="table5" border="1" width="100%">
        <tbody>
          <tr>
            <td width="586"><font face="Courier New"><b>direction_3d</b>(direction_2d_enum
val = WEST)</font></td>
            <td>Constructor defaults to LOW. </td>
          </tr>
          <tr>
            <td width="586"><b><font face="Courier New">direction_3d</font></b><font
 face="Courier New">(direction_3d_enum that)</font></td>
            <td>Construct from constant value</td>
          </tr>
          <tr>
            <td width="586"><b><font face="Courier New">direction_3d</font></b><font
 face="Courier New">(const direction_3d&amp; that)</font></td>
            <td>Copy construct</td>
          </tr>
          <tr>
            <td width="586"><b><font face="Courier New">direction_3d</font></b><font
 face="Courier New">(direction_2d that)</font></td>
            <td>Up cast direction_2d to direction_3d</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">direction_3d&amp; <b>operator=</b>(const
direction_3d dir)</font></td>
            <td>Assignment</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">direction_3d&amp; <b>operator==</b>(const
direction_3d dir) const</font></td>
            <td>Equivalence</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">direction_2d&amp; <b>operator!=</b>(const
direction_3d dir) const</font></td>
            <td>Inequivalence</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">unsigned int <b>to_int</b>()
const</font></td>
            <td>Convert to the integer enum value of current state to
use as index.&nbsp; Auto-cast to int is disallowed for type safety
reasons.</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">direction_1d&amp; <b>backward</b>()</font></td>
            <td>Inverts direction.</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">int <b>is_positive</b>()
const</font></td>
            <td>Returns true if direction is EAST, NORTH or UP.</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">int <b>is_negative</b>()
const</font></td>
            <td>Returns true if direction is WEST, SOUTH or DOWN</td>
          </tr>
          <tr>
            <td width="586"><font face="Courier New">int <b>get_sign</b>()
const</font></td>
            <td>Returns positive 1 if positive direction and negative
one if negative direction.</td>
          </tr>
        </tbody>
      </table>
      </td>
    </tr>
    <tr>
      <td style="background-color: rgb(238, 238, 238);" nowrap="1"
 valign="top"> &nbsp;</td>
      <td
 style="padding-left: 10px; padding-right: 10px; padding-bottom: 10px;"
 valign="top" width="100%">
      <table class="docinfo" id="table7" 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 © Intel Corporation 2008-2010.</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>
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      </td>
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