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| <title>Boost Polygon Library: Isotropy</title> |
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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_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> |
| </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> |
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| <br> |
| <p> |
| </p><h1>Isotropy</h1> |
| |
| <p> |
| <p align="left">What is isotropy?</p> |
| <P:COLORSCHEME |
| colors="#ffffff,#000000,#808080,#000000,#bbe0e3,#333399,#009999,#99cc00" /> |
| <div class="O" style="TEXT-ALIGN: center; mso-line-spacing: '90 0 0'; mso-margin-left-alt: 216; mso-char-wrap: 1; mso-kinsoku-overflow: 1" v:shape="_x0000_s1026"> |
| <p style="TEXT-ALIGN: left"> |
| <span style="mso-bidi-font-family: Arial">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 <an <i>isotropic</i> |
| crystal></span></div> |
| <p align="left">In computational geometry things are often symmetric and |
| invariant to direction and orientation. This invariance to direction is |
| called isotropy. 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. To do this effectively we provide an |
| internally consistent set of isotropic data types to represent program |
| data that describes orientations and directions. 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. Together they create a language for |
| describing isotropic situations programmatically. 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. Isotropic programming style is particularly applicable to working |
| with points, intervals and rectangles. 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. |
| 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 align="left">The direction_1d data type has two possible states. These |
| are the positive and negative directions on a continuum such as the number line. |
| These states can be described by one of several direction_1d_enum values: |
| We make clockwise and counterclockwise winding orientation of polygons a |
| direction 1d value instead of providing a separate winding_orientation data |
| type. This is because winding orientation can be thought of as positive |
| and negative directions in a 1d (although cyclic) space. 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> |
| |
| LEFT = 0, RIGHT = 1,<br> |
| |
| CLOCKWISE = 0, COUNTERCLOCKWISE = 1,<br> |
| |
| REVERSE = 0, FORWARD = 1,<br> |
| |
| NEGATIVE = 0, POSITIVE = 1 };</font></p> |
| <h2>Member Functions</h2> |
| <table border="1" width="100%" id="table1"> |
| <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& 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& 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& 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& <b>operator=</b>(const direction_1d dir)</font></td> |
| <td>Assignment</td> |
| </tr> |
| <tr> |
| <td width="586"> |
| <font face="Courier New">direction_1d& <b>operator==</b>(const direction_1d dir) |
| const</font></td> |
| <td>Equivalence</td> |
| </tr> |
| <tr> |
| <td width="586"> |
| <font face="Courier New">direction_1d& <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. |
| Auto-cast to int is disallowed for type safety reasons.</td> |
| </tr> |
| <tr> |
| <td width="586"> |
| <font face="Courier New">direction_1d& <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> |
| </table> |
| <h2>orientation_2d</h2> |
| |
| <p> |
| <p align="left">The orientation_2d data type has two possible states. |
| These are the horizontal and vertical axis of a 2d Cartesian coordinate system. |
| 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 border="1" width="100%" id="table2"> |
| <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& 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& 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& <b>operator=</b>(const orientation_2d |
| o)</font></td> |
| <td>Assignment</td> |
| </tr> |
| <tr> |
| <td width="586"> |
| <font face="Courier New">orientation_2d& <b>operator==</b>(const orientation_2d |
| o) const</font></td> |
| <td>Equivalence</td> |
| </tr> |
| <tr> |
| <td width="586"> |
| <font face="Courier New">orientation_2d& <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. |
| Auto-cast to int is disallowed for type safety reasons</td> |
| </tr> |
| <tr> |
| <td width="586"> |
| <font face="Courier New">orientation_2d& <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> |
| </table> |
| <h2>direction_2d</h2> |
| |
| <p> |
| <p align="left">The direction_2d data type has four possible states. These |
| are the cardinal directions of the 2D Cartesian coordinate system. |
| 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 border="1" width="100%" id="table3"> |
| <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& that)</font></td> |
| <td>Copy construct.</td> |
| </tr> |
| <tr> |
| <td width="586"> |
| <font face="Courier New">direction_1d& <b>operator=</b>(const direction_2d dir)</font></td> |
| <td>Assignment</td> |
| </tr> |
| <tr> |
| <td width="586"> |
| <font face="Courier New">direction_1d& <b>operator==</b>(const direction_2d dir) |
| const</font></td> |
| <td>Equivalence</td> |
| </tr> |
| <tr> |
| <td width="586"> |
| <font face="Courier New">direction_1d& <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. |
| Auto-cast to int is disallowed for type safety reasons.</td> |
| </tr> |
| <tr> |
| <td width="586"> |
| <font face="Courier New">direction_2d& <b>backward</b>()</font></td> |
| <td>Inverts direction.</td> |
| </tr> |
| <tr> |
| <td width="586"> |
| <font face="Courier New">direction_2d& <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& <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& <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> |
| </table> |
| <h2>orientation_3d</h2> |
| |
| <p> |
| <p align="left">The orientation_3d data type has three possible states. |
| These are the horizontal, vertical and proximal (x, y, z) axis of a 3d Cartesian |
| coordinate system. 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 border="1" width="100%" id="table6"> |
| <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& 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& 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& 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& 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& that)</font></td> |
| <td>Construct from constant value</td> |
| </tr> |
| <tr> |
| <td width="586"> |
| <font face="Courier New">orientation_3d& <b>operator=</b>(const orientation_3d |
| o)</font></td> |
| <td>Assignment</td> |
| </tr> |
| <tr> |
| <td width="586"> |
| <font face="Courier New">orientation_3d& <b>operator==</b>(const orientation_3d |
| o) const</font></td> |
| <td>Equivalence</td> |
| </tr> |
| <tr> |
| <td width="586"> |
| <font face="Courier New">orientation_3d& <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. |
| 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> |
| </table> |
| <h2>direction_3d</h2> |
| |
| <p> |
| <p align="left">The direction_3d data type has six possible states. These |
| are the cardinal directions of the 3D Cartesian coordinate system. |
| 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 border="1" width="100%" id="table5"> |
| <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& 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& <b>operator=</b>(const direction_3d dir)</font></td> |
| <td>Assignment</td> |
| </tr> |
| <tr> |
| <td width="586"> |
| <font face="Courier New">direction_3d& <b>operator==</b>(const direction_3d dir) |
| const</font></td> |
| <td>Equivalence</td> |
| </tr> |
| <tr> |
| <td width="586"> |
| <font face="Courier New">direction_2d& <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. |
| Auto-cast to int is disallowed for type safety reasons.</td> |
| </tr> |
| <tr> |
| <td width="586"> |
| <font face="Courier New">direction_1d& <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> |
| </table> |
| <tr> |
| <td style="background-color: rgb(238, 238, 238);" nowrap="1" valign="top"> |
| </td> |
| <td style="padding-left: 10px; padding-right: 10px; padding-bottom: 10px;" valign="top" width="100%"> |
| |
| |
| <table class="docinfo" rules="none" frame="void" id="table7"> |
| <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> |
| </table> |
| |
| </html> |
