there.</para>
</note>
</sect2>
+
+ <sect2 id="DE-9IM">
+ <title>Dimensionally Extended 9 Intersection Model (DE-9IM)</title>
+
+ <para>It is sometimes the case that the typical spatial predicates
+ (ST_Contains, ST_Crosses, ST_Intersects, ST_Touches, ...) are
+ insufficient in and of themselves to adequately provide that desired
+ spatial filter.</para>
+
+ <informaltable frame="none" border="0">
+ <tgroup cols="1">
+ <tbody>
+ <row>
+ <entry><para><informalfigure float="1" floatstyle="left">
+ <graphic align="left" fileref="images/de9im01.png" />
+ </informalfigure></para><para>For example, consider a linear
+ dataset representing a road network. It may be the task of a
+ GIS analyst to identify all road segments that cross
+ eachother, not at a point, but on a line, perhaps invalidating
+ some business rule. In this case, ST_Crosses does not
+ adequately provide the necessary spatial filter since, for
+ linear features, it returns <varname>true</varname> only where
+ they cross at a point.</para> <para>One two-step solution
+ might be to first perform the actual intersection
+ (ST_Intersection) of pairs of road segments that spatially
+ intersect (ST_Intersects), and then compare the intersection's
+ ST_GeometryType with '<varname>LINESTRING</varname>' (properly
+ dealing with cases that return
+ <varname>GEOMETRYCOLLECTION</varname>s of
+ <varname>[MULTI]POINT</varname>s,
+ <varname>[MULTI]LINESTRING</varname>s, etc.).</para> <para>A
+ more elegant / faster solution may indeed be
+ desireable.</para></entry>
+ </row>
+ </tbody>
+ </tgroup>
+ </informaltable>
+
+ <informaltable frame="none" border="0">
+ <tgroup cols="1">
+ <tbody>
+ <row>
+ <entry><para> <informalfigure float="1" floatstyle="right">
+ <graphic align="right" fileref="images/de9im02.png" />
+ </informalfigure></para> <para>A second [theoretical]
+ example may be that of a GIS analyst trying to locate all
+ wharfs or docks that intersect a lake's boundary on a line and
+ where only one end of the wharf is up on shore. In other
+ words, where a wharf is within, but not completely within a
+ lake, intersecting the boundary of a lake on a line, and where
+ the wharf's endpoints are both completely within and on the
+ boundary of the lake. The analyst may need to use a
+ combination of spatial predicates to isolate the sought after
+ features:</para> <itemizedlist>
+ <listitem>
+ <para>ST_Contains(lake, wharf) = TRUE</para>
+ </listitem>
+
+ <listitem>
+ <para>ST_ContainsProperly(lake, wharf) = FALSE</para>
+ </listitem>
+
+ <listitem>
+ <para>ST_GeometryType(ST_Intersection(wharf, lake)) =
+ 'LINESTRING'</para>
+ </listitem>
+
+ <listitem>
+ <para>ST_NumGeometries(ST_Multi(ST_Intersection(ST_Boundary(wharf),
+ ST_Boundary(lake)))) = 1</para>
+
+ <para>... (needless to say, this could get quite
+ complicated)</para>
+ </listitem>
+ </itemizedlist></entry>
+ </row>
+ </tbody>
+ </tgroup>
+ </informaltable>
+
+ <para>So enters the Dimensionally Extended 9 Intersection Model, or
+ DE-9IM for short.</para>
+
+ <sect3>
+ <title>Theory</title>
+
+ <para>According to the <ulink
+ url="http://www.opengeospatial.org/standards/sfs">OpenGIS Simple
+ Features Implementation Specification for SQL</ulink>, "the basic
+ approach to comparing two geometries is to make pair-wise tests of
+ the intersections between the Interiors, Boundaries and Exteriors of
+ the two geometries and to classify the relationship between the two
+ geometries based on the entries in the resulting 'intersection'
+ matrix."</para>
+
+ <glosslist>
+ <glossentry>
+ <glossterm>Boundary</glossterm>
+
+ <glossdef>
+ <para>The boundary of a geometry is the set of geometries of
+ the next lower dimension. For <varname>POINT</varname>s, which
+ have a dimension of 0, the boundary is the empty set. The
+ boundary of a <varname>LINESTRING</varname> are the two
+ endpoints. For <varname>POLYGON</varname>s, the boundary is
+ the linework that make up the exterior and interior
+ rings.</para>
+ </glossdef>
+ </glossentry>
+
+ <glossentry>
+ <glossterm>Interior</glossterm>
+
+ <glossdef>
+ <para>The interior of a geometry are those points of a
+ geometry that are left when the boundary is removed. For
+ <varname>POINT</varname>s, the interior is the
+ <varname>POINT</varname> itself. The interior of a
+ <varname>LINESTRING</varname> are the set of real points
+ between the endpoints. For <varname>POLYGON</varname>s, the
+ interior is the areal surface inside the polygon.</para>
+ </glossdef>
+ </glossentry>
+
+ <glossentry>
+ <glossterm>Exterior</glossterm>
+
+ <glossdef>
+ <para>The exterior of a geometry is the universe, an areal
+ surface, not on the interior or boundary of the
+ geometry.</para>
+ </glossdef>
+ </glossentry>
+ </glosslist>
+
+ <para>Given geometry <emphasis>a</emphasis>, where the
+ <emphasis>I(a)</emphasis>, <emphasis>B(a)</emphasis>, and
+ <emphasis>E(a)</emphasis> are the <emphasis>Interior</emphasis>,
+ <emphasis>Boundary</emphasis>, and <emphasis>Exterior</emphasis> of
+ a, the mathematical representation of the matrix is:</para>
+
+ <informaltable tabstyle="styledtable">
+ <tgroup align="center" cols="4">
+ <thead>
+ <row>
+ <entry></entry>
+
+ <entry><emphasis role="bold">Interior</emphasis></entry>
+
+ <entry><emphasis role="bold">Boundary</emphasis></entry>
+
+ <entry><emphasis role="bold">Exterior</emphasis></entry>
+ </row>
+ </thead>
+
+ <tbody>
+ <row>
+ <entry><emphasis role="bold">Interior</emphasis></entry>
+
+ <entry><emphasis><inlineequation>
+ <mml:math display="block"
+ xmlns:mml="http://www.w3.org/1998/Math/MathML">
+ <mml:mrow>
+ <mml:mtext mathvariant="italic">dim(
+ I(a)</mml:mtext>
+
+ <mml:mo>∩</mml:mo>
+
+ <mml:mtext mathvariant="italic">I(b) )</mml:mtext>
+ </mml:mrow>
+ </mml:math>
+ </inlineequation></emphasis></entry>
+
+ <entry><emphasis><inlineequation>
+ <mml:math display="block"
+ xmlns:mml="http://www.w3.org/1998/Math/MathML">
+ <mml:mrow>
+ <mml:mtext mathvariant="italic">dim(
+ I(a)</mml:mtext>
+
+ <mml:mo>∩</mml:mo>
+
+ <mml:mtext mathvariant="italic">B(b) )</mml:mtext>
+ </mml:mrow>
+ </mml:math>
+ </inlineequation></emphasis></entry>
+
+ <entry><emphasis><inlineequation>
+ <mml:math display="block"
+ xmlns:mml="http://www.w3.org/1998/Math/MathML">
+ <mml:mrow>
+ <mml:mtext mathvariant="italic">dim(
+ I(a)</mml:mtext>
+
+ <mml:mo>∩</mml:mo>
+
+ <mml:mtext mathvariant="italic">E(b) )</mml:mtext>
+ </mml:mrow>
+ </mml:math>
+ </inlineequation></emphasis></entry>
+ </row>
+
+ <row>
+ <entry><emphasis role="bold">Boundary</emphasis></entry>
+
+ <entry><emphasis><inlineequation>
+ <mml:math display="block"
+ xmlns:mml="http://www.w3.org/1998/Math/MathML">
+ <mml:mrow>
+ <mml:mtext mathvariant="italic">dim(
+ B(a)</mml:mtext>
+
+ <mml:mo>∩</mml:mo>
+
+ <mml:mtext mathvariant="italic">I(b) )</mml:mtext>
+ </mml:mrow>
+ </mml:math>
+ </inlineequation></emphasis></entry>
+
+ <entry><emphasis><inlineequation>
+ <mml:math display="block"
+ xmlns:mml="http://www.w3.org/1998/Math/MathML">
+ <mml:mrow>
+ <mml:mtext mathvariant="italic">dim(
+ B(a)</mml:mtext>
+
+ <mml:mo>∩</mml:mo>
+
+ <mml:mtext mathvariant="italic">B(b) )</mml:mtext>
+ </mml:mrow>
+ </mml:math>
+ </inlineequation></emphasis></entry>
+
+ <entry><emphasis><inlineequation>
+ <mml:math display="block"
+ xmlns:mml="http://www.w3.org/1998/Math/MathML">
+ <mml:mrow>
+ <mml:mtext mathvariant="italic">dim(
+ B(a)</mml:mtext>
+
+ <mml:mo>∩</mml:mo>
+
+ <mml:mtext mathvariant="italic">E(b) )</mml:mtext>
+ </mml:mrow>
+ </mml:math>
+ </inlineequation></emphasis></entry>
+ </row>
+
+ <row>
+ <entry><emphasis role="bold">Exterior</emphasis></entry>
+
+ <entry><emphasis><inlineequation>
+ <mml:math display="block"
+ xmlns:mml="http://www.w3.org/1998/Math/MathML">
+ <mml:mrow>
+ <mml:mtext mathvariant="italic">dim(
+ E(a)</mml:mtext>
+
+ <mml:mo>∩</mml:mo>
+
+ <mml:mtext mathvariant="italic">I(b) )</mml:mtext>
+ </mml:mrow>
+ </mml:math>
+ </inlineequation></emphasis></entry>
+
+ <entry><emphasis><inlineequation>
+ <mml:math display="block"
+ xmlns:mml="http://www.w3.org/1998/Math/MathML">
+ <mml:mrow>
+ <mml:mtext mathvariant="italic">dim(
+ E(a)</mml:mtext>
+
+ <mml:mo>∩</mml:mo>
+
+ <mml:mtext mathvariant="italic">B(b) )</mml:mtext>
+ </mml:mrow>
+ </mml:math>
+ </inlineequation></emphasis></entry>
+
+ <entry><emphasis><inlineequation>
+ <mml:math display="block"
+ xmlns:mml="http://www.w3.org/1998/Math/MathML">
+ <mml:mrow>
+ <mml:mtext mathvariant="italic">dim(
+ E(a)</mml:mtext>
+
+ <mml:mo>∩</mml:mo>
+
+ <mml:mtext mathvariant="italic">E(b) )</mml:mtext>
+ </mml:mrow>
+ </mml:math>
+ </inlineequation></emphasis></entry>
+ </row>
+ </tbody>
+ </tgroup>
+ </informaltable>
+
+ <para>Where <emphasis>dim(a)</emphasis> is the dimension of
+ <emphasis>a</emphasis> as specified by
+ <literal>ST_Dimension()</literal> but has the domain of
+ <literal>{0,1,2,T,F,*}</literal></para>
+
+ <itemizedlist spacing="compact">
+ <listitem>
+ <para><literal>0</literal> => point</para>
+ </listitem>
+
+ <listitem>
+ <para><literal>1</literal> => line</para>
+ </listitem>
+
+ <listitem>
+ <para><literal>2</literal> => area</para>
+ </listitem>
+
+ <listitem>
+ <para><literal>T</literal> =>
+ <literal>{0,1,2}</literal></para>
+ </listitem>
+
+ <listitem>
+ <para><literal>F</literal> => empty set</para>
+ </listitem>
+
+ <listitem>
+ <para><literal>*</literal> => don't care</para>
+ </listitem>
+ </itemizedlist>
+
+ <para>Visually, for two overlapping polygonal geometries, this looks
+ like:</para>
+
+ <informaltable frame="none" border="0">
+ <tgroup cols="2">
+ <colspec colwidth="80pt" />
+
+ <tbody>
+ <row>
+ <entry></entry>
+
+ <entry align="center"><para><informalfigure>
+ <graphic align="center" fileref="images/de9im04.png"
+ valign="middle" />
+ </informalfigure></para></entry>
+ </row>
+
+ <row>
+ <entry align="center" valign="middle"><para><informalfigure>
+ <graphic align="center" fileref="images/de9im03.png"
+ valign="middle" />
+ </informalfigure></para></entry>
+
+ <entry><para> <informaltable tabstyle="styledtable">
+ <tgroup align="center" cols="4">
+ <thead valign="middle">
+ <row>
+ <entry></entry>
+
+ <entry><emphasis
+ role="bold">Interior</emphasis></entry>
+
+ <entry><emphasis
+ role="bold">Boundary</emphasis></entry>
+
+ <entry><emphasis
+ role="bold">Exterior</emphasis></entry>
+ </row>
+ </thead>
+
+ <tbody valign="middle">
+ <row>
+ <entry spanname="de9im_a" style=""><emphasis
+ role="bold">Interior</emphasis></entry>
+
+ <entry><para><informalfigure>
+ <graphic fileref="images/de9im05.png" />
+ </informalfigure></para><para><emphasis>dim(...) =
+ </emphasis><emphasis
+ role="bold">2</emphasis></para></entry>
+
+ <entry><para><informalfigure>
+ <graphic fileref="images/de9im06.png" />
+ </informalfigure></para><para><emphasis>dim(...) =
+ </emphasis><emphasis
+ role="bold">1</emphasis></para></entry>
+
+ <entry><para><informalfigure>
+ <graphic fileref="images/de9im07.png" />
+ </informalfigure></para><para><emphasis>dim(...) =
+ </emphasis><emphasis
+ role="bold">2</emphasis></para></entry>
+ </row>
+
+ <row>
+ <entry><emphasis
+ role="bold">Boundary</emphasis></entry>
+
+ <entry><para><informalfigure>
+ <graphic fileref="images/de9im08.png" />
+ </informalfigure></para><para><emphasis>dim(...) =
+ </emphasis><emphasis
+ role="bold">1</emphasis></para></entry>
+
+ <entry><para><informalfigure>
+ <graphic fileref="images/de9im09.png" />
+ </informalfigure></para><para><emphasis>dim(...) =
+ </emphasis><emphasis
+ role="bold">0</emphasis></para></entry>
+
+ <entry><para><informalfigure>
+ <graphic fileref="images/de9im10.png" />
+ </informalfigure></para><para><emphasis>dim(...) =
+ </emphasis><emphasis
+ role="bold">1</emphasis></para></entry>
+ </row>
+
+ <row>
+ <entry><emphasis
+ role="bold">Exterior</emphasis></entry>
+
+ <entry><para><informalfigure>
+ <graphic fileref="images/de9im11.png" />
+ </informalfigure></para><para><emphasis>dim(...) =
+ </emphasis><emphasis
+ role="bold">2</emphasis></para></entry>
+
+ <entry><para><informalfigure>
+ <graphic fileref="images/de9im12.png" />
+ </informalfigure></para><para><emphasis>dim(...) =
+ </emphasis><emphasis
+ role="bold">1</emphasis></para></entry>
+
+ <entry><para><informalfigure>
+ <graphic fileref="images/de9im13.png" />
+ </informalfigure></para><para><emphasis>dim(...) =
+ </emphasis><emphasis
+ role="bold">2</emphasis></para></entry>
+ </row>
+ </tbody>
+ </tgroup>
+ </informaltable></para></entry>
+ </row>
+ </tbody>
+ </tgroup>
+ </informaltable>
+
+ <para>Read from left to right, the dimensional matrix is
+ represented, '<emphasis role="bold">212010212</emphasis>'.</para>
+
+ <para>A relate matrix that would therefore represent our first
+ example of two lines that intersect on a line would be: '<emphasis
+ role="bold">1*1***1**</emphasis>'</para>
+
+ <programlisting>-- Identify road segments that cross on a line
+SELECT a.id
+FROM roads a, roads b
+WHERE a.id != b.id
+AND a.geom && b.geom
+AND ST_Relate(a.geom, b.geom, '1*1***1**');</programlisting>
+
+ <para>A relate matrix that represents the second example of wharfs
+ partly on the lake's shoreline would be '<emphasis
+ role="bold">102101FF2</emphasis>'</para>
+
+ <programlisting>-- Identify wharfs partly on a lake's shoreline
+SELECT a.lake_id, b.wharf_id
+FROM lakes a, wharfs b
+WHERE a.geom && b.geom
+AND ST_Relate(a.geom, b.geom, '102101FF2');</programlisting>
+ </sect3>
+ </sect2>
+
</sect1>
<sect1>
projects / postgis / commitdiff