<!-- doc/src/sgml/cube.sgml -->

<sect1 id="cube" xreflabel="cube">
 <title>cube</title>

 <indexterm zone="cube">
  <primary>cube (extension)</primary>
 </indexterm>

 <para>
  This module implements a data type <type>cube</type> for
  representing multidimensional cubes.
 </para>

 <para>
  This module is considered <quote>trusted</quote>, that is, it can be
  installed by non-superusers who have <literal>CREATE</literal> privilege
  on the current database.
 </para>

 <sect2>
  <title>Syntax</title>

  <para>
   <xref linkend="cube-repr-table"/> shows the valid external
   representations for the <type>cube</type>
   type.  <replaceable>x</replaceable>, <replaceable>y</replaceable>, etc. denote
   floating-point numbers.
  </para>

  <table id="cube-repr-table">
   <title>Cube External Representations</title>
   <tgroup cols="2">
    <thead>
     <row>
      <entry>External Syntax</entry>
      <entry>Meaning</entry>
     </row>
    </thead>

    <tbody>
     <row>
      <entry><literal><replaceable>x</replaceable></literal></entry>
      <entry>A one-dimensional point
       (or, zero-length one-dimensional interval)
      </entry>
     </row>
     <row>
      <entry><literal>(<replaceable>x</replaceable>)</literal></entry>
      <entry>Same as above</entry>
     </row>
     <row>
      <entry><literal><replaceable>x1</replaceable>,<replaceable>x2</replaceable>,...,<replaceable>xn</replaceable></literal></entry>
      <entry>A point in n-dimensional space, represented internally as a
      zero-volume cube
      </entry>
     </row>
     <row>
      <entry><literal>(<replaceable>x1</replaceable>,<replaceable>x2</replaceable>,...,<replaceable>xn</replaceable>)</literal></entry>
      <entry>Same as above</entry>
     </row>
     <row>
      <entry><literal>(<replaceable>x</replaceable>),(<replaceable>y</replaceable>)</literal></entry>
      <entry>A one-dimensional interval starting at <replaceable>x</replaceable> and ending at <replaceable>y</replaceable> or vice versa; the
       order does not matter
      </entry>
     </row>
     <row>
      <entry><literal>[(<replaceable>x</replaceable>),(<replaceable>y</replaceable>)]</literal></entry>
      <entry>Same as above</entry>
     </row>
     <row>
      <entry><literal>(<replaceable>x1</replaceable>,...,<replaceable>xn</replaceable>),(<replaceable>y1</replaceable>,...,<replaceable>yn</replaceable>)</literal></entry>
      <entry>An n-dimensional cube represented by a pair of its diagonally
       opposite corners
      </entry>
     </row>
     <row>
      <entry><literal>[(<replaceable>x1</replaceable>,...,<replaceable>xn</replaceable>),(<replaceable>y1</replaceable>,...,<replaceable>yn</replaceable>)]</literal></entry>
      <entry>Same as above</entry>
     </row>
    </tbody>
   </tgroup>
  </table>

  <para>
   It does not matter which order the opposite corners of a cube are
   entered in.  The <type>cube</type> functions
   automatically swap values if needed to create a uniform
   <quote>lower left &mdash; upper right</quote> internal representation.
   When the corners coincide, <type>cube</type> stores only one corner
   along with an <quote>is point</quote> flag to avoid wasting space.
  </para>

  <para>
   White space is ignored on input, so
   <literal>[(<replaceable>x</replaceable>),(<replaceable>y</replaceable>)]</literal> is the same as
   <literal>[ ( <replaceable>x</replaceable> ), ( <replaceable>y</replaceable> ) ]</literal>.
  </para>
 </sect2>

 <sect2>
  <title>Precision</title>

  <para>
   Values are stored internally as 64-bit floating point numbers. This means
   that numbers with more than about 16 significant digits will be truncated.
  </para>
 </sect2>

 <sect2>
  <title>Usage</title>

  <para>
   <xref linkend="cube-operators-table"/> shows the specialized operators
   provided for type <type>cube</type>.
  </para>

  <table id="cube-operators-table">
   <title>Cube Operators</title>
    <tgroup cols="1">
     <thead>
      <row>
       <entry role="func_table_entry"><para role="func_signature">
        Operator
       </para>
       <para>
        Description
       </para></entry>
      </row>
     </thead>

     <tbody>
      <row>
       <entry role="func_table_entry"><para role="func_signature">
        <type>cube</type> <literal>&amp;&amp;</literal> <type>cube</type>
        <returnvalue>boolean</returnvalue>
       </para>
       <para>
        Do the cubes overlap?
       </para></entry>
      </row>

      <row>
       <entry role="func_table_entry"><para role="func_signature">
        <type>cube</type> <literal>@&gt;</literal> <type>cube</type>
        <returnvalue>boolean</returnvalue>
       </para>
       <para>
        Does the first cube contain the second?
       </para></entry>
      </row>

      <row>
       <entry role="func_table_entry"><para role="func_signature">
        <type>cube</type> <literal>&lt;@</literal> <type>cube</type>
        <returnvalue>boolean</returnvalue>
       </para>
       <para>
        Is the first cube contained in the second?
       </para></entry>
      </row>

      <row>
       <entry role="func_table_entry"><para role="func_signature">
        <type>cube</type> <literal>-&gt;</literal> <type>integer</type>
        <returnvalue>float8</returnvalue>
       </para>
       <para>
        Extracts the <parameter>n</parameter>-th coordinate of the cube
        (counting from 1).
       </para></entry>
      </row>

      <row>
       <entry role="func_table_entry"><para role="func_signature">
        <type>cube</type> <literal>~&gt;</literal> <type>integer</type>
        <returnvalue>float8</returnvalue>
       </para>
       <para>
        Extracts the <parameter>n</parameter>-th coordinate of the cube,
        counting in the following way: <parameter>n</parameter> = 2
        * <parameter>k</parameter> - 1 means lower bound
        of <parameter>k</parameter>-th dimension, <parameter>n</parameter> = 2
        * <parameter>k</parameter> means upper bound of
        <parameter>k</parameter>-th dimension.  Negative
        <parameter>n</parameter> denotes the inverse value of the corresponding
        positive coordinate.  This operator is designed for KNN-GiST support.
       </para></entry>
      </row>

      <row>
       <entry role="func_table_entry"><para role="func_signature">
        <type>cube</type> <literal>&lt;-&gt;</literal> <type>cube</type>
        <returnvalue>float8</returnvalue>
       </para>
       <para>
        Computes the Euclidean distance between the two cubes.
       </para></entry>
      </row>

      <row>
       <entry role="func_table_entry"><para role="func_signature">
        <type>cube</type> <literal>&lt;#&gt;</literal> <type>cube</type>
        <returnvalue>float8</returnvalue>
       </para>
       <para>
        Computes the taxicab (L-1 metric) distance between the two cubes.
       </para></entry>
      </row>

      <row>
       <entry role="func_table_entry"><para role="func_signature">
        <type>cube</type> <literal>&lt;=&gt;</literal> <type>cube</type>
        <returnvalue>float8</returnvalue>
       </para>
       <para>
        Computes the Chebyshev (L-inf metric) distance between the two cubes.
       </para></entry>
      </row>
     </tbody>
   </tgroup>
  </table>

  <para>
   (Before PostgreSQL 8.2, the containment operators <literal>@&gt;</literal> and <literal>&lt;@</literal> were
   respectively called <literal>@</literal> and <literal>~</literal>.  These names are still available, but are
   deprecated and will eventually be retired.  Notice that the old names
   are reversed from the convention formerly followed by the core geometric
   data types!)
  </para>

  <para>
   In addition to the above operators, the usual comparison
   operators shown in <xref linkend="functions-comparison-op-table"/> are
   available for type <type>cube</type>.  These
   operators first compare the first coordinates, and if those are equal,
   compare the second coordinates, etc.  They exist mainly to support the
   b-tree index operator class for <type>cube</type>, which can be useful for
   example if you would like a UNIQUE constraint on a <type>cube</type> column.
   Otherwise, this ordering is not of much practical use.
  </para>

  <para>
   The <filename>cube</filename> module also provides a GiST index operator class for
   <type>cube</type> values.
   A <type>cube</type> GiST index can be used to search for values using the
   <literal>=</literal>, <literal>&amp;&amp;</literal>, <literal>@&gt;</literal>, and
   <literal>&lt;@</literal> operators in <literal>WHERE</literal> clauses.
  </para>

  <para>
   In addition, a <type>cube</type> GiST index can be used to find nearest
   neighbors using the metric operators
   <literal>&lt;-&gt;</literal>, <literal>&lt;#&gt;</literal>, and
   <literal>&lt;=&gt;</literal> in <literal>ORDER BY</literal> clauses.
   For example, the nearest neighbor of the 3-D point (0.5, 0.5, 0.5)
   could be found efficiently with:
<programlisting>
SELECT c FROM test ORDER BY c &lt;-&gt; cube(array[0.5,0.5,0.5]) LIMIT 1;
</programlisting>
  </para>

  <para>
   The <literal>~&gt;</literal> operator can also be used in this way to
   efficiently retrieve the first few values sorted by a selected coordinate.
   For example, to get the first few cubes ordered by the first coordinate
   (lower left corner) ascending one could use the following query:
<programlisting>
SELECT c FROM test ORDER BY c ~&gt; 1 LIMIT 5;
</programlisting>
   And to get 2-D cubes ordered by the first coordinate of the upper right
   corner descending:
<programlisting>
SELECT c FROM test ORDER BY c ~&gt; 3 DESC LIMIT 5;
</programlisting>
  </para>

  <para>
   <xref linkend="cube-functions-table"/> shows the available functions.
  </para>

  <table id="cube-functions-table">
   <title>Cube Functions</title>
    <tgroup cols="1">
     <thead>
      <row>
       <entry role="func_table_entry"><para role="func_signature">
        Function
       </para>
       <para>
        Description
       </para>
       <para>
        Example(s)
       </para></entry>
      </row>
     </thead>

     <tbody>
      <row>
       <entry role="func_table_entry"><para role="func_signature">
        <function>cube</function> ( <type>float8</type> )
        <returnvalue>cube</returnvalue>
       </para>
       <para>
        Makes a one dimensional cube with both coordinates the same.
       </para>
       <para>
        <literal>cube(1)</literal>
        <returnvalue>(1)</returnvalue>
       </para></entry>
      </row>

      <row>
       <entry role="func_table_entry"><para role="func_signature">
        <function>cube</function> ( <type>float8</type>, <type>float8</type> )
        <returnvalue>cube</returnvalue>
       </para>
       <para>
        Makes a one dimensional cube.
       </para>
       <para>
        <literal>cube(1,2)</literal>
        <returnvalue>(1),(2)</returnvalue>
       </para></entry>
      </row>

      <row>
       <entry role="func_table_entry"><para role="func_signature">
        <function>cube</function> ( <type>float8[]</type> )
        <returnvalue>cube</returnvalue>
       </para>
       <para>
        Makes a zero-volume cube using the coordinates defined by the array.
       </para>
       <para>
        <literal>cube(ARRAY[1,2,3])</literal>
        <returnvalue>(1, 2, 3)</returnvalue>
       </para></entry>
      </row>

      <row>
       <entry role="func_table_entry"><para role="func_signature">
        <function>cube</function> ( <type>float8[]</type>, <type>float8[]</type> )
        <returnvalue>cube</returnvalue>
       </para>
       <para>
        Makes a cube with upper right and lower left coordinates as defined by
        the two arrays, which must be of the same length.
       </para>
       <para>
        <literal>cube(ARRAY[1,2], ARRAY[3,4])</literal>
        <returnvalue>(1, 2),(3, 4)</returnvalue>
       </para></entry>
      </row>

      <row>
       <entry role="func_table_entry"><para role="func_signature">
        <function>cube</function> ( <type>cube</type>, <type>float8</type> )
        <returnvalue>cube</returnvalue>
       </para>
       <para>
        Makes a new cube by adding a dimension on to an existing cube,
        with the same values for both endpoints of the new coordinate.  This
        is useful for building cubes piece by piece from calculated values.
       </para>
       <para>
        <literal>cube('(1,2),(3,4)'::cube, 5)</literal>
        <returnvalue>(1, 2, 5),(3, 4, 5)</returnvalue>
       </para></entry>
      </row>

      <row>
       <entry role="func_table_entry"><para role="func_signature">
        <function>cube</function> ( <type>cube</type>, <type>float8</type>, <type>float8</type> )
        <returnvalue>cube</returnvalue>
       </para>
       <para>
        Makes a new cube by adding a dimension on to an existing cube. This is
        useful for building cubes piece by piece from calculated values.
       </para>
       <para>
        <literal>cube('(1,2),(3,4)'::cube, 5, 6)</literal>
        <returnvalue>(1, 2, 5),(3, 4, 6)</returnvalue>
       </para></entry>
      </row>

      <row>
       <entry role="func_table_entry"><para role="func_signature">
        <function>cube_dim</function> ( <type>cube</type> )
        <returnvalue>integer</returnvalue>
       </para>
       <para>
        Returns the number of dimensions of the cube.
       </para>
       <para>
        <literal>cube_dim('(1,2),(3,4)')</literal>
        <returnvalue>2</returnvalue>
       </para></entry>
      </row>

      <row>
       <entry role="func_table_entry"><para role="func_signature">
        <function>cube_ll_coord</function> ( <type>cube</type>, <type>integer</type> )
        <returnvalue>float8</returnvalue>
       </para>
       <para>
        Returns the <parameter>n</parameter>-th coordinate value for the lower
        left corner of the cube.
       </para>
       <para>
        <literal>cube_ll_coord('(1,2),(3,4)', 2)</literal>
        <returnvalue>2</returnvalue>
       </para></entry>
      </row>

      <row>
       <entry role="func_table_entry"><para role="func_signature">
        <function>cube_ur_coord</function> ( <type>cube</type>, <type>integer</type> )
        <returnvalue>float8</returnvalue>
       </para>
       <para>
        Returns the <parameter>n</parameter>-th coordinate value for the
        upper right corner of the cube.
       </para>
       <para>
        <literal>cube_ur_coord('(1,2),(3,4)', 2)</literal>
        <returnvalue>4</returnvalue>
       </para></entry>
      </row>

      <row>
       <entry role="func_table_entry"><para role="func_signature">
        <function>cube_is_point</function> ( <type>cube</type> )
        <returnvalue>boolean</returnvalue>
       </para>
       <para>
        Returns true if the cube is a point, that is,
        the two defining corners are the same.
       </para>
       <para>
        <literal>cube_is_point(cube(1,1))</literal>
        <returnvalue>t</returnvalue>
       </para></entry>
      </row>

      <row>
       <entry role="func_table_entry"><para role="func_signature">
        <function>cube_distance</function> ( <type>cube</type>, <type>cube</type> )
        <returnvalue>float8</returnvalue>
       </para>
       <para>
        Returns the distance between two cubes. If both
        cubes are points, this is the normal distance function.
       </para>
       <para>
        <literal>cube_distance('(1,2)', '(3,4)')</literal>
        <returnvalue>2.8284271247461903</returnvalue>
       </para></entry>
      </row>

      <row>
       <entry role="func_table_entry"><para role="func_signature">
        <function>cube_subset</function> ( <type>cube</type>, <type>integer[]</type> )
        <returnvalue>cube</returnvalue>
       </para>
       <para>
        Makes a new cube from an existing cube, using a list of
        dimension indexes from an array. Can be used to extract the endpoints
        of a single dimension, or to drop dimensions, or to reorder them as
        desired.
       </para>
       <para>
        <literal>cube_subset(cube('(1,3,5),(6,7,8)'), ARRAY[2])</literal>
        <returnvalue>(3),(7)</returnvalue>
       </para>
       <para>
        <literal>cube_subset(cube('(1,3,5),(6,7,8)'), ARRAY[3,2,1,1])</literal>
        <returnvalue>(5, 3, 1, 1),(8, 7, 6, 6)</returnvalue>
       </para></entry>
      </row>

      <row>
       <entry role="func_table_entry"><para role="func_signature">
        <function>cube_union</function> ( <type>cube</type>, <type>cube</type> )
        <returnvalue>cube</returnvalue>
       </para>
       <para>
        Produces the union of two cubes.
       </para>
       <para>
        <literal>cube_union('(1,2)', '(3,4)')</literal>
        <returnvalue>(1, 2),(3, 4)</returnvalue>
       </para></entry>
      </row>

      <row>
       <entry role="func_table_entry"><para role="func_signature">
        <function>cube_inter</function> ( <type>cube</type>, <type>cube</type> )
        <returnvalue>cube</returnvalue>
       </para>
       <para>
        Produces the intersection of two cubes.
       </para>
       <para>
        <literal>cube_inter('(1,2)', '(3,4)')</literal>
        <returnvalue>(3, 4),(1, 2)</returnvalue>
       </para></entry>
      </row>

      <row>
       <entry role="func_table_entry"><para role="func_signature">
        <function>cube_enlarge</function> ( <parameter>c</parameter> <type>cube</type>, <parameter>r</parameter> <type>double</type>, <parameter>n</parameter> <type>integer</type> )
        <returnvalue>cube</returnvalue>
       </para>
       <para>
        Increases the size of the cube by the specified
        radius <parameter>r</parameter> in at least <parameter>n</parameter>
        dimensions.  If the radius is negative the cube is shrunk instead.
        All defined dimensions are changed by the
        radius <parameter>r</parameter>.  Lower-left coordinates are decreased
        by <parameter>r</parameter> and upper-right coordinates are increased
        by <parameter>r</parameter>.  If a lower-left coordinate is increased
        to more than the corresponding upper-right coordinate (this can only
        happen when <parameter>r</parameter> &lt; 0) than both coordinates are
        set to their average.  If <parameter>n</parameter> is greater than the
        number of defined dimensions and the cube is being enlarged
        (<parameter>r</parameter> &gt; 0), then extra dimensions are added to
        make <parameter>n</parameter> altogether; 0 is used as the initial
        value for the extra coordinates.  This function is useful for creating
        bounding boxes around a point for searching for nearby points.
       </para>
       <para>
        <literal>cube_enlarge('(1,2),(3,4)', 0.5, 3)</literal>
        <returnvalue>(0.5, 1.5, -0.5),(3.5, 4.5, 0.5)</returnvalue>
       </para></entry>
      </row>
     </tbody>
    </tgroup>
  </table>
 </sect2>

 <sect2>
  <title>Defaults</title>

  <para>
   I believe this union:
  </para>
<programlisting>
select cube_union('(0,5,2),(2,3,1)', '0');
cube_union
-------------------
(0, 0, 0),(2, 5, 2)
(1 row)
</programlisting>

   <para>
    does not contradict common sense, neither does the intersection
   </para>

<programlisting>
select cube_inter('(0,-1),(1,1)', '(-2),(2)');
cube_inter
-------------
(0, 0),(1, 0)
(1 row)
</programlisting>

   <para>
    In all binary operations on differently-dimensioned cubes, I assume the
    lower-dimensional one to be a Cartesian projection, i. e., having zeroes
    in place of coordinates omitted in the string representation. The above
    examples are equivalent to:
   </para>

<programlisting>
cube_union('(0,5,2),(2,3,1)','(0,0,0),(0,0,0)');
cube_inter('(0,-1),(1,1)','(-2,0),(2,0)');
</programlisting>

   <para>
    The following containment predicate uses the point syntax,
    while in fact the second argument is internally represented by a box.
    This syntax makes it unnecessary to define a separate point type
    and functions for (box,point) predicates.
   </para>

<programlisting>
select cube_contains('(0,0),(1,1)', '0.5,0.5');
cube_contains
--------------
t
(1 row)
</programlisting>
 </sect2>

 <sect2>
  <title>Notes</title>

  <para>
   For examples of usage, see the regression test <filename>sql/cube.sql</filename>.
  </para>

  <para>
   To make it harder for people to break things, there
   is a limit of 100 on the number of dimensions of cubes. This is set
   in <filename>cubedata.h</filename> if you need something bigger.
  </para>
 </sect2>

 <sect2>
  <title>Credits</title>

  <para>
   Original author: Gene Selkov, Jr. <email>selkovjr@mcs.anl.gov</email>,
   Mathematics and Computer Science Division, Argonne National Laboratory.
  </para>

  <para>
   My thanks are primarily to Prof. Joe Hellerstein
   (<ulink url="http://db.cs.berkeley.edu/jmh/"></ulink>) for elucidating the
   gist of the GiST (<ulink url="http://gist.cs.berkeley.edu/"></ulink>), and
   to his former student Andy Dong for his example written for Illustra.
   I am also grateful to all Postgres developers, present and past, for
   enabling myself to create my own world and live undisturbed in it. And I
   would like to acknowledge my gratitude to Argonne Lab and to the
   U.S. Department of Energy for the years of faithful support of my database
   research.
  </para>

  <para>
   Minor updates to this package were made by Bruno Wolff III
   <email>bruno@wolff.to</email> in August/September of 2002. These include
   changing the precision from single precision to double precision and adding
   some new functions.
  </para>

  <para>
   Additional updates were made by Joshua Reich <email>josh@root.net</email> in
   July 2006. These include <literal>cube(float8[], float8[])</literal> and
   cleaning up the code to use the V1 call protocol instead of the deprecated
   V0 protocol.
  </para>
 </sect2>

</sect1>
