<!-- 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>

 <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 operators provided for
   type <type>cube</type>.
  </para>

  <table id="cube-operators-table">
   <title>Cube Operators</title>
   <tgroup cols="3">
    <thead>
     <row>
      <entry>Operator</entry>
      <entry>Result</entry>
      <entry>Description</entry>
     </row>
    </thead>

    <tbody>
     <row>
      <entry><literal>a = b</literal></entry>
      <entry><type>boolean</type></entry>
      <entry>The cubes a and b are identical.</entry>
     </row>

     <row>
      <entry><literal>a &amp;&amp; b</literal></entry>
      <entry><type>boolean</type></entry>
      <entry>The cubes a and b overlap.</entry>
     </row>

     <row>
      <entry><literal>a @&gt; b</literal></entry>
      <entry><type>boolean</type></entry>
      <entry>The cube a contains the cube b.</entry>
     </row>

     <row>
      <entry><literal>a &lt;@ b</literal></entry>
      <entry><type>boolean</type></entry>
      <entry>The cube a is contained in the cube b.</entry>
     </row>

     <row>
      <entry><literal>a &lt; b</literal></entry>
      <entry><type>boolean</type></entry>
      <entry>The cube a is less than the cube b.</entry>
     </row>

     <row>
      <entry><literal>a &lt;= b</literal></entry>
      <entry><type>boolean</type></entry>
      <entry>The cube a is less than or equal to the cube b.</entry>
     </row>

     <row>
      <entry><literal>a &gt; b</literal></entry>
      <entry><type>boolean</type></entry>
      <entry>The cube a is greater than the cube b.</entry>
     </row>

     <row>
      <entry><literal>a &gt;= b</literal></entry>
      <entry><type>boolean</type></entry>
      <entry>The cube a is greater than or equal to the cube b.</entry>
     </row>

     <row>
      <entry><literal>a &lt;&gt; b</literal></entry>
      <entry><type>boolean</type></entry>
      <entry>The cube a is not equal to the cube b.</entry>
     </row>

     <row>
      <entry><literal>a -&gt; n</literal></entry>
      <entry><type>float8</type></entry>
      <entry>Get <replaceable>n</replaceable>-th coordinate of cube (counting from 1).</entry>
     </row>

     <row>
      <entry><literal>a ~&gt; n</literal></entry>
      <entry><type>float8</type></entry>
      <entry>
        Get <replaceable>n</replaceable>-th coordinate in <quote>normalized</quote> cube
        representation, in which the coordinates have been rearranged into
        the form <quote>lower left &mdash; upper right</quote>; that is, the
        smaller endpoint along each dimension appears first.
      </entry>
     </row>

     <row>
      <entry><literal>a &lt;-&gt; b</literal></entry>
      <entry><type>float8</type></entry>
      <entry>Euclidean distance between a and b.</entry>
     </row>

     <row>
      <entry><literal>a &lt;#&gt; b</literal></entry>
      <entry><type>float8</type></entry>
      <entry>Taxicab (L-1 metric) distance between a and b.</entry>
     </row>

     <row>
      <entry><literal>a &lt;=&gt; b</literal></entry>
      <entry><type>float8</type></entry>
      <entry>Chebyshev (L-inf metric) distance between a and b.</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>
   The scalar ordering operators (<literal>&lt;</literal>, <literal>&gt;=</literal>, etc)
   do not make a lot of sense for any practical purpose but sorting.  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.
  </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="4">
    <thead>
     <row>
      <entry>Function</entry>
      <entry>Result</entry>
      <entry>Description</entry>
      <entry>Example</entry>
     </row>
    </thead>

    <tbody>
     <row>
      <entry><literal>cube(float8)</literal></entry>
      <entry><type>cube</type></entry>
      <entry>Makes a one dimensional cube with both coordinates the same.
      </entry>
      <entry>
       <literal>cube(1) == '(1)'</literal>
      </entry>
     </row>

     <row>
      <entry><literal>cube(float8, float8)</literal></entry>
      <entry><type>cube</type></entry>
      <entry>Makes a one dimensional cube.
      </entry>
      <entry>
       <literal>cube(1,2) == '(1),(2)'</literal>
      </entry>
     </row>

     <row>
      <entry><literal>cube(float8[])</literal></entry>
      <entry><type>cube</type></entry>
      <entry>Makes a zero-volume cube using the coordinates
       defined by the array.
      </entry>
      <entry>
       <literal>cube(ARRAY[1,2]) == '(1,2)'</literal>
      </entry>
     </row>

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

     <row>
      <entry><literal>cube(cube, float8)</literal></entry>
      <entry><type>cube</type></entry>
      <entry>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.
      </entry>
      <entry>
       <literal>cube('(1,2),(3,4)'::cube, 5) == '(1,2,5),(3,4,5)'</literal>
      </entry>
     </row>

     <row>
      <entry><literal>cube(cube, float8, float8)</literal></entry>
      <entry><type>cube</type></entry>
      <entry>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.
      </entry>
      <entry>
       <literal>cube('(1,2),(3,4)'::cube, 5, 6) == '(1,2,5),(3,4,6)'</literal>
      </entry>
     </row>

     <row>
      <entry><literal>cube_dim(cube)</literal></entry>
      <entry><type>integer</type></entry>
      <entry>Returns the number of dimensions of the cube.
      </entry>
      <entry>
       <literal>cube_dim('(1,2),(3,4)') == '2'</literal>
      </entry>
     </row>

     <row>
      <entry><literal>cube_ll_coord(cube, integer)</literal></entry>
      <entry><type>float8</type></entry>
      <entry>Returns the <replaceable>n</replaceable>-th coordinate value for the lower
       left corner of the cube.
      </entry>
      <entry>
       <literal>cube_ll_coord('(1,2),(3,4)', 2) == '2'</literal>
      </entry>
     </row>

    <row>
      <entry><literal>cube_ur_coord(cube, integer)</literal></entry>
      <entry><type>float8</type></entry>
      <entry>Returns the <replaceable>n</replaceable>-th coordinate value for the
       upper right corner of the cube.
      </entry>
      <entry>
       <literal>cube_ur_coord('(1,2),(3,4)', 2) == '4'</literal>
      </entry>
     </row>

     <row>
      <entry><literal>cube_is_point(cube)</literal></entry>
      <entry><type>boolean</type></entry>
      <entry>Returns true if the cube is a point, that is,
       the two defining corners are the same.</entry>
      <entry>
      </entry>
     </row>

     <row>
      <entry><literal>cube_distance(cube, cube)</literal></entry>
      <entry><type>float8</type></entry>
      <entry>Returns the distance between two cubes. If both
       cubes are points, this is the normal distance function.
      </entry>
      <entry>
      </entry>
     </row>

     <row>
      <entry><literal>cube_subset(cube, integer[])</literal></entry>
      <entry><type>cube</type></entry>
      <entry>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.
      </entry>
      <entry>
       <literal>cube_subset(cube('(1,3,5),(6,7,8)'), ARRAY[2]) == '(3),(7)'</literal>
       <literal>cube_subset(cube('(1,3,5),(6,7,8)'), ARRAY[3,2,1,1]) ==
        '(5,3,1,1),(8,7,6,6)'</literal>
      </entry>
     </row>

     <row>
      <entry><literal>cube_union(cube, cube)</literal></entry>
      <entry><type>cube</type></entry>
      <entry>Produces the union of two cubes.
      </entry>
      <entry>
      </entry>
     </row>

     <row>
      <entry><literal>cube_inter(cube, cube)</literal></entry>
      <entry><type>cube</type></entry>
      <entry>Produces the intersection of two cubes.
      </entry>
      <entry>
      </entry>
     </row>

     <row>
      <entry><literal>cube_enlarge(c cube, r double, n integer)</literal></entry>
      <entry><type>cube</type></entry>
      <entry>Increases the size of the cube by the specified
       radius <replaceable>r</replaceable> in at least <replaceable>n</replaceable> dimensions.
       If the radius is negative the cube is shrunk instead.
       All defined dimensions are changed by the radius <replaceable>r</replaceable>.
       Lower-left coordinates are decreased by <replaceable>r</replaceable> and
       upper-right coordinates are increased by <replaceable>r</replaceable>.  If a
       lower-left coordinate is increased to more than the corresponding
       upper-right coordinate (this can only happen when <replaceable>r</replaceable>
       &lt; 0) than both coordinates are set to their average.
       If <replaceable>n</replaceable> is greater than the number of defined dimensions
       and the cube is being enlarged (<replaceable>r</replaceable> &gt; 0), then extra
       dimensions are added to make <replaceable>n</replaceable> 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.
      </entry>
      <entry>
       <literal>cube_enlarge('(1,2),(3,4)', 0.5, 3) ==
        '(0.5,1.5,-0.5),(3.5,4.5,0.5)'</literal>
      </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>
