
<sect1 id="cube">
 <title>cube</title>
 
 <indexterm zone="cube">
  <primary>cube</primary>
 </indexterm>

 <para>
  This module contains the user-defined type, CUBE, representing 
  multidimensional cubes.
 </para>

 <sect2>
  <title>Syntax</title>

  <para>
   The following are valid external representations for the CUBE type:
  </para>

  <table>
   <title>Cube external representations</title>
   <tgroup cols="2">
    <tbody>
     <row>
      <entry>'x'</entry>
      <entry>A floating point value representing a one-dimensional point or 
       one-dimensional zero length cubement
      </entry>
     </row>
     <row>
      <entry>'(x)'</entry>
      <entry>Same as above</entry>
     </row>
     <row>
      <entry>'x1,x2,x3,...,xn'</entry>
      <entry>A point in n-dimensional space, represented internally as a zero 
       volume box
      </entry>
     </row>
     <row>
      <entry>'(x1,x2,x3,...,xn)'</entry>
      <entry>Same as above</entry>
     </row>
     <row>
      <entry>'(x),(y)'</entry>
      <entry>1-D cubement starting at x and ending at y or vice versa; the 
       order does not matter
      </entry>
     </row>
     <row>
      <entry>'(x1,...,xn),(y1,...,yn)'</entry>
      <entry>n-dimensional box represented by a pair of its opposite corners, no 
       matter which.  Functions take care of swapping to achieve "lower left -- 
       upper right" representation before computing any values
      </entry>
     </row>
    </tbody>
   </tgroup>
  </table>
   </sect2>
  
   <sect2>
    <title>Grammar</title>
    <table>
     <title>Cube Grammar Rules</title>
     <tgroup cols="2">
      <tbody>
       <row>
        <entry>rule 1</entry>
        <entry>box -> O_BRACKET paren_list COMMA paren_list C_BRACKET</entry>
       </row>
       <row>
        <entry>rule 2</entry>
        <entry>box -> paren_list COMMA paren_list</entry>
       </row>
       <row>
        <entry>rule 3</entry>
        <entry>box -> paren_list</entry>
       </row>
       <row>
        <entry>rule 4</entry>
        <entry>box -> list</entry>
       </row>
       <row>
        <entry>rule 5</entry>
        <entry>paren_list -> O_PAREN list C_PAREN</entry>
       </row>
       <row>
        <entry>rule 6</entry>
        <entry>list -> FLOAT</entry>
       </row>
       <row>
        <entry>rule 7</entry>
        <entry>list -> list COMMA FLOAT</entry>
       </row>
      </tbody>
     </tgroup>
    </table>
   </sect2>

   <sect2>
    <title>Tokens</title>
    <table>
     <title>Cube Grammar Rules</title>
     <tgroup cols="2">
      <tbody>
       <row>
        <entry>n</entry>
        <entry>[0-9]+</entry>
       </row>
       <row>
        <entry>i</entry>
        <entry>nteger                [+-]?{n}</entry>
       </row>
       <row>
        <entry>real</entry>
        <entry>[+-]?({n}\.{n}?|\.{n})</entry>
       </row>
       <row>
        <entry>FLOAT</entry>
        <entry>({integer}|{real})([eE]{integer})?</entry>
       </row>
       <row>
        <entry>O_BRACKET</entry>        
        <entry>\[</entry>
       </row>
       <row>
        <entry>C_BRACKET</entry>
        <entry>\]</entry>
       </row>
       <row>
        <entry>O_PAREN</entry>
        <entry>\(</entry>
       </row>
       <row>
        <entry>C_PAREN</entry>
        <entry>\)</entry>
       </row>
       <row>
        <entry>COMMA</entry>
        <entry>\,</entry>
       </row>
      </tbody>
     </tgroup>
    </table>
   </sect2>

 <sect2>
  <title>Examples</title>
  <table>
   <title>Examples</title>
   <tgroup cols="2">
    <tbody>
     <row>
      <entry>'x'</entry>
      <entry>A floating point value representing a one-dimensional point 
       (or, zero-length one-dimensional interval)
      </entry>
     </row>
     <row>
      <entry>'(x)'</entry>
      <entry>Same as above</entry>
     </row>
     <row>
      <entry>'x1,x2,x3,...,xn'</entry>
      <entry>A point in n-dimensional space,represented internally as a zero 
       volume cube
      </entry>
     </row>
     <row>
      <entry>'(x1,x2,x3,...,xn)'</entry>
      <entry>Same as above</entry>
     </row>
     <row>
      <entry>'(x),(y)'</entry>
      <entry>A 1-D interval starting at x and ending at y or vice versa; the
       order does not matter
      </entry>
     </row>
     <row>
      <entry>'[(x),(y)]'</entry>
      <entry>Same as above</entry>
     </row>
     <row>
      <entry>'(x1,...,xn),(y1,...,yn)'</entry>
      <entry>An n-dimensional box represented by a pair of its diagonally 
       opposite corners, regardless of order. Swapping is provided
       by all comarison routines to ensure the 
       "lower left -- upper right" representation
       before actaul comparison takes place.
      </entry>
     </row>
     <row>
      <entry>'[(x1,...,xn),(y1,...,yn)]'</entry>
      <entry>Same as above</entry>
     </row>
    </tbody>
   </tgroup>
  </table>
  <para>
   White space is ignored, so '[(x),(y)]' can be: '[ ( x ), ( y ) ]'
  </para>
 </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 to the 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 sized boxes, I assume the smaller
    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 the special 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>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>
   The access method for CUBE is a GiST index (gist_cube_ops), which is a
   generalization of R-tree. GiSTs allow the postgres implementation of
   R-tree, originally encoded to support 2-D geometric types such as
   boxes and polygons, to be used with any data type whose data domain
   can be partitioned using the concepts of containment, intersection and
   equality. In other words, everything that can intersect or contain
   its own kind can be indexed with a GiST. That includes, among other
   things, all geometric data types, regardless of their dimensionality
   (see also contrib/seg).
  </para>

  <para>
   The operators supported by the GiST access method include:
  </para>

  <programlisting>
a = b                Same as
  </programlisting>
  <para>
   The cubements a and b are identical.
  </para>

  <programlisting>
a && b                Overlaps
  </programlisting>
  <para>
   The cubements a and b overlap.
  </para>

  <programlisting>
a @> b                Contains
  </programlisting>
  <para>
   The cubement a contains the cubement b.
  </para>

  <programlisting>
a <@ b                Contained in
  </programlisting>
  <para>
   The cubement a is contained in b.
  </para>

  <para>
   (Before PostgreSQL 8.2, the containment operators @> and <@ were
   respectively called @ and ~.  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
   datatypes!)
  </para>

  <para>
   Although the mnemonics of the following operators is questionable, I
   preserved them to maintain visual consistency with other geometric
   data types defined in Postgres.
  </para>

  <para>
   Other operators:
  </para>

  <programlisting>
[a, b] < [c, d]                Less than
[a, b] > [c, d]                Greater than
  </programlisting>
 
  <para>
   These operators do not make a lot of sense for any practical
   purpose but sorting. These operators first compare (a) to (c),
   and if these are equal, compare (b) to (d). That accounts for
   reasonably good sorting in most cases, which is useful if
   you want to use ORDER BY with this type
  </para>

  <para>
   The following functions are available:
  </para>

  <table>
   <title>Functions available</title>
   <tgroup cols="2">
    <tbody>
     <row>
      <entry><literal>cube_distance(cube, cube) returns double</literal></entry>
      <entry>cube_distance returns the distance between two cubes. If both 
       cubes are points, this is the normal distance function.
      </entry>
     </row>
     <row>
      <entry><literal>cube(float8) returns cube</literal></entry>
      <entry>This makes a one dimensional cube with both coordinates the same.
       If the type of the argument is a numeric type other than float8 an
       explicit cast to float8 may be needed.
       <literal>cube(1) == '(1)'</literal>
      </entry>
     </row>

     <row>
      <entry><literal>cube(float8, float8) returns cube</literal></entry>
      <entry>
       This makes a one dimensional cube.
       <literal>cube(1,2) == '(1),(2)'</literal>
      </entry>
     </row>

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

     <row>
      <entry><literal>cube(float8[], float8[]) returns cube</literal></entry>
      <entry>This makes a cube, with upper right and lower left 
       coordinates as defined by the 2 float arrays. Arrays must be of the 
       same length.
       <literal>cube('{1,2}'::float[], '{3,4}'::float[]) == '(1,2),(3,4)'
       </literal>
      </entry>
     </row>

     <row>
      <entry><literal>cube(cube, float8) returns cube</literal></entry>
      <entry>This builds a new cube by adding a dimension on to an 
       existing cube with the same values for both parts of the new coordinate. 
       This is useful for building cubes piece by piece from calculated values.
       <literal>cube('(1)',2) == '(1,2),(1,2)'</literal>
      </entry>
     </row>

     <row>
      <entry><literal>cube(cube, float8, float8) returns cube</literal></entry>
      <entry>This builds a new cube by adding a dimension on to an 
       existing cube. This is useful for building cubes piece by piece from 
       calculated values. <literal>cube('(1,2)',3,4) == '(1,3),(2,4)'</literal>
      </entry>
     </row>

     <row>
      <entry><literal>cube_dim(cube) returns int</literal></entry>
      <entry>cube_dim returns the number of dimensions stored in the 
       the data structure
       for a cube. This is useful for constraints on the dimensions of a cube.
      </entry>
     </row>

     <row>
      <entry><literal>cube_ll_coord(cube, int) returns double </literal></entry>
      <entry>
       cube_ll_coord returns the nth coordinate value for the lower left 
       corner of a cube. This is useful for doing coordinate transformations.
      </entry>
     </row>

    <row>
      <entry><literal>cube_ur_coord(cube, int) returns double
      </literal></entry>
      <entry>cube_ur_coord returns the nth coordinate value for the
       upper right corner of a cube. This is useful for doing coordinate 
       transformations.
      </entry>
     </row>

     <row>
      <entry><literal>cube_subset(cube, int[]) returns cube
      </literal></entry>
      <entry>Builds a new cube from an existing cube, using a list of 
       dimension indexes
       from an array. Can be used to find both the ll and ur coordinate of single
       dimenion, e.g.: cube_subset(cube('(1,3,5),(6,7,8)'), ARRAY[2]) = '(3),(7)'
       Or can be used to drop dimensions, or reorder them as desired, e.g.:
       cube_subset(cube('(1,3,5),(6,7,8)'), ARRAY[3,2,1,1]) = 
       '(5, 3, 1, 1),(8, 7, 6, 6)'
      </entry>
     </row>

     <row>
      <entry><literal>cube_is_point(cube) returns bool</literal></entry>
      <entry>cube_is_point returns true if a cube is also a point. 
       This is true when the two defining corners are the same.</entry>
     </row>
 
     <row>
      <entry><literal>cube_enlarge(cube, double, int) returns cube</literal></entry>
      <entry>
       cube_enlarge increases the size of a cube by a specified 
       radius in at least
       n dimensions. If the radius is negative the box is shrunk instead. This
       is useful for creating bounding boxes around a point for searching for
       nearby points. All defined dimensions are changed by the radius. If n
       is greater than the number of defined dimensions and the cube is being
       increased (r >= 0) then 0 is used as the base for the extra coordinates.
       LL coordinates are decreased by r and UR coordinates are increased by r. 
       If a LL coordinate is increased to larger than the corresponding UR 
       coordinate (this can only happen when r < 0) than both coordinates are 
       set to their average. To make it harder for people to break things there 
       is an effective maximum on the dimension of cubes of 100. This is set 
       in cubedata.h if you need something bigger.
      </entry>
     </row>
    </tbody>
   </tgroup>
  </table>
 
  <para>
   There are a few other potentially useful functions defined in cube.c 
   that vanished from the schema because I stopped using them. Some of 
   these were meant to support type casting. Let me know if I was wrong: 
   I will then add them back to the schema. I would also appreciate 
   other ideas that would enhance the type and make it more useful.
  </para>

  <para>
   For examples of usage, see sql/cube.sql
  </para>
 </sect2>

 <sect2>
  <title>Credits</title>
  <para>
   This code is essentially based on the example written for
   Illustra, <ulink url="http://garcia.me.berkeley.edu/~adong/rtree"></ulink>
  </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 
   (<ulink url="http://best.me.berkeley.edu/~adong/"></ulink>), for his exemplar.
   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>
   Gene Selkov, Jr.
   Computational Scientist
   Mathematics and Computer Science Division
   Argonne National Laboratory
   9700 S Cass Ave.
   Building 221
   Argonne, IL 60439-4844
   <email>selkovjr@mcs.anl.gov</email>
  </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 
   form.
  </para>
 </sect2>
</sect1>

