14.16.9.2 Spatial Relation Functions That Use Minimum Bounding Rectangles

MySQL provides several MySQL-specific functions that test the relationship between minimum bounding rectangles (MBRs) of two geometries g1 and g2. The return values 1 and 0 indicate true and false, respectively.

The MBR (also known as the bounding box) for a two-dimensional geometry is the smallest rectangle which holds all points in the geometry, and so encloses the area between its greatest extents in both coordinate directions. In other words, it is the rectangle bounded by the points (min(x), min(y)), (min(x), max(y)), (max(x), max(y)), and (max(x), min(y)), where min() and max() represent the geometry's minimum and maximum x-coordinate or y-coordinate, respectively.

When speaking of relationships between geometries, it is important to distinguish between containment and covering, as described here:

• A geometry g1 contains another geometry g2 if and only if all points in g2 are also in g1, and their boundaries do not intersect. That is, all points (a, b) in g2 must satisfy the conditions min(x) < a < max(x) and min(y) < b < max(y). In this case, ST_Contains(g1, g2) and MBRContains(g1, g2) both return true, as does ST_Within(g2, g1).

• We say that g1 covers g2 if all points in g2 are also in g1, including any boundary points. That is, all points (a, b) in g2 must satisfy the conditions min(x) <= a <= max(x) and min(y) <= b <= max(y). In this case, MBRCovers(g1, g2) and MBRCoveredBy(g2, g1) both return true.

Let us define a rectangle g1 and points p1, p2, and p3 using the SQL statements shown here:

SET 
  @g1 = ST_GeomFromText('Polygon((0 0,0 3,3 3,3 0,0 0))'),

  @p1 = ST_GeomFromText('Point(1 1)'),
  @p2 = ST_GeomFromText('Point(3 3)'),
  @p3 = ST_GeomFromText('Point(5 5)');

g1 contains and covers p1; p1 is entirely within g1 and does not touch any of its boundaries, as we can see from the SELECT statement shown here:

mysql> SELECT
    ->   ST_Contains(@g1, @p1), ST_Within(@p1, @g1),
    ->   MBRContains(@g1, @p1),
    ->   MBRCovers(@g1, @p1), MBRCoveredBy(@p1, @g1), 
    ->   ST_Disjoint(@g1, @p1), ST_Intersects(@g1, @p1)\G
*************************** 1. row ***************************
  ST_Contains(@g1, @p1): 1
    ST_Within(@p1, @g1): 1
  MBRContains(@g1, @p1): 1
    MBRCovers(@g1, @p1): 1
 MBRCoveredBy(@p1, @g1): 1
  ST_Disjoint(@g1, @p1): 0
ST_Intersects(@g1, @p1): 1
1 row in set (0.01 sec)

Using the same query with @p2 in place of @p1, we can see that g2 covers p2, but does not contain it, because p2 is included in the boundary of g2, but does not lie within its interior. (That is, min(x) <= a <= max(x) and min(y) <= b <= max(y) are true, but min(x) < a < max(x) and min(y) < b < max(y) are not.)

mysql> SELECT
    ->   ST_Contains(@g1, @p2), ST_Within(@p2, @g1),
    ->   MBRContains(@g1, @p2),
    ->   MBRCovers(@g1, @p2), MBRCoveredBy(@p2, @g1), 
    ->   ST_Disjoint(@g1, @p2), ST_Intersects(@g1, @p2)\G
*************************** 1. row ***************************
  ST_Contains(@g1, @p2): 0
    ST_Within(@p2, @g1): 0
  MBRContains(@g1, @p2): 0
    MBRCovers(@g1, @p2): 1
 MBRCoveredBy(@p2, @g1): 1
  ST_Disjoint(@g1, @p2): 0
ST_Intersects(@g1, @p2): 1
1 row in set (0.00 sec)

Executing the query—this time using @p3 rather than @p2 or @p1—shows us that p3 is disjoint from g1; the two geometries have no points in common, and g1 neither contains nor covers p3. ST_Disjoint(g1, p3) returns true; ST_Intersects(g1, p3) returns false.

mysql> SELECT
    ->   ST_Contains(@g1, @p3), ST_Within(@p3, @g1),
    ->   MBRContains(@g1, @p3),
    ->   MBRCovers(@g1, @p3), MBRCoveredBy(@p3, @g1), 
    ->   ST_Disjoint(@g1, @p3), ST_Intersects(@g1, @p3)\G
*************************** 1. row ***************************
  ST_Contains(@g1, @p3): 0
    ST_Within(@p3, @g1): 0
  MBRContains(@g1, @p3): 0
    MBRCovers(@g1, @p3): 0
 MBRCoveredBy(@p3, @g1): 0
  ST_Disjoint(@g1, @p3): 1
ST_Intersects(@g1, @p3): 0
1 row in set (0.00 sec)

The function descriptions shown later in this section and in Section 14.16.9.1, “Spatial Relation Functions That Use Object Shapes” provide additional examples.

The bounding box of a point is interpreted as a point that is both boundary and interior.

The bounding box of a straight horizontal or vertical line is interpreted as a line where the interior of the line is also boundary. The endpoints are boundary points.

If any of the parameters are geometry collections, the interior, boundary, and exterior of those parameters are those of the union of all elements in the collection.

Functions in this section detect arguments in either Cartesian or geographic spatial reference systems (SRSs), and return results appropriate to the SRS.

Unless otherwise specified, functions in this section handle their geometry arguments as follows:

• If any argument is NULL or an empty geometry, the return value is NULL.

• If any geometry argument is not a syntactically well-formed geometry, an ER_GIS_INVALID_DATA error occurs.

• If any geometry argument is a syntactically well-formed geometry in an undefined spatial reference system (SRS), an ER_SRS_NOT_FOUND error occurs.

• For functions that take multiple geometry arguments, if those arguments are not in the same SRS, an ER_GIS_DIFFERENT_SRIDS error occurs.

• If any argument is geometrically invalid, either the result is true or false (it is undefined which), or an error occurs.

• For geographic SRS geometry arguments, if any argument has a longitude or latitude that is out of range, an error occurs:

  • If a longitude value is not in the range (−180, 180], an ER_GEOMETRY_PARAM_LONGITUDE_OUT_OF_RANGE error occurs.

  • If a latitude value is not in the range [−90, 90], an ER_GEOMETRY_PARAM_LATITUDE_OUT_OF_RANGE error occurs.

  Ranges shown are in degrees. If an SRS uses another unit, the range uses the corresponding values in its unit. The exact range limits deviate slightly due to floating-point arithmetic.

• Otherwise, the return value is non-NULL.

These MBR functions are available for testing geometry relationships:

• MBRContains(g1, g2)

  Returns 1 or 0 to indicate whether the minimum bounding rectangle of g1 contains the minimum bounding rectangle of g2. This tests the opposite relationship as MBRWithin().

  MBRContains() handles its arguments as described in the introduction to this section.

  mysql> SET
      ->   @g1 = ST_GeomFromText('Polygon((0 0,0 3,3 3,3 0,0 0))'),
      ->   @g2 = ST_GeomFromText('Polygon((1 1,1 2,2 2,2 1,1 1))'),
      ->   @g3 = ST_GeomFromText('Polygon((0 0,0 5,5 5,5 0,0 0))'),
      ->   @g4 = ST_GeomFromText('Polygon((5 5,5 10,10 10,10 5,5 5))'),
      ->   @p1 = ST_GeomFromText('Point(1 1)'),
      ->   @p2 = ST_GeomFromText('Point(3 3)');
      ->   @p3 = ST_GeomFromText('Point(5 5)');
  Query OK, 0 rows affected (0.00 sec)
  
  mysql> SELECT
      ->   MBRContains(@g1, @g2), MBRContains(@g1, @g4),
      ->   MBRContains(@g2, @g1), MBRContains(@g2, @g4),
      ->   MBRContains(@g2, @g3), MBRContains(@g3, @g4),
      ->   MBRContains(@g3, @g1), MBRContains(@g1, @g3),
      ->   MBRContains(@g1, @p1), MBRContains(@p1, @g1),
      ->   MBRContains(@g1, @p1), MBRContains(@p1, @g1),
      ->   MBRContains(@g2, @p2), MBRContains(@g2, @p3),
      ->   MBRContains(@g3, @p1), MBRContains(@g3, @p2),
      ->   MBRContains(@g3, @p3), MBRContains(@g4, @p1),
      ->   MBRContains(@g4, @p2), MBRContains(@g4, @p3)\G
  *************************** 1. row ***************************
  MBRContains(@g1, @g2): 1
  MBRContains(@g1, @g4): 0
  MBRContains(@g2, @g1): 0
  MBRContains(@g2, @g4): 0
  MBRContains(@g2, @g3): 0
  MBRContains(@g3, @g4): 0
  MBRContains(@g3, @g1): 1
  MBRContains(@g1, @g3): 0
  MBRContains(@g1, @p1): 1
  MBRContains(@p1, @g1): 0
  MBRContains(@g1, @p1): 1
  MBRContains(@p1, @g1): 0
  MBRContains(@g2, @p2): 0
  MBRContains(@g2, @p3): 0
  MBRContains(@g3, @p1): 1
  MBRContains(@g3, @p2): 1
  MBRContains(@g3, @p3): 0
  MBRContains(@g4, @p1): 0
  MBRContains(@g4, @p2): 0
  MBRContains(@g4, @p3): 0
  1 row in set (0.00 sec)

• MBRCoveredBy(g1, g2)

  Returns 1 or 0 to indicate whether the minimum bounding rectangle of g1 is covered by the minimum bounding rectangle of g2. This tests the opposite relationship as MBRCovers().

  MBRCoveredBy() handles its arguments as described in the introduction to this section.

  mysql> SET @g1 = ST_GeomFromText('Polygon((0 0,0 3,3 3,3 0,0 0))');
  mysql> SET @g2 = ST_GeomFromText('Point(1 1)');
  mysql> SELECT MBRCovers(@g1,@g2), MBRCoveredby(@g1,@g2);
  +--------------------+-----------------------+
  | MBRCovers(@g1,@g2) | MBRCoveredby(@g1,@g2) |
  +--------------------+-----------------------+
  |                  1 |                     0 |
  +--------------------+-----------------------+
  mysql> SELECT MBRCovers(@g2,@g1), MBRCoveredby(@g2,@g1);
  +--------------------+-----------------------+
  | MBRCovers(@g2,@g1) | MBRCoveredby(@g2,@g1) |
  +--------------------+-----------------------+
  |                  0 |                     1 |
  +--------------------+-----------------------+

  See the description of the MBRCovers() function for additional examples.

• MBRCovers(g1, g2)

  Returns 1 or 0 to indicate whether the minimum bounding rectangle of g1 covers the minimum bounding rectangle of g2. This tests the opposite relationship as MBRCoveredBy(). See the description of MBRCoveredBy() for additional examples.

  MBRCovers() handles its arguments as described in the introduction to this section.

  mysql> SET 
      ->   @g1 = ST_GeomFromText('Polygon((0 0,0 3,3 3,3 0,0 0))'),
      ->   @g2 = ST_GeomFromText('Polygon((1 1,1 2,2 2,2 1,1 1))'),
      ->   @p1 = ST_GeomFromText('Point(1 1)'),
      ->   @p2 = ST_GeomFromText('Point(3 3)'),
      ->   @p3 = ST_GeomFromText('Point(5 5)');
  Query OK, 0 rows affected (0.02 sec)
  
  mysql> SELECT 
      ->   MBRCovers(@g1, @p1), MBRCovers(@g1, @p2), 
      ->   MBRCovers(@g1, @g2), MBRCovers(@g1, @p3)\G
  *************************** 1. row ***************************
  MBRCovers(@g1, @p1): 1
  MBRCovers(@g1, @p2): 1
  MBRCovers(@g1, @g2): 1
  MBRCovers(@g1, @p3): 0
  1 row in set (0.00 sec)

• MBRDisjoint(g1, g2)

  Returns 1 or 0 to indicate whether the minimum bounding rectangles of the two geometries g1 and g2 are disjoint (do not intersect).

  MBRDisjoint() handles its arguments as described in the introduction to this section.

  mysql> SET 
      ->   @g1 = ST_GeomFromText('Polygon((0 0,0 3,3 3,3 0,0 0))'),
      ->   @g2 = ST_GeomFromText('Polygon((1 1,1 2,2 2,2 1,1 1))'),
      ->   @g3 = ST_GeomFromText('Polygon((0 0,0 5,5 5,5 0,0 0))'),
      ->   @g4 = ST_GeomFromText('Polygon((5 5,5 10,10 10,10 5,5 5))'),
      ->   @p1 = ST_GeomFromText('Point(1 1)'),
      ->   @p2 = ST_GeomFromText('Point(3 3)'),
      ->   @p3 = ST_GeomFromText('Point(5 5)');
  Query OK, 0 rows affected (0.00 sec)
  
  mysql> SELECT
      ->   MBRDisjoint(@g1, @g4), MBRDisjoint(@g2, @g4),
      ->   MBRDisjoint(@g3, @g4), MBRDisjoint(@g4, @g4),
      ->   MBRDisjoint(@g1, @p1), MBRDisjoint(@g1, @p2), 
      ->   MBRDisjoint(@g1, @p3)\G
  *************************** 1. row ***************************
  MBRDisjoint(@g1, @g4): 1
  MBRDisjoint(@g2, @g4): 1
  MBRDisjoint(@g3, @g4): 0
  MBRDisjoint(@g4, @g4): 0
  MBRDisjoint(@g1, @p1): 0
  MBRDisjoint(@g1, @p2): 0
  MBRDisjoint(@g1, @p3): 1
  1 row in set (0.00 sec)

• MBREquals(g1, g2)

  Returns 1 or 0 to indicate whether the minimum bounding rectangles of the two geometries g1 and g2 are the same.

  MBREquals() handles its arguments as described in the introduction to this section, except that it does not return NULL for empty geometry arguments.

  mysql> SET
      ->   @g1 = ST_GeomFromText('Polygon((0 0,0 3,3 3,3 0,0 0))'),
      ->   @g2 = ST_GeomFromText('Polygon((1 1,1 2,2 2,2 1,1 1))'),
      ->   @p1 = ST_GeomFromText('Point(1 1)'),
      ->   @p2 = ST_GeomFromText('Point(3 3)'),
      ->   @p3 = ST_GeomFromText('Point(5 5)');
  Query OK, 0 rows affected (0.00 sec)
  
  mysql> SELECT 
      ->   MBREquals(@g1, @g1), MBREquals(@g1, @g2), 
      ->   MBREquals(@g1, @p1), MBREquals(@g1, @p2), MBREquals(@g2, @g2), 
      ->   MBREquals(@p1, @p1), MBREquals(@p1, @p2), MBREquals(@p2, @p2)\G
  *************************** 1. row ***************************
  MBREquals(@g1, @g1): 1
  MBREquals(@g1, @g2): 0
  MBREquals(@g1, @p1): 0
  MBREquals(@g1, @p2): 0
  MBREquals(@g2, @g2): 1
  MBREquals(@p1, @p1): 1
  MBREquals(@p1, @p2): 0
  MBREquals(@p2, @p2): 1
  1 row in set (0.00 sec)

• MBRIntersects(g1, g2)

  Returns 1 or 0 to indicate whether the minimum bounding rectangles of the two geometries g1 and g2 intersect.

  MBRIntersects() handles its arguments as described in the introduction to this section.

  mysql> SET
      ->   @g1 = ST_GeomFromText('Polygon((0 0,0 3,3 3,3 0,0 0))'),
      ->   @g2 = ST_GeomFromText('Polygon((1 1,1 2,2 2,2 1,1 1))'),
      ->   @g3 = ST_GeomFromText('Polygon((0 0,0 5,5 5,5 0,0 0))'),
      ->   @g4 = ST_GeomFromText('Polygon((5 5,5 10,10 10,10 5,5 5))'),
      ->   @g5 = ST_GeomFromText('Polygon((2 2,2 8,8 8,8 2,2 2))'),
      ->   @p1 = ST_GeomFromText('Point(1 1)'),
      ->   @p2 = ST_GeomFromText('Point(3 3)'),
      ->   @p3 = ST_GeomFromText('Point(5 5)');
  Query OK, 0 rows affected (0.00 sec)
  
  mysql> SELECT
      ->   MBRIntersects(@g1, @g1), MBRIntersects(@g1, @g2),
      ->   MBRIntersects(@g1, @g3), MBRIntersects(@g1, @g4), MBRIntersects(@g1, @g5), 
      ->   MBRIntersects(@g1, @p1), MBRIntersects(@g1, @p2), MBRIntersects(@g1, @p3), 
      ->   MBRIntersects(@g2, @p1), MBRIntersects(@g2, @p2), MBRIntersects(@g2, @p3)\G
  *************************** 1. row ***************************
  MBRIntersects(@g1, @g1): 1
  MBRIntersects(@g1, @g2): 1
  MBRIntersects(@g1, @g3): 1
  MBRIntersects(@g1, @g4): 0
  MBRIntersects(@g1, @g5): 1
  MBRIntersects(@g1, @p1): 1
  MBRIntersects(@g1, @p2): 1
  MBRIntersects(@g1, @p3): 0
  MBRIntersects(@g2, @p1): 1
  MBRIntersects(@g2, @p2): 0
  MBRIntersects(@g2, @p3): 0
  1 row in set (0.00 sec)

• MBROverlaps(g1, g2)

  Two geometries spatially overlap if they intersect and their intersection results in a geometry of the same dimension but not equal to either of the given geometries.

  This function returns 1 or 0 to indicate whether the minimum bounding rectangles of the two geometries g1 and g2 overlap.

  MBROverlaps() handles its arguments as described in the introduction to this section.

• MBRTouches(g1, g2)

  Two geometries spatially touch if their interiors do not intersect, but the boundary of one of the geometries intersects either the boundary or the interior of the other.

  This function returns 1 or 0 to indicate whether the minimum bounding rectangles of the two geometries g1 and g2 touch.

  MBRTouches() handles its arguments as described in the introduction to this section.

• MBRWithin(g1, g2)

  Returns 1 or 0 to indicate whether the minimum bounding rectangle of g1 is within the minimum bounding rectangle of g2. This tests the opposite relationship as MBRContains().

  MBRWithin() handles its arguments as described in the introduction to this section.

  mysql> SET
      ->   @g1 = ST_GeomFromText('Polygon((0 0,0 3,3 3,3 0,0 0))'),
      ->   @g2 = ST_GeomFromText('Polygon((1 1,1 2,2 2,2 1,1 1))'),
      ->   @g3 = ST_GeomFromText('Polygon((0 0,0 5,5 5,5 0,0 0))'),
      ->   @g4 = ST_GeomFromText('Polygon((5 5,5 10,10 10,10 5,5 5))'),
      ->   @p1 = ST_GeomFromText('Point(1 1)'),
      ->   @p2 = ST_GeomFromText('Point(3 3)');
      ->   @p3 = ST_GeomFromText('Point(5 5)');
  Query OK, 0 rows affected (0.00 sec)
  
  mysql> SELECT
      ->   MBRWithin(@g1, @g2), MBRWithin(@g1, @g4),
      ->   MBRWithin(@g2, @g1), MBRWithin(@g2, @g4),
      ->   MBRWithin(@g2, @g3), MBRWithin(@g3, @g4),
      ->   MBRWithin(@g1, @p1), MBRWithin(@p1, @g1),
      ->   MBRWithin(@g1, @p1), MBRWithin(@p1, @g1),
      ->   MBRWithin(@g2, @p2), MBRWithin(@g2, @p3)\G
  *************************** 1. row ***************************
  MBRWithin(@g1, @g2): 0
  MBRWithin(@g1, @g4): 0
  MBRWithin(@g2, @g1): 1
  MBRWithin(@g2, @g4): 0
  MBRWithin(@g2, @g3): 1
  MBRWithin(@g3, @g4): 0
  MBRWithin(@g1, @p1): 0
  MBRWithin(@p1, @g1): 1
  MBRWithin(@g1, @p1): 0
  MBRWithin(@p1, @g1): 1
  MBRWithin(@g2, @p2): 0
  MBRWithin(@g2, @p3): 0
  1 row in set (0.00 sec)