14.16.9.1 Spatial Relation Functions That Use Object Shapes

The OpenGIS specification defines the following functions to test the relationship between two geometry values g1 and g2, using precise object shapes. The return values 1 and 0 indicate true and false, respectively, except that distance functions return distance values.

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 any geometry argument is 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 geometry 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.

Some functions in this section permit a unit argument that specifies the length unit for the return value. Unless otherwise specified, functions handle their unit argument as follows:

A unit is supported if it is found in the INFORMATION_SCHEMA ST_UNITS_OF_MEASURE table. See Section 28.3.37, “The INFORMATION_SCHEMA ST_UNITS_OF_MEASURE Table”.
If a unit is specified but not supported by MySQL, an ER_UNIT_NOT_FOUND error occurs.
If a supported linear unit is specified and the SRID is 0, an ER_GEOMETRY_IN_UNKNOWN_LENGTH_UNIT error occurs.
If a supported linear unit is specified and the SRID is not 0, the result is in that unit.
If a unit is not specified, the result is in the unit of the SRS of the geometries, whether Cartesian or geographic. Currently, all MySQL SRSs are expressed in meters.

These object-shape functions are available for testing geometry relationships:

ST_Contains(g1, g2)

Returns 1 or 0 to indicate whether g1 completely contains g2 (this means that g1 and g2 must not intersect). This relationship is the inverse of that tested by ST_Within().

ST_Contains() 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))'),
    ->     @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
    ->   ST_Contains(@g1, @p1), ST_Within(@p1, @g1), 
    ->   ST_Disjoint(@g1, @p1), ST_Intersects(@g1, @p1)\G
*************************** 1. row ***************************
  ST_Contains(@g1, @p1): 1
    ST_Within(@p1, @g1): 1
  ST_Disjoint(@g1, @p1): 0
ST_Intersects(@g1, @p1): 1
1 row in set (0.00 sec)

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

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

ST_Crosses(g1, g2)
Two geometries spatially cross if their spatial relation has the following properties:
- Unless g1 and g2 are both of dimension 1: g1 crosses g2 if the interior of g2 has points in common with the interior of g1, but g2 does not cover the entire interior of g1.
- If both g1 and g2 are of dimension 1: If the lines cross each other in a finite number of points (that is, no common line segments, only single points in common).
This function returns 1 or 0 to indicate whether g1 spatially crosses g2.
ST_Crosses() handles its arguments as described in the introduction to this section except that the return value is NULL for these additional conditions:
- g1 is of dimension 2 (Polygon or MultiPolygon).
- g2 is of dimension 1 (Point or MultiPoint).
ST_Disjoint(g1, g2)
Returns 1 or 0 to indicate whether g1 is spatially disjoint from (does not intersect) g2.
ST_Disjoint() handles its arguments as described in the introduction to this section.
ST_Distance(g1, g2 [, unit])
Returns the distance between g1 and g2, measured in the length unit of the spatial reference system (SRS) of the geometry arguments, or in the unit of the optional unit argument if that is specified.
This function processes geometry collections by returning the shortest distance among all combinations of the components of the two geometry arguments.
ST_Distance() handles its geometry arguments as described in the introduction to this section, with these exceptions:
- ST_Distance() detects arguments in a geographic (ellipsoidal) spatial reference system and returns the geodetic distance on the ellipsoid. ST_Distance() supports distance calculations for geographic SRS arguments of all geometry types.
- If any argument is geometrically invalid, either the result is an undefined distance (that is, it can be any number), or an error occurs.
- If an intermediate or final result produces NaN or a negative number, an ER_GIS_INVALID_DATA error occurs.
ST_Distance() permits specifying the linear unit for the returned distance value with an optional unit argument which ST_Distance() handles as described in the introduction to this section.
```
mysql> SET @g1 = ST_GeomFromText('POINT(1 1)');
mysql> SET @g2 = ST_GeomFromText('POINT(2 2)');
mysql> SELECT ST_Distance(@g1, @g2);
+-----------------------+
| ST_Distance(@g1, @g2) |
+-----------------------+
|    1.4142135623730951 |
+-----------------------+

mysql> SET @g1 = ST_GeomFromText('POINT(1 1)', 4326);
mysql> SET @g2 = ST_GeomFromText('POINT(2 2)', 4326);
mysql> SELECT ST_Distance(@g1, @g2);
+-----------------------+
| ST_Distance(@g1, @g2) |
+-----------------------+
|     156874.3859490455 |
+-----------------------+
mysql> SELECT ST_Distance(@g1, @g2, 'metre');
+--------------------------------+
| ST_Distance(@g1, @g2, 'metre') |
+--------------------------------+
|              156874.3859490455 |
+--------------------------------+
mysql> SELECT ST_Distance(@g1, @g2, 'foot');
+-------------------------------+
| ST_Distance(@g1, @g2, 'foot') |
+-------------------------------+
|             514679.7439273146 |
+-------------------------------+
```
For the special case of distance calculations on a sphere, see the ST_Distance_Sphere() function.

ST_Equals(g1, g2)

Returns 1 or 0 to indicate whether g1 is spatially equal to g2.

ST_Equals() 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 = Point(1,1), @g2 = Point(2,2);
mysql> SELECT ST_Equals(@g1, @g1), ST_Equals(@g1, @g2);
+---------------------+---------------------+
| ST_Equals(@g1, @g1) | ST_Equals(@g1, @g2) |
+---------------------+---------------------+
|                   1 |                   0 |
+---------------------+---------------------+

ST_FrechetDistance(g1, g2 [, unit])
Returns the discrete Fréchet distance between two geometries, reflecting how similar the geometries are. The result is a double-precision number measured in the length unit of the spatial reference system (SRS) of the geometry arguments, or in the length unit of the unit argument if that argument is given.
This function implements the discrete Fréchet distance, which means it is restricted to distances between the points of the geometries. For example, given two LineString arguments, only the points explicitly mentioned in the geometries are considered. Points on the line segments between these points are not considered.
ST_FrechetDistance() handles its geometry arguments as described in the introduction to this section, with these exceptions:
- The geometries may have a Cartesian or geographic SRS, but only LineString values are supported. If the arguments are in the same Cartesian or geographic SRS, but either is not a LineString, an ER_NOT_IMPLEMENTED_FOR_CARTESIAN_SRS or ER_NOT_IMPLEMENTED_FOR_GEOGRAPHIC_SRS error occurs, depending on the SRS type.
ST_FrechetDistance() handles its optional unit argument as described in the introduction to this section.
```
mysql> SET @ls1 = ST_GeomFromText('LINESTRING(0 0,0 5,5 5)');
mysql> SET @ls2 = ST_GeomFromText('LINESTRING(0 1,0 6,3 3,5 6)');
mysql> SELECT ST_FrechetDistance(@ls1, @ls2);
+--------------------------------+
| ST_FrechetDistance(@ls1, @ls2) |
+--------------------------------+
|             2.8284271247461903 |
+--------------------------------+

mysql> SET @ls1 = ST_GeomFromText('LINESTRING(0 0,0 5,5 5)', 4326);
mysql> SET @ls2 = ST_GeomFromText('LINESTRING(0 1,0 6,3 3,5 6)', 4326);
mysql> SELECT ST_FrechetDistance(@ls1, @ls2);
+--------------------------------+
| ST_FrechetDistance(@ls1, @ls2) |
+--------------------------------+
|              313421.1999416798 |
+--------------------------------+
mysql> SELECT ST_FrechetDistance(@ls1, @ls2, 'foot');
+----------------------------------------+
| ST_FrechetDistance(@ls1, @ls2, 'foot') |
+----------------------------------------+
|                     1028284.7767115477 |
+----------------------------------------+
```
ST_HausdorffDistance(g1, g2 [, unit])
Returns the discrete Hausdorff distance between two geometries, reflecting how similar the geometries are. The result is a double-precision number measured in the length unit of the spatial reference system (SRS) of the geometry arguments, or in the length unit of the unit argument if that argument is given.
This function implements the discrete Hausdorff distance, which means it is restricted to distances between the points of the geometries. For example, given two LineString arguments, only the points explicitly mentioned in the geometries are considered. Points on the line segments between these points are not considered.
ST_HausdorffDistance() handles its geometry arguments as described in the introduction to this section, with these exceptions:
- If the geometry arguments are in the same Cartesian or geographic SRS, but are not in a supported combination, an ER_NOT_IMPLEMENTED_FOR_CARTESIAN_SRS or ER_NOT_IMPLEMENTED_FOR_GEOGRAPHIC_SRS error occurs, depending on the SRS type. These combinations are supported:
  - LineString and LineString
  - Point and MultiPoint
  - LineString and MultiLineString
  - MultiPoint and MultiPoint
  - MultiLineString and MultiLineString
ST_HausdorffDistance() handles its optional unit argument as described in the introduction to this section.
```
mysql> SET @ls1 = ST_GeomFromText('LINESTRING(0 0,0 5,5 5)');
mysql> SET @ls2 = ST_GeomFromText('LINESTRING(0 1,0 6,3 3,5 6)');
mysql> SELECT ST_HausdorffDistance(@ls1, @ls2);
+----------------------------------+
| ST_HausdorffDistance(@ls1, @ls2) |
+----------------------------------+
|                                1 |
+----------------------------------+

mysql> SET @ls1 = ST_GeomFromText('LINESTRING(0 0,0 5,5 5)', 4326);
mysql> SET @ls2 = ST_GeomFromText('LINESTRING(0 1,0 6,3 3,5 6)', 4326);
mysql> SELECT ST_HausdorffDistance(@ls1, @ls2);
+----------------------------------+
| ST_HausdorffDistance(@ls1, @ls2) |
+----------------------------------+
|               111319.49079326246 |
+----------------------------------+
mysql> SELECT ST_HausdorffDistance(@ls1, @ls2, 'foot');
+------------------------------------------+
| ST_HausdorffDistance(@ls1, @ls2, 'foot') |
+------------------------------------------+
|                        365221.4264870815 |
+------------------------------------------+
```
ST_Intersects(g1, g2)
Returns 1 or 0 to indicate whether g1 spatially intersects g2.
ST_Intersects() handles its arguments as described in the introduction to this section.
ST_Overlaps(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 g1 spatially overlaps g2.
ST_Overlaps() handles its arguments as described in the introduction to this section except that the return value is NULL for the additional condition that the dimensions of the two geometries are not equal.
ST_Touches(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 g1 spatially touches g2.
ST_Touches() handles its arguments as described in the introduction to this section except that the return value is NULL for the additional condition that both geometries are of dimension 0 (Point or MultiPoint).
ST_Within(g1, g2)
Returns 1 or 0 to indicate whether g1 is spatially within g2. This tests the opposite relationship as ST_Contains().
ST_Within() handles its arguments as described in the introduction to this section.