Crosses(g1, g2)

The term spatially crosses denotes a spatial relation between two given
geometries that has the following properties:

o The two geometries intersect.

o Their intersection results in a geometry that has a dimension that is
  one less than the maximum dimension of the two given geometries.

o Their intersection is not equal to either of the two given
  geometries.

This function returns 1 or 0 to indicate whether g1 spatially crosses
g2. If g1 is a Polygon or a MultiPolygon, or if g2 is a Point or a
MultiPoint, the return value is NULL.

URL: http://dev.mysql.com/doc/refman/5.5/en/spatial-relation-functions-object-shapes.html

