MySQL does not allow the definition of custom operators for specific types like PostgresSQL does. Therefore operators are implemented as function calls. Operator functions will return 0 if they are evaluated as false and 1 if true.
mysql_sphere provides different casting mechanisms than possible with pgSphere. These casting methods have been described in the section before, however will be summarised here again. So, you can transform an object to another data type. A cast in mysql_sphere can only be done using a typename(x) construct.
Table 2. Castings
| casting argument | type target | returns |
|---|---|---|
| spoint | scircle | circle with center position spoint and radius 0.0 |
| spoint | sellipse | an ellipse at position spoint and radius 0.0 |
| spoint | sline | a line with length 0.0 at position spoint |
| scircle | sellipse | the scircle as sellipse |
| sline | strans | the Euler transformation of sline |
| sellipse | scircle | the bounding circle of sellipse |
| sellipse | strans | the Euler transformation of sellipse |
| all available types | sstr | a string representation of the according mysql_sphere type |
Example 23. Cast a spherical point as a circle
mysql> SELECT sstr( scircle ( spoint( '(10d,20d)' ) ) ); +-------------------------------------------+ | sstr( scircle ( spoint( '(10d,20d)' ) ) ) | +-------------------------------------------+ | <(10d , 20d) , 0d> | +-------------------------------------------+ 1 row in set (0.01 sec)
All data types of mysql_sphere have the sequal( x, y ) equality function. The corresponding negation function is snotequal( x, y ).
Example 24. Equality of two spherical points
mysql> SELECT sequal( spoint( '(10d,20d)' ), spoint( '(370d,20d)' ) ); +---------------------------------------------------------+ | sequal( spoint( '(10d,20d)' ), spoint( '(370d,20d)' ) ) | +---------------------------------------------------------+ | 1 | +---------------------------------------------------------+ 1 row in set (0.01 sec)
On spheres, an equality relationship is rarely used. There are frequently questions like Is object a contained by object b? or Does object a overlap object b? mysql_sphere supports such queries using special functions returning 1 for true or 0 for false:
Table 3. Contain and overlap operators
| mysql_sphere operator | pgSphere equivalent | operator returns true, if |
|---|---|---|
| scontainsl | @ | the left object is contained by right object |
| slcontainsr | ˜ | the left object contains right object |
| srnotcontainsl | !@ | the left object is not contained by right object |
| slnotcontainsr | !˜ | the left object does not contain right object |
| soverlaps | && | the objects overlap each other |
| snotoverlaps | !&& | the objects do not overlap each other |
An overlap or contain operator does not exist for all combinations of data types. For instance, srcontainsl( scircle, spoint ) is useless because a spherical point can never contain a spherical circle.
Another binary relationship is crossing. mysql_sphere supports only crossing of lines. Therefor the correlative function is named sline_crosses.
The distance function sdist is a non-boolean function returning the distance between two objects in radians. Currently, mysql_sphere supports only distances between points, circles, and between point and circle. If the objects are overlapping, the distance operator returns zero (0.0).
The length/circumference function scircum or slength is a non-boolean unary operator returning the cirumference or length of an object. In the current implementation, mysql_sphere supports only circumferences of circles, polygons, and boxes. It supports lengths of lines and paths too.
The center function scenter is a non-boolean unary function returning the center of an object. In the current implementation of mysql_sphere, only centers of circles and ellipses are supported.
Example 31. Center of a circle
mysql> SET sphere_outmode="DEG";
Query OK, 0 rows affected (0.02 sec)
mysql> SELECT sstr( scenter( scircle( '<(0d,20d),30d>' ) ) );
+------------------------------------------------+
| sstr( scenter( scircle( '<(0d,20d),30d>' ) ) ) |
+------------------------------------------------+
| (0d , 20d) |
+------------------------------------------------+
1 row in set (0.00 sec)
The unary function sswap changes the direction of sline or spath objects. You can use it with a Euler transformation object in the figurative sense, too (Section 5.10).
Example 32. Swap begin and end of a sline
mysql> SET sphere_outmode="DEG";
Query OK, 0 rows affected (0.02 sec)
mysql> SELECT sstr( sswap( sline ( spoint( '(0d,0d)' ), spoint( '(10d,0d)' ) ) ) );
+----------------------------------------------------------------------+
| sstr( sswap( sline ( spoint( '(0d,0d)' ), spoint( '(10d,0d)' ) ) ) ) |
+----------------------------------------------------------------------+
| ( 350d, 180d, 0d, ZXZ ), 10d |
+----------------------------------------------------------------------+
1 row in set (0.02 sec)
The unary function sline_turn turns the path of sline objects, but preserves begin and end of the spherical line. You could also imagine the resulting line as the complementary of the input line on a great circle. Thus, the length of the returned line will be 360° minus the line length of the operator's argument.
The operator sline_turn returns NULL, if the length of sline argument is 0, because the path of returned sline is undefined. One could also
Example 33. Return length and check if north pole on slines
mysql> SET sphere_outmode="DEG"; Query OK, 0 rows affected (0.02 sec) mysql> SELECT slength( sline( spoint( '(0d,0d)' ), spoint( '(0d,10d)' ) ) ) * 180.0 / pi() AS length; +--------------------+ | length | +--------------------+ | 10.000000000000012 | +--------------------+ 1 row in set (0.02 sec) mysql> SELECT srcontainsl( spoint( '(0d,90d)' ), sline( spoint( '(0d,0d)' ), spoint( '(0d,10d)' ) ) ) AS test; +------+ | test | +------+ | 0 | +------+ 1 row in set (0.00 sec) mysql> SELECT slength( sline_turn( sline( spoint( '(0d,0d)' ), spoint( '(0d,10d)' ) ) ) ) * 180.0 / pi() AS length; +--------------------+ | length | +--------------------+ | 350.00000000000006 | +--------------------+ 1 row in set (0.00 sec) mysql> SELECT srcontainsl( spoint( '(0d,90d)' ), sline_turn( sline( spoint( '(0d,0d)' ), spoint( '(0d,10d)' ) ) ) ) AS test; +------+ | test | +------+ | 1 | +------+ 1 row in set (0.00 sec)
As in a plane, translations and rotations are needed to do object or coordinate transformations. With mysql_sphere, it is done using Euler transformations (strans). On a sphere, there aren't real translations. All movements on a sphere are rotations around axes.
The general syntax for a transformation is always:
functions( object, euler )
where functions are stransform for a usual transformation, sinverse for an inverse transformation. You can transform any object having a mysql_sphere data type, except the data type sbox.
Example 34. Transformation of a point
Rotate a point on a sphere counterclockwise, first 90° around the x-axis, second 90° around the z-axis, and last 40.5° around the x-axis.
mysql> SET sphere_outmode="DEG"; Query OK, 0 rows affected (0.02 sec) sql> SELECT sstr( stransform( spoint( '(30d,0d)' ), strans( '90d, 90d, 40.5d, XZX' ) ) ); +------------------------------------------------------------------------------+ | sstr( stransform( spoint( '(30d,0d)' ), strans( '90d, 90d, 40.5d, XZX' ) ) ) | +------------------------------------------------------------------------------+ | (90d , 70.5d) | +------------------------------------------------------------------------------+
To obtain the inverse of a transformation, use sinverse instead of stransform:
Example 35.1 An inverse transformation
mysql> SET sphere_outmode="DEG"; Query OK, 0 rows affected (0.02 sec) mysql> SELECT sstr( sinverse( spoint( '(90d,70.5d)' ), strans( '90d, 90d, 40.5d, XZX' ) ) ); +-------------------------------------------------------------------------------+ | sstr( sinverse( spoint( '(90d,70.5d)' ), strans( '90d, 90d, 40.5d, XZX' ) ) ) | +-------------------------------------------------------------------------------+ | (30d , -1.130303475e-16d) | +-------------------------------------------------------------------------------+ 1 row in set (0.00 sec)
Alternatively, one can also use the inverse of the Euler transformation with strans_invert, which takes an strans-object as argument, as shown in the following example:
Example 35.2 Inverse transformation by using inverted Euler
Transform the point back to the original point by using strans_invert to invert the Euler-expression:
mysql> SET sphere_outmode="DEG";
Query OK, 0 rows affected (0.02 sec)
mysql> SELECT sstr( stransform( spoint( '(90d,70.5d)' ), strans_invert( strans('90d, 90d, 40.5d, XZX') ) ) );
+------------------------------------------------------------------------------------------------+
| sstr( stransform( spoint( '(90d,70.5d)' ), strans_invert( strans('90d, 90d, 40.5d, XZX') ) ) ) |
+------------------------------------------------------------------------------------------------+
| (30d , -1.130303475e-16d) |
+------------------------------------------------------------------------------------------------+
1 row in set (0.02 sec)
And here is another example, equivalent to Example 35 of the pgSphere documentation:
mysql> SET sphere_outmode="DEG"; Query OK, 0 rows affected (0.02 sec) mysql> SELECT sstr( strans_invert( strans( '20d, 50d, 80d, XYZ' ) ) ) AS inverted; +-----------------------+ | inverted | +-----------------------+ | 280d, 310d, 340d, ZYX | +-----------------------+ 1 row in set (0.02 sec)