Data types

mysql_sphere 0.1 Based on pgSphere 1.1.1
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3.1. Overview

mysql_sphere provides spherical data types for storing with MySQL / MariaDB. This is achieved by storing the according data types as base64 encoded binary BLOBs. I.e. in order to store mysql_sphere objects, you need to store them in BLOB columns! Furthermore, there is a data type to do transformations.

Table 1. Data types

`SQL` type name	spherical type
spoint	point (position)
strans	Euler transformation
scircle	circle
sline	line
sellipse	ellipse
spoly	polygon
spath	path
sbox	coordinate range

3.2. Point

A spherical point is an object without expanse but with a position.

Use cases are:

sites on earth
star positions on the sky sphere
spherical positions on planets

A spherical point (or position) is given by two values: longitude and latitude. Longitude is a floating point value between 0 and 2π. Latitude is a floating point value, too, but between -π/2 and π/2. It is possible to give a spherical position in degrees (DEG) or with a triple value of degrees, minutes and seconds (DMS). Degrees and minutes are integer values. The seconds are represented using a floating point value. A fourth method is specifying a longitude value as a triple value of hours, minutes and seconds (HMS). But, you cannot use it with latitude values.

Example 1. A position specified using longitude and latitude in radians

mysql> SELECT spoint( '(0.1,-0.2)' );
+---------------------------------------+
| spoint( '(0.1,-0.2)' )                |
+---------------------------------------+
| PHB0PpqZmZmZmbk/oJmZmZmZyb88dHA+AA==  |
+---------------------------------------+
1 row in set (0.02 sec)

The mysql_sphere data type can be converted to a readable string using sstr:

mysql> SELECT sstr(spoint( '(0.1,-0.2)' ));
+------------------------------+
| sstr(spoint( '(0.1,-0.2)' )) |
+------------------------------+
| (0.1 , -0.2)                 |
+------------------------------+
1 row in set (0.00 sec)

or

mysql> SELECT sstr('PHB0PpqZmZmZmbk/oJmZmZmZyb88dHA+AA==')
+----------------------------------------------+
| sstr('PHB0PpqZmZmZmbk/oJmZmZmZyb88dHA+AA==') |
+----------------------------------------------+
| (0.1 , -0.2)                                 |
+----------------------------------------------+
1 row in set (0.00 sec)

Example 2. A position specified using longitude and latitude in degrees

mysql> SELECT spoint( '( 10.1d, -90d)' );
+---------------------------------------+
| spoint( '( 10.1d, -90d)' )            |
+---------------------------------------+
| PHB0Pkuz7ixJkMY/GC1EVPsh+b88dHA+AA==  |
+---------------------------------------+
1 row in set (0.01 sec)

Example 3. A position specified using longitude and latitude (DMS)

mysql> SELECT spoint( '( 10d 12m 11.3s, -13d 14m)' );
+----------------------------------------+
| spoint( '( 10d 12m 11.3s, -13d 14m)' ) |
+----------------------------------------+
| PHB0Pmvvlp5Fy8Y/oOws10SQzb88dHA+AA==   |
+----------------------------------------+
1 row in set (0.00 sec)

Example 4. A position specified using longitude in HMS, and latitude in RAD

mysql> SELECT spoint( '( 23h 44m 10s, -1.4321 )' );
+---------------------------------------+
| spoint( '( 23h 44m 10s, -1.4321 )' )  |
+---------------------------------------+
| PHB0Pr9QrNw82xhAKKCJsOHp9r88dHA+AA==  |
+---------------------------------------+
1 row in set (0.00 sec)

As you can see you can combine the input format for longitude and latitude. The value pairs are always enclosed within braces. Spaces are optional.

3.3. Euler transformation

A Euler transformation is done with three counterclockwise object rotations around the following axes: x-axis, y-axis, or z-axis.

Use cases are:

spherical object transformations
spherical coordinates transformations

The input syntax of a Euler transformation is:

angle1, angle2, angle3 [, axes ]

where axes is an optional 3 letter code with letters : X, Y, or Z. Default is ZXZ. angleN is any valid angle with the input format RAD, DEG, or DMS.

To do a transformation, you have to use a transformation operator (see Section 5.10).

Example 5. Create a transformation object

Create a transformation object to rotate a spherical object counterclockwise, first 20° around the x-axis, second -270° around the z-axis and last 70.5° around the y-axis.

mysql> SELECT strans( '20d, -270d, 70.5d, XZY' );
+-----------------------------------------------------------+
| strans( '20d, -270d, 70.5d, XZY' )                        |
+-----------------------------------------------------------+
| PGV0Pi0AAAAAAAAAh0TnShhX1j8YLURU+yH5PzEBz87zr/M/PHRlPgA=  |
+-----------------------------------------------------------+
1 row in set (0.04 sec)

Example 6. Create a second transformation object

Create a transformation object to rotate a spherical object counterclockwise, first 2° 20' around the z-axis, second 10° around the x-axis, and last 0° around the z-axis.

mysql> SELECT strans( '2d 20m, 10d, 0' );
+-----------------------------------------------------------+
| strans( '2d 20m, 10d, 0' )                                |
+-----------------------------------------------------------+
| PGV0PjcAAAAAAAAAwwwLaNLZpD+HROdKGFfGPwAAAAAAAAAAPHRlPgA=  |
+-----------------------------------------------------------+
1 row in set (0.00 sec)

3.4. Circle

A spherical circle is an area around a point, where all points inside the circle have a distance less than or equal to the radius of the circle.

Use cases are:

sites on earth having a maximum distance from another site
round cluster or nebula on sky sphere
a position with an undirected position error

A circle is specified using a spherical point (spoint) and a radius:

< point , radius >

The radius is given as the "opening angle" of the circle (when viewed from the center of the sphere). Valid radius units are RAD, DEG, and DMS. The circle radius must be less than or equal to 90° and cannot be less than zero.

Example 7. A circle around the North Pole with a radius of 5°.

mysql> SELECT scircle( '< (0d, 90d), 5d >' );
+-----------------------------------------------+
| scircle( '< (0d, 90d), 5d >' )                |
+-----------------------------------------------+
| PGNsPgAAAAAAAAAAGC1EVPsh+T+HROdKGFe2PzxsYz4A  |
+-----------------------------------------------+
1 row in set (0.00 sec)

3.5. Line

A spherical line is part of a great circle (meridian) that has a beginning and an end and hence, a direction.

Use cases are:

direct connection of two points on a sphere
meteors on the sky sphere

To allow lines with a length larger than 180°, the input i syntax is somewhat complex.

A general located line with a length length is defined as a line starting at position (0d,0d) and ending at position (length,0d) transformed with a Euler transformation euler. The input syntax is :

( euler ), length

Note:

For a simpler line input, use casting operators (Section 5.1) or constructor functions (Section 4).
If the length is larger than 360°, the line length is truncated to 360°.
The transformation euler will always be converted to a Euler transformation using axes Z, X, and Z.

Example 8. A line input

A line starting at position (200d,+20d) and ending at position (200d,-10d).

mysql> SELECT sline( '( -90d, -20d, 200d, XYZ ), 30d ' );
+-----------------------------------------------------------+
| sline( '( -90d, -20d, 200d, XYZ ), 30d ' )                |
+-----------------------------------------------------------+
| PGxpPodE50oYVwZAGC1EVPsh+T9gofLYbIcaQGVzLThSweA/PGlsPgA=  |
+-----------------------------------------------------------+
1 row in set (0.07 sec)

3.6. Ellipses

Within mysql_sphere, ellipses are defined as :

If the center of any spherical ellipse is the North Pole, the perpendicular projection into the x-y-plane gives an ellipse as in two-dimensional space.

Use cases are:

cluster or nebula on the sky sphere where the `height' is lower than the `width'
to describe a position error

An ellipse always has:

a major radius rad_1
a minor radius rad_2
a center center
a position angle pos (inclination)

Hence, the input syntax is:

< { rad_1, rad_2 }, center, pos >

Note:

The radii rad_1 and rad_2 have to be less than 90°.
If rad_1 is less than rad_2 , the values will be swapped.
The position angle pos is defined within mysql_sphere as a counterclockwise rotation around the ellipse center and is zero, if the ellipse is “parallel to the equator”

Example 9. Input of a spherical ellipse

An ellipse has a center at 20° longitude and 0° latitude. The minor radius is part of the equator and 5°, the major radius is 10°.

mysql> SELECT sellipse( '< { 10d, 5d } , ( 20d, 0d ), 90d >' );
+-----------------------------------------------------------------------+
| sellipse( '< { 10d, 5d } , ( 20d, 0d ), 90d >' )                      |
+-----------------------------------------------------------------------+
| PGVsPodE50oYV8Y/h0TnShhXtj8YLURU+yH5PwAAAAAAAAAAh0TnShhX1j88bGU+AA==  |
+-----------------------------------------------------------------------+
1 row in set (0.03 sec)

3.7. Path

A spherical path is a concatination of spherical lines. Use cases are:

rivers on earth
trajectories of planets or comets on the sky plane

Paths within mysql_sphere are simplified lists of positions. The input syntax is :

{pos1,pos2[,pos3[,pos4[,...]]]}

Note:

The distance between 2 sequent positions has to be less than 180° and greater than zero.
At least 2 positions are required.

Example 10. Path input example

A path going from (10d,0d) to (80d,30d) via (45d,15d).

mysql> SELECT spath( '{ (10d,0d),(45d,15d),(80d,30d) } ' );
+--------------------------------------------------------------------------------------------------------+
| spath( '{ (10d,0d),(45d,15d),(80d,30d) } ' )                                                           |
+--------------------------------------------------------------------------------------------------------+
| PHBhPkAAAAAAAAAAAwAAAAAAAACHROdKGFfGPwAAAAAAAAAAGC1EVPsh6T9lcy04UsHQP4dE50oY
V/Y/ZXMtOFLB4D88YXA+AA==  |
+--------------------------------------------------------------------------------------------------------+
1 row in set (0.02 sec)

3.8. Polygon

A spherical polygon is a closed spherical path where line segments cannot be crossed. One main use case are areas on the earth and sky sphere. Polygons within mysql_sphere have the same input syntax as paths:

{pos1,pos2,pos3[,pos4[,... ]]}

Note:

A spherical polygon has the same restrictions as a spherical path (see Section 3.7). Except that a polygon needs at least 3 positions.
The line segments can not be crossed.
The maximum dimension of a polygon must be less than 180°.

Example 11. Input of polygon

A polygon going from (270d,-10d). via (270d,30d) to (290d,10d) back to (270d,-10d)

mysql> SELECT spoly( '{ (270d,-10d), (270d,30d), (290d,10d) } ');
+--------------------------------------------------------------------------------------------------------+
| spoly( '{ (270d,-10d), (270d,30d), (290d,10d) } ')                                                     |
+--------------------------------------------------------------------------------------------------------+
| PHBnPkAAAAAAAAAAAwAAAAAAAADSITN/fNkSQIBE50oYV8a/0iEzf3zZEkBlcy04UsHgPxqW4QPu
PhRAh0TnShhXxj88Z3A+AA==  |
+--------------------------------------------------------------------------------------------------------+

3.9. Coordinates range

A spherical box is a coordinates range. Hence, you can select objects within a longitude range and latitude range. The box is represented using two spherical points: the southwest (pos_sw) and the northeast (pos_ne) edge of the box. The input syntax is:

( pos_sw, pos_ne )

or

pos_sw, pos_ne

Note:

If the latitude of the southwest edge is larger than the latitude of the northeast edge, mysql_sphere swaps the edges.
If the longitude of the southwest edge is equal to the longitude of the northeast edge, mysql_sphere assumes a full latitude range, except that the latitudes are equal, too.

Example 12. Input of a full latitude range

A full latitude range between +20° and +23°.

mysql> SELECT sbox( '( (0d,20d), (0d,23d) )' );
+-----------------------------------------------------------+
| sbox( '( (0d,20d), (0d,23d) )' )                          |
+-----------------------------------------------------------+
| PGJ4PgAAAAAAAAAAh0TnShhX1j8YLURU+yEZQALCVon1sNk/PHhiPgA=  |
+-----------------------------------------------------------+
1 row in set (0.01 sec)

Example 13. A simple coordinates range

A coordinate range between -10° and +10° in latitude and 350° and 10° in longitude.

mysql> SELECT sbox( '( (350d,-10d), (10d,+10d) )' );
+-----------------------------------------------------------+
| sbox( '( (350d,-10d), (10d,+10d) )' )                     |
+-----------------------------------------------------------+
| PGJ4PvTy7JFCbxhAgETnShhXxr+HROdKGFfGP4dE50oYV8Y/PHhiPgA=  |
+-----------------------------------------------------------+
1 row in set (0.00 sec)

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