## 9.11. Geometric Functions and Operators

The geometric types `point`, `box`, `lseg`, `line`, `path`, `polygon`, and `circle` have a large set of native support functions and operators, shown in Table 9.33, Table 9.34, and Table 9.35.

### Caution

Note that the “same as” operator, `~=`, represents the usual notion of equality for the `point`, `box`, `polygon`, and `circle` types. Some of these types also have an `=` operator, but `=` compares for equal areas only. The other scalar comparison operators (`<=` and so on) likewise compare areas for these types.

Table 9.33. Geometric Operators

Operator | Description | Example
--- | --- | ---
`+` | Translation | `box '((0,0),(1,1))' + point '(2.0,0)'`
`-` | Translation | `box '((0,0),(1,1))' - point '(2.0,0)'`
`*` | Scaling/rotation | `box '((0,0),(1,1))' * point '(2.0,0)'`
`/` | Scaling/rotation | `box '((0,0),(2,2))' / point '(2.0,0)'`
`#` | Point or box of intersection | `box '((1,-1),(-1,1))' # box '((1,1),(-2,-2))'`
`#` | Number of points in path or polygon | `# path '((1,0),(0,1),(-1,0))'`
`@-@` | Length or circumference | `@-@ path '((0,0),(1,0))'`
`@@` | Center | `@@ circle '((0,0),10)'`
`##` | Closest point to first operand on second operand | `point '(0,0)' ## lseg '((2,0),(0,2))'`
`<->` | Distance between | `circle '((0,0),1)' <-> circle '((5,0),1)'`
`&&` | Overlaps? (One point in common makes this true.) | `box '((0,0),(1,1))' && box '((0,0),(2,2))'`
`<<` | Is strictly left of? | `circle '((0,0),1)' << circle '((5,0),1)'`
`>>` | Is strictly right of? | `circle '((5,0),1)' >> circle '((0,0),1)'`
`&<` | Does not extend to the right of? | `box '((0,0),(1,1))' &< box '((0,0),(2,2))'`
`&>` | Does not extend to the left of? | `box '((0,0),(3,3))' &> box '((0,0),(2,2))'`
`<<|` | Is strictly below? | `box '((0,0),(3,3))' <<| box '((3,4),(5,5))'`
`|>>` | Is strictly above? | `box '((3,4),(5,5))' |>> box '((0,0),(3,3))'`
`&<|` | Does not extend above? | `box '((0,0),(1,1))' &<| box '((0,0),(2,2))'`
`|&>` | Does not extend below? | `box '((0,0),(3,3))' |&> box '((0,0),(2,2))'`
`<^` | Is below (allows touching)? | `circle '((0,0),1)' <^ circle '((0,5),1)'`
`>^` | Is above (allows touching)? | `circle '((0,5),1)' >^ circle '((0,0),1)'`
`?#` | Intersects? | `lseg '((-1,0),(1,0))' ?# box '((-2,-2),(2,2))'`
`?-` | Is horizontal? | `?- lseg '((-1,0),(1,0))'`
`?-` | Are horizontally aligned? | `point '(1,0)' ?- point '(0,0)'`
`?|` | Is vertical? | `?| lseg '((-1,0),(1,0))'`
`?|` | Are vertically aligned? | `point '(0,1)' ?| point '(0,0)'`
`?-|` | Is perpendicular? | `lseg '((0,0),(0,1))' ?-| lseg '((0,0),(1,0))'`
`?||` | Are parallel? | `lseg '((-1,0),(1,0))' ?|| lseg '((-1,2),(1,2))'`
`@>` | Contains? | `circle '((0,0),2)' @> point '(1,1)'`
`<@` | Contained in or on? | `point '(1,1)' <@ circle '((0,0),2)'`
`~=` | Same as? | `polygon '((0,0),(1,1))' ~= polygon '((1,1),(0,0))'`

### Note

Before PostgreSQL 8.2, the containment operators `@>` and `<@` were respectively called `~` and `@`. These names are still available, but are deprecated and will eventually be removed.

Table 9.34. Geometric Functions

Function | Return Type | Description | Example
--- | --- | --- | ---
``area(`object`)`` | `double precision` | area | `area(box '((0,0),(1,1))')`
``center(`object`)`` | `point` | center | `center(box '((0,0),(1,2))')`
``diameter(`circle`)`` | `double precision` | diameter of circle | `diameter(circle '((0,0),2.0)')`
``height(`box`)`` | `double precision` | vertical size of box | `height(box '((0,0),(1,1))')`
``isclosed(`path`)`` | `boolean` | a closed path? | `isclosed(path '((0,0),(1,1),(2,0))')`
``isopen(`path`)`` | `boolean` | an open path? | `isopen(path '[(0,0),(1,1),(2,0)]')`
``length(`object`)`` | `double precision` | length | `length(path '((-1,0),(1,0))')`
``npoints(`path`)`` | `int` | number of points | `npoints(path '[(0,0),(1,1),(2,0)]')`
``npoints(`polygon`)`` | `int` | number of points | `npoints(polygon '((1,1),(0,0))')`
``pclose(`path`)`` | `path` | convert path to closed | `pclose(path '[(0,0),(1,1),(2,0)]')`
``popen(`path`)`` | `path` | convert path to open | `popen(path '((0,0),(1,1),(2,0))')`
``radius(`circle`)`` | `double precision` | radius of circle | `radius(circle '((0,0),2.0)')`
``width(`box`)`` | `double precision` | horizontal size of box | `width(box '((0,0),(1,1))')`

Table 9.35. Geometric Type Conversion Functions

Function | Return Type | Description | Example
--- | --- | --- | ---
``box(`circle`)`` | `box` | circle to box | `box(circle '((0,0),2.0)')`
``box(`point`)`` | `box` | point to empty box | `box(point '(0,0)')`
``box(`point`, `point`)`` | `box` | points to box | `box(point '(0,0)', point '(1,1)')`
``box(`polygon`)`` | `box` | polygon to box | `box(polygon '((0,0),(1,1),(2,0))')`
``bound_box(`box`, `box`)`` | `box` | boxes to bounding box | `bound_box(box '((0,0),(1,1))', box '((3,3),(4,4))')`
``circle(`box`)`` | `circle` | box to circle | `circle(box '((0,0),(1,1))')`
``circle(`point`, `double precision`)`` | `circle` | center and radius to circle | `circle(point '(0,0)', 2.0)`
``circle(`polygon`)`` | `circle` | polygon to circle | `circle(polygon '((0,0),(1,1),(2,0))')`
``line(`point`, `point`)`` | `line` | points to line | `line(point '(-1,0)', point '(1,0)')`
``lseg(`box`)`` | `lseg` | box diagonal to line segment | `lseg(box '((-1,0),(1,0))')`
``lseg(`point`, `point`)`` | `lseg` | points to line segment | `lseg(point '(-1,0)', point '(1,0)')`
``path(`polygon`)`` | `path` | polygon to path | `path(polygon '((0,0),(1,1),(2,0))')`
``point`(`double precision`, `double precision`)` | `point` | construct point | `point(23.4, -44.5)`
``point(`box`)`` | `point` | center of box | `point(box '((-1,0),(1,0))')`
``point(`circle`)`` | `point` | center of circle | `point(circle '((0,0),2.0)')`
``point(`lseg`)`` | `point` | center of line segment | `point(lseg '((-1,0),(1,0))')`
``point(`polygon`)`` | `point` | center of polygon | `point(polygon '((0,0),(1,1),(2,0))')`
``polygon(`box`)`` | `polygon` | box to 4-point polygon | `polygon(box '((0,0),(1,1))')`
``polygon(`circle`)`` | `polygon` | circle to 12-point polygon | `polygon(circle '((0,0),2.0)')`
``polygon(`npts`, `circle`)`` | `polygon` | circle to `npts`-point polygon | `polygon(12, circle '((0,0),2.0)')`
``polygon(`path`)`` | `polygon` | path to polygon | `polygon(path '((0,0),(1,1),(2,0))')`

It is possible to access the two component numbers of a `point` as though the point were an array with indexes 0 and 1. For example, if `t.p` is a `point` column then `SELECT p[0] FROM t` retrieves the X coordinate and `UPDATE t SET p[1] = ...` changes the Y coordinate. In the same way, a value of type `box` or `lseg` can be treated as an array of two `point` values.

The `area` function works for the types `box`, `circle`, and `path`. The `area` function only works on the `path` data type if the points in the `path` are non-intersecting. For example, the `path` `'((0,0),(0,1),(2,1),(2,2),(1,2),(1,0),(0,0))'::PATH` will not work; however, the following visually identical `path` `'((0,0),(0,1),(1,1),(1,2),(2,2),(2,1),(1,1),(1,0),(0,0))'::PATH` will work. If the concept of an intersecting versus non-intersecting `path` is confusing, draw both of the above `path`s side by side on a piece of graph paper.