9.11. Geometric Functions and Operators
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## 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 9.36, Table 9.37, and Table 9.38.

Table 9.36. Geometric Operators

Operator Description Example(s)
`geometric_type` `+` point `geometric_type` Adds the coordinates of the second point to those of each point of the first argument, thus performing translation. Available for point, box, path, circle. `box '(1,1),(0,0)' + point '(2,0)'` (3,1),(2,0)
path `+` path path Concatenates two open paths (returns NULL if either path is closed). `path '[(0,0),(1,1)]' + path '[(2,2),(3,3),(4,4)]'` [(0,0),(1,1),(2,2),(3,3),(4,4)]
`geometric_type` `-` point `geometric_type` Subtracts the coordinates of the second point from those of each point of the first argument, thus performing translation. Available for point, box, path, circle. `box '(1,1),(0,0)' - point '(2,0)'` (-1,1),(-2,0)
`geometric_type` `*` point `geometric_type` Multiplies each point of the first argument by the second point (treating a point as being a complex number represented by real and imaginary parts, and performing standard complex multiplication). If one interprets the second point as a vector, this is equivalent to scaling the object's size and distance from the origin by the length of the vector, and rotating it counterclockwise around the origin by the vector's angle from the `x` axis. Available for point, box,[a] path, circle. `path '((0,0),(1,0),(1,1))' * point '(3.0,0)'` ((0,0),(3,0),(3,3)) `path '((0,0),(1,0),(1,1))' * point(cosd(45), sind(45))` ((0,0),​(0.7071067811865475,0.7071067811865475),​(0,1.414213562373095))
`geometric_type` `/` point `geometric_type` Divides each point of the first argument by the second point (treating a point as being a complex number represented by real and imaginary parts, and performing standard complex division). If one interprets the second point as a vector, this is equivalent to scaling the object's size and distance from the origin down by the length of the vector, and rotating it clockwise around the origin by the vector's angle from the `x` axis. Available for point, box,[a] path, circle. `path '((0,0),(1,0),(1,1))' / point '(2.0,0)'` ((0,0),(0.5,0),(0.5,0.5)) `path '((0,0),(1,0),(1,1))' / point(cosd(45), sind(45))` ((0,0),​(0.7071067811865476,-0.7071067811865476),​(1.4142135623730951,0))
`@-@` `geometric_type` double precision Computes the total length. Available for lseg, path. `@-@ path '[(0,0),(1,0),(1,1)]'` 2
`@@` `geometric_type` point Computes the center point. Available for box, lseg, polygon, circle. `@@ box '(2,2),(0,0)'` (1,1)
`#` `geometric_type` integer Returns the number of points. Available for path, polygon. `# path '((1,0),(0,1),(-1,0))'` 3
`geometric_type` `#` `geometric_type` point Computes the point of intersection, or NULL if there is none. Available for lseg, line. `lseg '[(0,0),(1,1)]' # lseg '[(1,0),(0,1)]'` (0.5,0.5)
box `#` box box Computes the intersection of two boxes, or NULL if there is none. `box '(2,2),(-1,-1)' # box '(1,1),(-2,-2)'` (1,1),(-1,-1)
`geometric_type` `##` `geometric_type` point Computes the closest point to the first object on the second object. Available for these pairs of types: (point, box), (point, lseg), (point, line), (lseg, box), (lseg, lseg), (line, lseg). `point '(0,0)' ## lseg '[(2,0),(0,2)]'` (1,1)
`geometric_type` `<->` `geometric_type` double precision Computes the distance between the objects. Available for all seven geometric types, for all combinations of point with another geometric type, and for these additional pairs of types: (box, lseg), (lseg, line), (polygon, circle) (and the commutator cases). `circle '<(0,0),1>' <-> circle '<(5,0),1>'` 3
`geometric_type` `@>` `geometric_type` boolean Does first object contain second? Available for these pairs of types: (`box`, `point`), (`box`, `box`), (`path`, `point`), (`polygon`, `point`), (`polygon`, `polygon`), (`circle`, `point`), (`circle`, `circle`). `circle '<(0,0),2>' @> point '(1,1)'` t
`geometric_type` `<@` `geometric_type` boolean Is first object contained in or on second? Available for these pairs of types: (`point`, `box`), (`point`, `lseg`), (`point`, `line`), (`point`, `path`), (`point`, `polygon`), (`point`, `circle`), (`box`, `box`), (`lseg`, `box`), (`lseg`, `line`), (`polygon`, `polygon`), (`circle`, `circle`). `point '(1,1)' <@ circle '<(0,0),2>'` t
`geometric_type` `&&` `geometric_type` boolean Do these objects overlap? (One point in common makes this true.) Available for box, polygon, circle. `box '(1,1),(0,0)' && box '(2,2),(0,0)'` t
`geometric_type` `<<` `geometric_type` boolean Is first object strictly left of second? Available for point, box, polygon, circle. `circle '<(0,0),1>' << circle '<(5,0),1>'` t
`geometric_type` `>>` `geometric_type` boolean Is first object strictly right of second? Available for point, box, polygon, circle. `circle '<(5,0),1>' >> circle '<(0,0),1>'` t
`geometric_type` `&<` `geometric_type` boolean Does first object not extend to the right of second? Available for box, polygon, circle. `box '(1,1),(0,0)' &< box '(2,2),(0,0)'` t
`geometric_type` `&>` `geometric_type` boolean Does first object not extend to the left of second? Available for box, polygon, circle. `box '(3,3),(0,0)' &> box '(2,2),(0,0)'` t
`geometric_type` `<<|` `geometric_type` boolean Is first object strictly below second? Available for point, box, polygon, circle. `box '(3,3),(0,0)' <<| box '(5,5),(3,4)'` t
`geometric_type` `|>>` `geometric_type` boolean Is first object strictly above second? Available for point, box, polygon, circle. `box '(5,5),(3,4)' |>> box '(3,3),(0,0)'` t
`geometric_type` `&<|` `geometric_type` boolean Does first object not extend above second? Available for box, polygon, circle. `box '(1,1),(0,0)' &<| box '(2,2),(0,0)'` t
`geometric_type` `|&>` `geometric_type` boolean Does first object not extend below second? Available for box, polygon, circle. `box '(3,3),(0,0)' |&> box '(2,2),(0,0)'` t
box `<^` box boolean Is first object below second (allows edges to touch)? `box '((1,1),(0,0))' <^ box '((2,2),(1,1))'` t
box `>^` box boolean Is first object above second (allows edges to touch)? `box '((2,2),(1,1))' >^ box '((1,1),(0,0))'` t
`geometric_type` `?#` `geometric_type` boolean Do these objects intersect? Available for these pairs of types: (box, box), (lseg, box), (lseg, lseg), (lseg, line), (line, box), (line, line), (path, path). `lseg '[(-1,0),(1,0)]' ?# box '(2,2),(-2,-2)'` t
`?-` line boolean `?-` lseg boolean Is line horizontal? `?- lseg '[(-1,0),(1,0)]'` t
point `?-` point boolean Are points horizontally aligned (that is, have same y coordinate)? `point '(1,0)' ?- point '(0,0)'` t
`?|` line boolean `?|` lseg boolean Is line vertical? `?| lseg '[(-1,0),(1,0)]'` f
point `?|` point boolean Are points vertically aligned (that is, have same x coordinate)? `point '(0,1)' ?| point '(0,0)'` t
line `?-|` line boolean lseg `?-|` lseg boolean Are lines perpendicular? `lseg '[(0,0),(0,1)]' ?-| lseg '[(0,0),(1,0)]'` t
line `?||` line boolean lseg `?||` lseg boolean Are lines parallel? `lseg '[(-1,0),(1,0)]' ?|| lseg '[(-1,2),(1,2)]'` t
`geometric_type` `~=` `geometric_type` boolean Are these objects the same? Available for point, box, polygon, circle. `polygon '((0,0),(1,1))' ~= polygon '((1,1),(0,0))'` t
[a] “Rotating” a box with these operators only moves its corner points: the box is still considered to have sides parallel to the axes. Hence the box's size is not preserved, as a true rotation would do.

### Caution

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

### Note

Before PostgreSQL™ 14, the point is strictly below/above comparison operators point `<<|` point and point `|>>` point were respectively called `<^` and `>^`. These names are still available, but are deprecated and will eventually be removed.

Table 9.37. Geometric Functions

Function Description Example(s)
`area` ( `geometric_type` ) double precision Computes area. Available for box, path, circle. A path input must be closed, else NULL is returned. Also, if the path is self-intersecting, the result may be meaningless. `area(box '(2,2),(0,0)')` 4
`center` ( `geometric_type` ) point Computes center point. Available for box, circle. `center(box '(1,2),(0,0)')` (0.5,1)
`diagonal` ( box ) lseg Extracts box's diagonal as a line segment (same as `lseg(box)`). `diagonal(box '(1,2),(0,0)')` [(1,2),(0,0)]
`diameter` ( circle ) double precision Computes diameter of circle. `diameter(circle '<(0,0),2>')` 4
`height` ( box ) double precision Computes vertical size of box. `height(box '(1,2),(0,0)')` 2
`isclosed` ( path ) boolean Is path closed? `isclosed(path '((0,0),(1,1),(2,0))')` t
`isopen` ( path ) boolean Is path open? `isopen(path '[(0,0),(1,1),(2,0)]')` t
`length` ( `geometric_type` ) double precision Computes the total length. Available for lseg, path. `length(path '((-1,0),(1,0))')` 4
`npoints` ( `geometric_type` ) integer Returns the number of points. Available for path, polygon. `npoints(path '[(0,0),(1,1),(2,0)]')` 3
`pclose` ( path ) path Converts path to closed form. `pclose(path '[(0,0),(1,1),(2,0)]')` ((0,0),(1,1),(2,0))
`popen` ( path ) path Converts path to open form. `popen(path '((0,0),(1,1),(2,0))')` [(0,0),(1,1),(2,0)]
`radius` ( circle ) double precision Computes radius of circle. `radius(circle '<(0,0),2>')` 2
`slope` ( point, point ) double precision Computes slope of a line drawn through the two points. `slope(point '(0,0)', point '(2,1)')` 0.5
`width` ( box ) double precision Computes horizontal size of box. `width(box '(1,2),(0,0)')` 1

Table 9.38. Geometric Type Conversion Functions

Function Description Example(s)
`box` ( circle ) box Computes box inscribed within the circle. `box(circle '<(0,0),2>')` (1.414213562373095,1.414213562373095),​(-1.414213562373095,-1.414213562373095)
`box` ( point ) box Converts point to empty box. `box(point '(1,0)')` (1,0),(1,0)
`box` ( point, point ) box Converts any two corner points to box. `box(point '(0,1)', point '(1,0)')` (1,1),(0,0)
`box` ( polygon ) box Computes bounding box of polygon. `box(polygon '((0,0),(1,1),(2,0))')` (2,1),(0,0)
`bound_box` ( box, box ) box Computes bounding box of two boxes. `bound_box(box '(1,1),(0,0)', box '(4,4),(3,3)')` (4,4),(0,0)
`circle` ( box ) circle Computes smallest circle enclosing box. `circle(box '(1,1),(0,0)')` <(0.5,0.5),0.7071067811865476>
`circle` ( point, double precision ) circle Constructs circle from center and radius. `circle(point '(0,0)', 2.0)` <(0,0),2>
`circle` ( polygon ) circle Converts polygon to circle. The circle's center is the mean of the positions of the polygon's points, and the radius is the average distance of the polygon's points from that center. `circle(polygon '((0,0),(1,3),(2,0))')` <(1,1),1.6094757082487299>
`line` ( point, point ) line Converts two points to the line through them. `line(point '(-1,0)', point '(1,0)')` {0,-1,0}
`lseg` ( box ) lseg Extracts box's diagonal as a line segment. `lseg(box '(1,0),(-1,0)')` [(1,0),(-1,0)]
`lseg` ( point, point ) lseg Constructs line segment from two endpoints. `lseg(point '(-1,0)', point '(1,0)')` [(-1,0),(1,0)]
`path` ( polygon ) path Converts polygon to a closed path with the same list of points. `path(polygon '((0,0),(1,1),(2,0))')` ((0,0),(1,1),(2,0))
`point` ( double precision, double precision ) point Constructs point from its coordinates. `point(23.4, -44.5)` (23.4,-44.5)
`point` ( box ) point Computes center of box. `point(box '(1,0),(-1,0)')` (0,0)
`point` ( circle ) point Computes center of circle. `point(circle '<(0,0),2>')` (0,0)
`point` ( lseg ) point Computes center of line segment. `point(lseg '[(-1,0),(1,0)]')` (0,0)
`point` ( polygon ) point Computes center of polygon (the mean of the positions of the polygon's points). `point(polygon '((0,0),(1,1),(2,0))')` (1,0.3333333333333333)
`polygon` ( box ) polygon Converts box to a 4-point polygon. `polygon(box '(1,1),(0,0)')` ((0,0),(0,1),(1,1),(1,0))
`polygon` ( circle ) polygon Converts circle to a 12-point polygon. `polygon(circle '<(0,0),2>')` ((-2,0),​(-1.7320508075688774,0.9999999999999999),​(-1.0000000000000002,1.7320508075688772),​(-1.2246063538223773e-16,2),​(0.9999999999999996,1.7320508075688774),​(1.732050807568877,1.0000000000000007),​(2,2.4492127076447545e-16),​(1.7320508075688776,-0.9999999999999994),​(1.0000000000000009,-1.7320508075688767),​(3.673819061467132e-16,-2),​(-0.9999999999999987,-1.732050807568878),​(-1.7320508075688767,-1.0000000000000009))
`polygon` ( integer, circle ) polygon Converts circle to an `n`-point polygon. `polygon(4, circle '<(3,0),1>')` ((2,0),​(3,1),​(4,1.2246063538223773e-16),​(3,-1))
`polygon` ( path ) polygon Converts closed path to a polygon with the same list of points. `polygon(path '((0,0),(1,1),(2,0))')` ((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.

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