9.11. Geometric Functions and Operators

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,¹ 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, 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

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

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,-1.00000000

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))