DE-9IM Intersection Model
Documentation | Other Specifications | DE-9IM
TatukGIS supports a powerful, yet easy to manage, mechanism for querying against spatial relationships between two shapes known as the Dimensionally Extended 9 Intersection Model (DE-9IM). See also at Wikipedia.
The idea is to manage a nine-element mechanism in which each item defines relationship possibilities between the interiors, exteriors, and boundaries of two shapes. A relationship is defined in terms of intersection types, or lack of an intersection.
The meanings of the interior, exterior, and boundary, as they relate to shape types, are defined as follows:
| Interior | Boundary | Exterior | |
|---|---|---|---|
| TGIS_Point | Point itself | Nothing | All except Interior |
| TGIS_Multipoint | Points themselves | Nothing | All except Interior |
| TGIS_Arc (lines) | Line itself except starting/ending points | Starting & ending points of line | All except Interior & Boundary |
| TGIS_Polygon (areas) | Area of polygon | Outline of polygon | All except Interior & Boundary |
The nature of the shape intersections is defined by a set of the interior, boundary, and exterior relationship possibilities, forming a 3×3 table.
| A\B | Interior | Boundary | Exterior |
|---|---|---|---|
| Interior | * | * | * |
| Boundary | * | * | * |
| Exterior | * | * | * |
The table can be represented by a string (row-by-row) like: T*F**FFF*
Each element of the table represents an intersection possibility:
| Type | Description |
|---|---|
| T | Intersection exists |
| F | Intersection does not exist |
| * | Does not matter |
| 0 | Intersection is a point (dimension = 0) |
| 1 | Intersection is a line(dimension = 1) |
| 2 | Intersection is a polygon (dimension = 2) |
Common cases
Contains
Case description
A contains B if the interiors intersect and the exterior of A is not intersected by the Interior or boundary of B.
| A\B | Interior | Boundary | Exterior |
|---|---|---|---|
| Interior | T | * | * |
| Boundary | * | * | * |
| Exterior | F | F | * |
String representation is: T*****FF*
Predefined constant: RELATE_CONTAINS
Completely Contains
There is also a possible additional requirement that A completely contains B. In this case, the boundary of A must not intersect the interior or boundary of B.
String representation for this case is: T**FF*FF*
Predefined constant: RELATE_CONTAINS_COMPLETELY
Cross
For relating shapes of different types. So, polygon can be crossed by line, but not with another polygon. (See Overlap).
Case description
A crosses B if the interiors intersect and the exterior of B intersects with the interior of A. General formula (except line-to-line situations).
| A\B | Interior | Boundary | Exterior |
|---|---|---|---|
| Interior | T | * | T |
| Boundary | * | * | * |
| Exterior | * | * | * |
String representation is: "T*T******
Predefined constant: RELATE_CROSS
Case description
A crosses B (and B crosses A) if the dimension of the intersection equals 0 (i.e., is a point). Only for line-to-line relationships.
| A\B | Interior | Boundary | Exterior |
|---|---|---|---|
| Interior | 0 | * | * |
| Boundary | * | * | * |
| Exterior | * | * | * |
String representation is: 0********
Predefined constant: RELATE_CROSS_LINE
Disjoint
Case description
There is no intersection of any kind.
| A\B | Interior | Boundary | Exterior |
|---|---|---|---|
| Interior | F | F | * |
| Boundary | F | F | * |
| Exterior | * | * | * |
String representation is: FF*FF****
Predefined constant: RELATE_DISJOINT
Equality
Case description
A & B interiors intersect, and no interior or boundary of one shape intersects with the exterior or boundary of the other shape.
| A\B | Interior | Boundary | Exterior |
|---|---|---|---|
| Interior | T | * | F |
| Boundary | * | * | F |
| Exterior | F | F | * |
String representation is: T*F**FFF*
Predefined constant: RELATE_EQUALITY
Intersect
Case description
By interiors.
| A\B | Interior | Boundary | Exterior |
|---|---|---|---|
| Interior | T | * | * |
| Boundary | * | * | * |
| Exterior | * | * | * |
String representation is: T********
Predefined constant: RELATE_INTERSECT
Case description
The interior of A intersects the boundary of B.
| A\B | Interior | Boundary | Exterior |
|---|---|---|---|
| Interior | * | T | * |
| Boundary | * | * | * |
| Exterior | * | * | * |
String representation is: “*T*******”
Predefined constant: RELATE_INTERSECT1
Case description
The boundary of A intersects the interior of B.
| A\B | Interior | Boundary | Exterior |
|---|---|---|---|
| Interior | * | * | * |
| Boundary | T | * | * |
| Exterior | * | * | * |
String representation is: ***T*****
Predefined constant: RELATE_INTERSECT2
Case description
The boundary of A intersects the boundary of B.
| A\B | Interior | Boundary | Exterior |
|---|---|---|---|
| Interior | * | * | * |
| Boundary | * | T | * |
| Exterior | * | * | * |
String representation is: ****T****
Predefine constant: RELATE_INTERSECT3
Overlap
For relating shapes of the same type. So, a polygon can overlap with another polygon, but not with a line. (See Cross).
Case description
A overlaps B (and B overlaps A) if both the interiors and exteriors intersect. General formula (except line-to-line situations).
| A\B | Interior | Boundary | Exterior |
|---|---|---|---|
| Interior | T | * | T |
| Boundary | * | * | * |
| Exterior | T | * | * |
String representation is: T*T***T**
Predefined constant: RELATE_OVERLAP
Case description
A overlaps B (and B overlap A) if the interior intersection has a dimension equal to 1 (is a line) and the exteriors intersect. Only for line-to-line relationships.
| A\B | Interior | Boundary | Exterior |
|---|---|---|---|
| Interior | 1 | * | T |
| Boundary | * | * | * |
| Exterior | T | * | * |
String representation is: 1*T***T**
Predefined constant: RELATE_OVERLAP_LINE
Touch
Case description
The boundary of A intersects with the interior of B.
| A\B | Interior | Boundary | Exterior |
|---|---|---|---|
| Interior | F | * | * |
| Boundary | T | * | * |
| Exterior | * | * | * |
String representation is: F**T*****
Predefined constant: RELATE_TOUCH_INTERIOR
Case description
The boundary of A intersects the boundary of B (but not the interior of B).
| A\B | Interior | Boundary | Exterior |
|---|---|---|---|
| Interior | F | * | * |
| Boundary | * | T | * |
| Exterior | * | * | * |
String representation is: F***T****
Predefined constant: RELATE_TOUCH
Within
Case description
A is within B if the interiors intersect and the exterior of B is not intersected by the Interior or boundary of A.
| A\B | Interior | Boundary | Exterior |
|---|---|---|---|
| Interior | T | * | F |
| Boundary | * | * | F |
| Exterior | * | * | * |
String representation is: T*F**F***
Predefined constant: RELATE_WITHIN
Completely Within
There is also a possible additional requirement that A is completely within B.
String representation for this case is: 'TFF*FF***
Predefined constant: RELATE_WITHIN_COMPLETELY