Spatial Relations: Co-location

3. Spatial Relations: Co-location#

3.1. Topics#

  • Co-locational spatial relationship

  • Readings: Lloyd Chp. 5

3.2. Fundamental/Basic Spatial Relationships#

  • Relationships between two attribute values

    • , <

    • difference, ratio

  • Relationships between two locations (spatial relationships)

    • = (occupy the exact same space)

    • Co-locate (share some space, i.e., overlap in space)

    • Distance (no co-locate)

  • Mapping unit as location in practice

3.3. Co-locational Relationships#

  • Share some space

  • Qualitative co-locational relationship

    • Intersect, touch, contain, …, disjoint

    • Independent of coordinates

    • Topological relationship

  • Quantitative co-locational relationship

    • Where is the shared (or not shared) space located?

    • Based on and derived from coordinates

3.4. Qualitative Locational (Topological) Relationship#

  • Locations that share some space

    • intersect, touch, contain, …

  • Most GIS support some common spatial relationships

    • Hard to define precisely

    • intersect, touch, contain, …

  • Dimensionally Extended 9-Intersection Model (DE-9IM)

    • Decompose a MU into interior, boundary and exterior

    • Relationship base on the dimensionality of the intersection of boundary, interior, and exterior

    • 9-intersection model (9IM) is a special case (Egenhofer and Herring, 1991)

3.5. DE-9IM#

  • A feature decomposed into interior (I), boundary (B) and exterior (E)

Feature Type

Interior (I)

Boundary (B)

Exterior (E)

Polygon

points within the polygon

the rings

points not in the polygon

Line

points of the line except endpoints

the endpoints

points not on the line

Point

the point

empty set

points other than the point

  • Dimensionality of intersection: {None (-1), Point (0), Line (1), Area (2)}

3.6. DE-9IM Examples#

  • A contains B: \(I(B) \cap E(A) = \varnothing\) and \(I(A) \cap I(B) \neq \varnothing\)

  • A disjoint B: \(I(A) \cap I(B) = \varnothing\) and \(I(A) \cap B(B) = \varnothing\) and \(B(A) \cap I(B) = \varnothing\) and \(B(A) \cap B(B) = \varnothing\)

3.7. Quantitative Co-locational Relationship#

  • Where is the shared/not shared space located?

  • Use mapping units (MUs) as the location

    • New MUs created as the result of overlapping existing MUs

    • Also called Map Overlay

    • Attributes of new MUs are associated with original attributes

3.8. Map Overlay#

  • Create new maps (MUs) based on existing maps (MUs)

  • Vertical integration

  • Two steps

    1. Determine the geometry of new mapping units

      • Overlap existing mapping units

    2. Assign attribute values to the new mapping units

      • Inherit from existing mapping units

3.9. Geometry of New Mapping Units#

  • Intersect boundaries of original MUs to create new MUs

  • New MUs are more “homogeneous” than original MUs

3.10. Identify Relations between MUs#

  • For two maps A and B, four kinds of new mapping units (at most) can be created:

    • MUs in A and also in B (A AND B)

    • MUs in A but not in B (A NOT B)

    • MUs in B but not in A (B NOT A)

    • MUs neither in A nor in B (NOT A AND NOT B)

  • Total space = (A AND B) + (A NOT B) + (B NOT A) + (NOT A AND NOT B)

3.11. Map Overlay on Raster Maps#

  • Determine the geometry of new MUs

    • Two raster maps must have the same “geometry”

      • Same resolution, same orientation, same extent

    • New MUs are just the original cells

  • Assign attribute values to the new MUs

    • Combine values of original cells at the same location

    • Map Algebra (Tomlin, 1990)

  • Relation between cells is 1-to-1

3.12. Map Overlay on Vector Maps#

  • Determine the geometry of new MUs

    • Intersect boundaries of original MUs to create new MUs

  • Assign attribute values to the new MUs

    • Combine values of original MUs

  • Relation between MUs is typically M-to-N

3.13. Attributes of the New Mapping Units#

  • Typically a combination of the attributes of the original mapping units

    • Multi-valued map

    • \(Value_{new} = \{Value_A, Value_B\}\)

  • A new attribute value can be calculated based on original values

    • \(Value_{new} = f(Value_A, Value_B)\)

    • Boolean (0, 1), Arithmetic (+, -, *, /)

3.14. Overlay Tools in ArcGIS#

  • Different tools keep different kinds of new MUs

  • Intersect, Union, Identity, Erase, Symmetrical Difference, Update

3.15. Map ArcGIS Overlay Tools to Kinds of MUs Kept#

  • One kind of MUs

    • A AND B (intersect)

    • A NOT B (erase)

    • B NOT A (erase)

  • Two kinds of MUs

    • A NOT B, A AND B (identity)

    • B NOT A, A AND B (identity)

    • A NOT B and B NOT A (symmetrical difference)

  • All kinds of MUs

    • A AND B, A NOT B, B NOT A (union)

3.16. Clip Tool in ArcGIS#

  • What’s the difference between Intersect and Clip tool?

    • Attributes of the new MUs

    • Clip tool in the Extract toolset—Why?

  • How about Erase tool?

3.17. Other Overlay Tools in ArcGIS#

  • Two steps

    • Choose output MUs for the new map

    • Process/handle related MUs (including attributes)

3.18. Overlay and Attribute Table#

  • Number of new MUs (rows)

    • Vector: Usually increases

    • Raster: Stays the same

  • Number of attributes (columns)

    • Usually increases (attributes from both original maps)

3.19. Dimensionally Extended Nine-Intersection Model (DE-9IM)#

Author: Christian Strobl, German Remote Sensing Data Center (DFD), German Aerospace Center (DLR)

3.20. Synonyms#

Dimensionally Extended Nine-Intersection Model (DE-9IM), Nine-Intersection Model (9IM), Four-Intersection Model (4IM), Egenhofer Operators, Clementini Operators; Topological Operators.

3.21. Definition#

The Dimensionally Extended Nine-Intersection Model (DE-9IM) or Clementini-Matrix is specified by the OGC “Simple Features for SQL” specification for computing the spatial relationships between geometries.

It is based on the Nine-Intersection Model (9IM) or Egenhofer-Matrix, which is an extension of the Four-Intersection Model (4IM). The DE-9IM considers the two objects’ interiors, boundaries, and exteriors and analyzes the intersections of these nine parts for their relationships. The model records the maximum dimension (-1, 0, 1, or 2) of the intersection geometries, where -1 corresponds to no intersection (empty set).

The primary spatial relationships described by the DE-9IM are:

  • Equals

  • Disjoint

  • Intersects

  • Touches

  • Crosses

  • Within

  • Contains

  • Overlaps

3.22. Main Text#

There are three common approaches for describing topological relationships between geodata, each based on an intersection matrix.

3.22.1. Four-Intersection Model (4IM)#

The 4IM considers only the interior and the boundary of two objects. It creates a 2x2 matrix that describes whether the intersections between the interiors and boundaries of objects A and B are empty (0) or non-empty (1).

3.22.2. Nine-Intersection Model (9IM)#

The 9IM extends the 4IM by including the exterior of the objects. This results in a 3x3 matrix. Like the 4IM, it traditionally uses boolean values (empty or non-empty) to describe the intersections of Interior (I), Boundary (B), and Exterior (E).

3.22.3. Dimensionally Extended Nine-Intersection Model (DE-9IM)#

The DE-9IM further refines the 9IM by recording the dimension of the intersection rather than just a boolean state.

The intersection of any two parts results in a set of points. The value in the DE-9IM matrix is the maximum dimension of this point set:

  • -1: The intersection is empty (\(\emptyset\)).

  • 0: The intersection contains only points (0-dimensional).

  • 1: The intersection contains lines (1-dimensional).

  • 2: The intersection contains polygons/areas (2-dimensional).

The matrix is structured as follows:

Interior(B)

Boundary(B)

Exterior(B)

Interior(A)

dim(I(A)∩I(B))

dim(I(A)∩B(B))

dim(I(A)∩E(B))

Boundary(A)

dim(B(A)∩I(B))

dim(B(A)∩B(B))

dim(B(A)∩E(B))

Exterior(A)

dim(E(A)∩I(B))

dim(E(A)∩B(B))

dim(E(A)∩E(B))

3.23. Geometric Operators and Predicates#

3.23.1. Disjoint#

\(A \cap B = \emptyset\) The interiors and boundaries of the two geometries do not intersect at all.

  • Pattern: FF*FF****

3.23.2. Intersects#

\(A \cap B \neq \emptyset\) The complement of Disjoint. The geometries share at least one point.

  • Pattern: T******** or *T******* or ***T***** or ****T****

3.23.3. Touches#

\(A \cap B \neq \emptyset\), but \(Interior(A) \cap Interior(B) = \emptyset\). The geometries have at least one point in common, but their interiors do not overlap. This applies to Area/Area, Line/Line, Line/Area, and Point/Area, but not Point/Point.

  • Pattern: FT******* or F**T***** or F***T****

3.23.4. Crosses#

The geometries share some but not all interior points, and the dimension of the intersection is less than the maximum dimension of the two geometries.

  • Pattern: T*T****** (for Line/Area) or 0******** (for Line/Line)

3.23.5. Within#

The geometry A lies entirely within the interior of B.

  • Pattern: T*F**F***

3.23.6. Contains#

The inverse of Within. Geometry B is within Geometry A.

  • Pattern: T*****FF*

3.23.7. Overlaps#

The geometries share some but not all points in common, and the intersection has the same dimension as the geometries themselves.

  • Pattern: T*T***T** (for Area/Area) or 1*T***T** (for Line/Line)

3.23.8. Equals#

The two geometries are topologically equal; they occupy the same space.

  • Pattern: T*F**FFF*

3.24. Summary#

The DE-9IM is a powerful mathematical framework that allows GIS systems to query spatial relationships between complex objects rigorously. By using a 3x3 matrix of dimensions, it provides a unique “fingerprint” for every possible topological configuration between two geometric features.

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