- Use of different terminology
- Use of different concepts of space

- Spatial concepts are structures that humans use to organize their perception of geographic space.
- Spatial data models are formalizations of the semantics of spatial concepts.
- Spatial data structures are implementations of spatial data models.

Informal conceptual tools to comprehend and structure our perception and cognition of reality.

Different geographic applications require different spatial concepts:

- Networks for navigation tasks or utilities
- Euclidean geometry for surveying and cadastre
- Block units for census or land use mapping

Comprehensive sets of conceptual tools to describe and structure spatial information.

- Formally defined
- Implementable
- Widely applicable.

Examples: raster model, simplicial model

Detailed, low-level descriptions of structures and algorithms for the storage and retrieval of data.

Primary concerns

- performance
- storage utilization

Examples: quad trees, strip trees, winged edges

Remote sensing data as an example

The data source is conceptualized as a continuous measurable field of real values ("geographic reality" [Goodchild 1990]).

Data collection is based on the geometric data model of a square raster.

Data storage is often done in a quad tree data structure.

Models for Spatial Relations

Spatial relations are more complex than relations between standard data types such as integers or strings.

Main problem: modeling the semantics of spatial relations.

- quantitative relations (distances, bearings, angles)
- qualitative relations (topology, cardinal directions, approximate distances)

Formal semantics are crucial for

- query processing (what to retrieve when a user asks for "all cities north of Vienna?")
- query evaluation (is a query executable or has it internal contradictions?)
- optimization of complex spatial queries

Our investigations identified that the topological relations between two point sets, A and B, can be described by the 9 set intersections of the boundaries, interiors, and exteriors of A and B.

interior A /\ interior B interior A /\ boundary B interior A /\ exterior B

I (A, B) = boundary A /\ interior B boundary A /\ boundary B boundary A /\ exterior B

exterior A /\ interior B exterior A /\ boundary B° exterior A /\ exterior B

Restriction: all objects have "sharp" boundaries.

Considering empty (ø) and non-empty (¬ø) intersections, we found that we could realize

- 8 relations between two regions in 2-D
- 57 relations between two lines in 2-D
- 20 relations between a region and a line in 2-D.

A geometric data model for capturing complete topological information in arbitrary dimensions.

The model is based on simplicial complexes, in particular closed surfaces.

A simplex is the elementary geometric building block in a given dimension:

- 0-simplex is a point (node)
- 1-simplex is a line segment (edge)
- 2-simplex is a triangle (face)
- etc.

A simplicial complex is a collection K of simplexes such that

- with every simplex, all its bounding simplexes are contained in K
- the intersection of two simplexes is either empty or a simplex in K.

A closed surface is a simplicial complex partitioning the plane.

In practice, a closed surface is a triangulation of the plane resulting from the complete intersection of all geometric objects and its subdivision into simplexes.

Properties of Geographic Data | User Interfaces and Spatial Query Languages |

Practical Issues of Geographic Databases | Literature ]

*Last updated on July 26, 1996.*

[ Max J. Egenhofer | NCGIA Maine | Department of Spatial Information Science and Engineering ]