Topological Relationships Between Complex Spatial Objects

1. Topological Relationships Between Complex Spatial Objects Daniel Hess and Yun Zhang

2. Problems • To distinguish between simplified and complex spatial objects •To define topological relationships between complex spatial objects • To ensure that two different topological relationships cannot hold for the same two spatial objects

3. Major contributions of this paper • Introduction of definitions for complex points, complex lines and complex regions • Determination of complete sets of mutually exclusive topological relationships for combinations of all complex spatial data types • Prove the completeness and mutual exclusion of the topological relationship predicates • Provide for the user concepts of topological cluster predicates and topological predicate groups

4. Complex spatial objects (Schneider and Behr, 2006, p. 46) • A complex point object may include several points • A complex line may be a spatially embedded network possibly consisting of several components • A complex region may be a multipart region possibly consisting of multiple faces and holes

5. Complex spatial objects • Example: -A complex region with two faces in which the upper face has two holes: -A complex region with five faces and three holes: (Schneider and Behr, 2006, p. 53) (Schneider and Behr, 2006, p. 53)

6. Validation methodology • The authors use a two-step proof technique called proof by constraint and drawing -Step 1: For each possible data type combination, a complete set of topological constraint rules are collected and then applied to the topological matrix -Step 2: The existence of the remaining assignments of the topological matrix indicates a possible topological relationship between the two data types

7. Validation methodology • The proof technique is suitable for validating the approaches used in this paper, as it is generally abstract and precise • The proof technique may miss some valid topological relationships and can be time consuming and labor intensive • This paper extends the 9-intersection model from simple to complex spatial objects

8. Validation methodology • Proof example: (Schneider and Behr, 2006, p. 68) (Schneider and Behr, 2006, p. 45) (Schneider and Behr, 2006, p. 74)

9. Assumptions • The authors assume the existence of the Euclidean distance function when making the definition for complex lines: • When discussing topological relationships between two complex lines, non-emptiness of a line object is assumed • When discussing topological relationships between complex lines and complex points; complex lines and complex regions; complex points and complex regions, the authors assume that all objects are non-empty

10. If rewriting this paper, we would… • Keep the key ideas of the approach. We would still apply the 9-intersection model to complex spatial objects • Keep the clustering of topological predicates in order to reduce the large predicates set and to make the topological relationships more manageable • Change step 2 of the proof method and apply a math formula to define valid topological relationships between specific data types, in order to improve efficiency • Extend the spatial data types to three dimensions