1. 1 13. Spatiotemporal Databases Extreme Point Data Models Parametric Extreme Point Data Models Geometric Transformation Data Models Queries

2. 2 Spatiotemporal objects - have spatial and temporal extents Spatial extent- the set of points in space that belong to an object Temporal extent- the set of time instances when an object exists

3. 3 13.1 Extreme Point Data Models Extreme points – the endpoints of intervals and the corner vertices of polygonal or polyhedral objects Examples: extreme points data models include: Rectangle data model and Worboys ’ data model

4. 4 Extreme Point Data Models Rectangles data model --- for each object Spatial extent : a set of rectangles. Temporal extent: a set of time intervals.

5. 5

6. 6 Rectangles Data Model Archaeological Site (Figure 13.1) IdXYT 1[3,6] [100,200] 2[8,11][3,7][150,350] 3[2,4][5,10][250,400] 3[2,10][8,10][250,400]

7. 7 Worboys ’ Data Model --- for each object Spatial extent: a set of triangles, represented by corner vertices Temporal extent: a set of time intervals, represented by From and To endpoints

8. 8

9. 9 Worboys ’ Data Model Park (Figure 13.2) IdAxAyBxByCxCyFromTo Fountain104 4 419801986 Road510969619951996 Road96939319951996 Tulip23276319751990 Park12111121119741996 ………………………

10. 10 13.2 Parametric Extreme Point Data Models Extend the extreme point data models by specifying the extreme points as linear, polynomial, or periodic functions of time Examples: parametric rectangles and parametric 2-spaghetti data models

11. 11 Parametric Rectangles Data Model --- for each object Spatial extent: a set of intervals, whose endpoints are represented by functions of time (time t is the only parameter) Temporal extent: a time interval, whose endpoints are represented by From and To constants

12. 12 Example: Plankton X Y T [5+t, 10+2t] [5+t, 15+3t] [0, 20]

13. 13 The Parametric 2-Spaghetti Data Model--- for each object Spatial Extent: set of triangles, whose corner vertices represented as functions of time Temporal Extent: A constant time interval Example: Net Ax Ay Bx By Cx Cy From To 3 3-t4+0.5t 4-0.5t 5+t 3 0 10

14. 14 13.2.1 Periodic Parametric Data Models Periodic Parametric Rectangles Data Model --- Spatial Extent: a set of triangles, whose corner vertices are represented as periodic functions of time Temporal Extent: Periodic intervals

15. 15 123 1- 2- 3- 4- 12:00 am 3:00 am 5:00 am Parking Lot

16. 16 Example: Tide (Figure 13.6) Ax Ay Bx By Cx Cy From To P End 1 4 1 4-t ’ t ’ +1 4 0 2 11.5 +∞ 1 4 1 2 3 4 2 9.5 11.5 +∞ 1 2 3 4 3 6-t ’ 2 3 11.5 +∞ 1 2 1 4-t ’ 3 6-t ’ 2 3 11.5 +∞ 1 2 3 4 3 3 3 8.5 11.5 +∞ 1 2 1 1 3 3 3 8.5 11.5 +∞ 1 1 3 3 3 6-t ’ 3 5 11.5 +∞

17. 17 13.3 Geometric Transformation Data Models Generalize geometric transformations by using a time parameter. Types of geometric transformations: scaling, translation, linear, affine.

18. 18 13.3 Geometric Transformation Data Models Geometric Transformation -- bijection of d-dimensional space into itself. Example: Affine Motion: x ’ = Ax + B Linear Motion: x ’ = Ax Scaling:x ’ = Ax where A is diagonal Translation:x ’ = x + B Identity:x ’ = x

19. 19 Geometric Transformation Data Model --- defines each spatiotemporal object as some spatial object together with a continuous transformation that produces an image of the spatial object for every time instant

20. 20 13.4 Queries Querying Parametric Extreme Point Databases --- allow only the constraints of the type x=c, x = c. Example: Find where and when will it snow given Clouds(X, Y, T, humidity) Region(X, Y, T, temperature) (SELECT x, y, t FROM Clouds WHERE humidity >= 80) INTERSECT (SELECT x, y, t FROM Region WHERE temperature <= 32)

21. 21 Example: Window(id, x, y, t) -- open windows on a computer screen, where id is the identifier, x, y spatial points of the window, and t is the time when it is active. Which windows are completely hidden by other windows? Seen(i) :- Window(i, x, y, t), not Window(i2, x, y, t2), t2 > t. Hidden(i) :- Window(i, x, y, t), not Seen(i).