Representation of Heterogeneous Tessellation Structures by Graphs

1. Representation of Heterogeneous Tessellation Structures by Graphs Takeo YAKU (Nihon U.) WAAP 108 Dec. 1st, 2001

2. 1. INTRUDUCTION • Tabular Forms • Tree Representation based on Partition of Tabular Forms • Graph Representation by Edge to Edge Correspondence • Graph Representation by Edge to Node Correspondence

3. 2. Definitions of Tables • Definition. • An (n, m)-support is a set {(I, j) | 1 =< I =< n, 1 =< j =< m} of integer pairs. • A support is a (n, m)-support for some n and m. For (n, m)-support, • Row locationloc-lrow：{1, 2, ..., n} →R. • Column locationloc-column : {1, 2, ..., m} →R. • Location：　a tuple (loc-row, loc-column, O), where O in R denotes the left upper most location of the table. • A partial support : a subset P of an (n, m)-support, whereP is of the form {(i, j) | k <= i <= l, s <= j <= t} for some integers k, l, s, t.

4. Definition. • A support partition P over a support S is a pair wise disjoint set {S1, S2, ..., SN} of partial supports, where S1 ∪ S2 ∪ ... ∪ SN = S. • A table over a support S is a 3-tuple (S, P, loc} of a support S, a support partition P over S and the location of S.

5. Location of each cell c = is defined by: • Ceiling (north wall) of c is max I - 1 ... • Floor (south wall) of c is min i ... • Left wall (west wall) of c is min j - 1 ... • Right wall (east wall) of c is max j ... • Each partial support is called by cell.

6. Example 1. The following figure denotes a support partition {{(1,1), (2,1)}, {(1, 2)}, {(1, 3)}, {(2, 2), (2, 3)}} over (2, 3)-support.

7. Example 2.

8. 3. TABLES AND QUASI TESSELLATION GRAPHS

10. East wall and west wall coordinates are same of x and y West wall coordinates are same of x and y East wall coordinates are same of x and y

11. Representation of table by tessellation graphs (continued) • Peripheral cells and corresponding nodes • Cell a is represented by node • Ceiling of the cell a is represented by node z • Virtual cells x and y has no valid heights • Ceiling of the cell z is covered by cells x and y

12. a x y example z

13. General rules x y Cells x and y are of same ceiling coordinates x y Cells x and y are of same floor coordinates Cells x and y are of same ceiling and floor coordinates x y

14. a • Tables and tessellation graphs(3) • A large inner cell and corresponding nodes • The cell a is represented by the node x. x

15. y x North wall and south wall coordinates are same of x and y North wall coordinates are same of x and y south wall coordinates are same of x and y

16. East wall and west wall coordinates are same of x and y East wall coordinates are same of x and y West wall coordinates are same of x and y

17. 4. CONCLUSION • Future Works • Other Methods • Relation to Mathematical Theory • Graph Grammars