Representation of Heterogeneous Tessellation Structures by Graphs

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Representation of Heterogeneous Tessellation
Structures by Graphs

• Takeo YAKU (Nihon U.)
• WAAP 108
• Dec. 1st, 2001

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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

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2. Definitions of Tables

• Definition.
• An (n, m)-support is a set (I, j) 1 lt I lt n,
  1 lt j lt m of integer pairs.
• A support is a (n, m)-support for some n and m.
• For (n, m)-support,
• Row location loc-lrow1, 2, ..., n ? R.
• Column location loc-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, where P is of the form (i, j) k lt
  i lt l, s lt j lt t for some integers k, l, s,
  t.

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• 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.

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• 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.
•  

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• 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.

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• Example 2.
•  

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3. TABLES AND QUASI TESSELLATION GRAPHS
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(No Transcript)
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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
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• 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

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

x
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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
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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
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 4. CONCLUSION

• Future Works
• Other Methods
• Relation to Mathematical Theory
• Graph Grammars