17
|
Union |
Intersection | ||||||
|
⅛ι |
k2 |
к |
s |
k1 |
k2 |
к |
s |
|
A |
A |
A |
max(sι, s2) |
A |
A |
A |
min(sι, s2) |
|
A |
-∣A |
A |
Sl |
A |
-ιA |
-A |
s2 |
|
-∣A |
A |
A |
s2 |
-∏A |
A |
-ιA |
Sl |
|
-∣A |
-∣A |
-∣A |
min(sɪ, s2) |
-A |
-∏A |
-∣A |
max(sι, s2) |
Table 2.1 : Rules for performing Intersection and Union operations. We consider
the pairs (sɪ,ftɪ) and S2,k2, as the input. The output of Union and Intersection is
denoted as pair (s, fc).
Similar argument can be used to derive the formulas given for the remaining cases.
Another interpretation of these operations on the scalars sɪ and s2 is tθ view these
numbers as estimate of the distance from the grid point to the boundary of the region
A. In the case of Union, the rule is that if both grid points lie in A, a good estimate
of the distance from the grid point to the boundary of the union is the maximum of
these two distances. Similar arguments again apply in the other cases.
2.2.2 Operations for Three or More Materials
Given the method for union and intersection defined above, the generalization of these
operations to three or more materials is relatively easy. We suggest two operations
analogous to Union and Intersection for the multi-material case. The first operation
Overwrite takes a multi-material map and a two-material map (with material A and
-ιA) and performs the multi-material analog of Union. In particular, it treats the
material in the first multi-material map as either A or -∣A and applies the two material
rules for Union described above. The result of an Overwrite operation is that the
material A in the second map overwritten onto any existing materials in the first
map. The resulting map contains the union of the materials A in both maps.
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