1. Computing the Visibility Polygon of an Island in a Polygonal Domain
Danny Z. Chen and Haitao Wang
University of Notre Dame
Indiana, USA
ICALP 2012

2. The visibility polygon
The point case

3. The visibility polygon (the segment case)
the weak visibility polygon

4. The visibility polygon (the island case)
light

5. Defining “visible”
Two points p and q are visible to each other if the line segment joining them is in the polygon p q

6. Defining “visible”
A points p is (weakly) visible to a region R if p is visible to at least one point of R q R w p

7. Previous work The point case: O(n) time The segment case: O(n) time
ElGindy and Avis 81’, Lee 83’, Joe 90’, Joe and Simpson 87’
n: the number of vertices of the polygon
The segment case: O(n) time
Guibas et al. 87’
The island case: O(n+m) time
Ghosh 91’
m: the number of vertices of the island

8. A polygonal domain
A set of h disjoint polygonal obstacles of totally n vertices
Free space: the space outside the obstacles

9. The visibility polygon in a polygonal domain
The point case

10. The visibility polygon in a polygonal domain (cont.)
The segment case

11. The visibility polygon in a polygonal domain (cont.)
The island case
island

12. Previous work and our result
The point case
O(nlog n) time, Suri and O’Rourke 86’, Asano et al. 86’
O(n+hlogh) time, Heffernan and Mitchell 95’
The segment case
O(n4) time, Suri and O’Rourke 86’
Lower bound: Ω(n+h4) time
The island case
No previous work
Our result:
A first-known algorithm: O(n2 h2) time, worst-case optimal
If all obstacles are convex, O(n+h4) time

13. Our approaches
A generalization of Suri and O’Rourke’s algorithm (the SO algorithm) to convex chains
A new bound on the SO algorithm: O(n4) -> O(n2h2)
Goal: computing the visibility polygon of P*
P*: the island
Reduce the problem to computing the visibility polygon of convex chains

14. The SO algorithm Do a rotational sweeping around each vertex v
Generate some triangles visible to s w s v w u

15. The SO algorithm Do a rotational sweeping around each vertex v
Generate some triangles visible to s
The visibility polygon of s is the union of all such triangles w s v u

16. The time analysis of the SO algorithm
A rotational sweeping on each vertex generates O(n) triangles
O(n2) triangles in total
A new bound: O(nh)
O(n2 h2) time to compute the union of the triangles

17. Generalize the SO algorithm to a convex chain
s is a convex chain of O(n) vertices
A naive approach: apply the SO algorithm on each edge
Sweep the tangent lines of s, generating O(n) triangles u
The number of triangles generated
is O(nh) s v w

18. Reducing P* to convex chains
If the island P* is convex
Compute the visibility polygon of the boundary of P* P*

19. Reducing P* to convex chains (cont.)
P* is not convex but the convex hull of P* does not intersect any obstacle
Compute the visibility polygon of the convex hull of P* P*

20. The convex hull of P* intersecting other obstacles
Key: how to determine the visibility polygon of the purple side?
The visibility polygon of P* is the union of the visibility polygons of the
two boundary portions
Observation: A point outside the canal is visible to the purple side
if and only if it is visible to a canal gate
canal gate x
bays e P* P
canal q f y q
canal gate p

21. Determining the canal canal gate a c x e P* P d canal f y
the corridor path
canal gate b

22. Determining a, b, c, and d a c
junction
triangles P* P d b

23. Conclusion
Compute the visibility polygon of an island in a polygonal domain
O(n2h2) – worst case optimal
O(n+h4) for the convex case
Match the lower bound

24. Thank You