1. Visibility and Ray Shooting Queries in Polygonal Domains
Danny Z. Chen1 and Haitao Wang2
1University of Notre Dame
2Utah State University
WADS 2013

2. A polygonal domain
A set of h disjoint polygonal obstacles with a total of n vertices
Free space: the space outside the obstacles
h<<n is possible

3. The visibility polygon from a pointq

4. Visibility queries Given a query point q,
report Vis(q): the visibility polygon of q
k: = |Vis(q)|
Goal: design data structures to support efficient visibility queries

5. A special case – ray-shooting queries

6. Previous work – ray shootings
preprocessing
time
space
query time
Chazelle, Edelsbrunner, Grigni, Guibas, Hershberger, Sharir, Snoeyink, Suri, 94’ n
log n
simple polygons

7. Previous work and our result – ray shootings
preprocessing
time
space
query time
Pocchiola, 90’ n2
logn
Chazelle et al. 94’
nh1/2+nlogn
+h3/2logh n
h1/2logn
Agarwal and Sharir, 96’
(nlogn+h2)logh
(n+h2)logh
log2nlog2h
Our result
n+h2poly(logh)
n+h2
polygonal domains

8. Previous work – visibility queries
preprocessing
time
space
query time
Bose, Lubiw, and Munro, 02’
n3 logn n3
k + log n
Aronov, Guibas, Teichmann, and Zhang, 02’
n2 log n n2
k + log2 n
simple polygons

9. Previous work and our results – visibility queries (m=min{k,h})
preprocessing
time
space
query time
Zarei and Ghodsi, 08’
n3 log n n3
k+mlog n
Inkulu and Kapoor, 09’
n2 log n n2
k+h+mlog2 n
n2 h3
n2 h2
klog n
Lu et al., 11’
k+log2 n+hlogn/h
Our result 1
k+log2n+mlogn
Our result 2
n+h2logh
n+h2
polygonal domains

10. Cone visibility
The visibility is restricted by a cone q

11. Our approach An extended corridor structure
partitions the free space into an “ocean” M, and multiple “bays” and “canals”
M is bounded by a set of convex chains
Use the visibility complex of convex objects (Pocchiola and Vegter 96’)
Each bay or canal is a simply polygon
Use data structures on simply polygons
The query result is the combination of that on M and those on bays and canals

12. The convex case: all obstacles are convex
Ray-shootings (Pocchiola and Vegter 96’)
Preprocessing: O(n+h2 poly(log n)) time and O(n+h2) space
Query: O(log n) time
Visibility queries (our result, using visibility complex)
Preprocessing: O(n+h2 log h) time and O(n+h2) space
Query: O(k+ h’ log n) time
h’: the number of obstacles visibile to the query point

13. Non-convex case Computing the convex hull for each obstacle
Two sub-cases:
The convex hulls are pairwise disjoint
Not pairwise disjoint

14. Not pairwise disjoint
Two convex hulls intersect each other

15. First sub-case: convex hulls pairwise disjoint
Bays:
The regions between the convex hulls and the obstacles
Each bay is a simple polygon
Each bay has a gate
The ocean M:
the free space minus the bays
bays
bays

16. The query algorithmic scheme
Given a query:
solve the query on the ocean M
solve the query on the bays
combine the above solutions

17. Ray-shootings Given a ray r, let q be the origin, which in M or a bay
If q is in M
determine the first point p on the convex hulls hit by r
two cases: p is on an obstacle or a bay gate
p on an obstacle: done
p on a bay gate: enter the bay p q p q

18. Ray-shootings (cont.) If q is in a bay B
determine the first point p on the boundary of B hit by r
if p is on an obstacle
done
else (p is on the gate)
goes into M
the case where q is in M p p q

19. Ray-shootings (cont.) Preprocessing
Visibility complex based approach on M (PV 96’)
Preprocessing: O(n+h2 poly(log n)) time and O(n+h2) space
Query: O(log n) time
The data structures for simple polygons on all bays (CEGGHSSS 94’)
Preprocessing: O(n) time and space

20. Visibility queries Given a query point q, in M or a bay If q is in M
compute Vis(q,M): the visibility region in M
For each portion of the boundary of Vis(q,M) on a gate
compute the visibility region through the cone q

21. Visibility queries If q is in bay B
Compute Vis(q,B): the visibility region in B
If a portion of the gate is on the boundary
compute the cone visibility outside the bay
solved in the similar way as the previous case q

22. Visibility queries – the preprocessing
The preprocessing on M
Build our cone visibility query data structure on M
Preprocessing: O(n+h2 log h) time and O(n+h2) space
Query: O(k+ h’ log n) time
The preprocessing on all bays
First approach
Exterior visibility decomposition (Aronov et al. 02’)
preprocessing: O(n2logn) time and O(n2) space
query: O(k+logn)
Second approach:
The ray-shooting data structure on simple polygons
preprocessing: O(n) time and space
query: O(klogn)

23. Exterior visibility decomposition
Critical constraints of the bay
all points in the same cell has the same combinatorial representation of the visibility polygon in the bay through the gate
Overlap the critical constraint arrangement of all bays of the same obstacle
a single point location is sufficient q

24. An example By using the overlapped critical constraints arrangement
one point location operation is sufficient instead of two q

25. The remaining case: the convex hulls are not disjoint

26. The remaining case: the convex hulls are not disjoint
The four vertices x, e, f, y can be determined by triangulation
Two canal gates: xe and fy x
bays e
canal
bays f y

27. A key property of canals
For any observer q outside the canal
q cannot see “through” the canal via the two gates
the visibility polygons of q in the canal through the two gates do not intersect x e
canal f y q

28. Determining the canal: first determine four obstacle vertices
junction
triangles d b

29. Determining the canal canal gate a c x e canal d f y
their intersection
canal gate b

30. Thank You