Point Location Problem for Polygon and Planar Subdivision'

1
Point Location Problem for Polygon and Planar
Subdivision.

• I4B05 ???
• I4B18 ???
• Ref
• Computational Geometry, Chap. 6 Preparata and
  Shamos, Chap. 2

2
Point Location Problem in Polygon

• Given a polygon P and a point z, determine
  whether or not z is internal to P.

3
Point Location Problem in Polygon

• The single-shot approach
• Division Wedges for Convex Inclusion

4
The single-shot approach
Assume A horizontal line L intersects
polygon P, but L doesnt pass though any vertex
of P.
5
The single-shot approach

• Algorithm
• Begin CL 0
• For i 1 until N do if edge ( i ) is not
  horizontal then
• if( lower extreme of edge ( I )
  intersects L to the left of z )
• then L 1 then L L 1
• if( L is odd) then z is internal else z is
  external
• End.

6
The single-shot approach

• Whether a point Z is internal to a simple N-gon P
  can be determined in O( N ) time, without
  preprocessing.
• The result holds for nonconvex polygons as well.

7
Division Wedges for Convex Inclusion

• The method relies on the convexity of P.
• Treating Q as the origin of polar coordinates.
• The vertices of a polygon occur in angular order
  about any internal point Q.

8
Division Wedges for Convex Inclusion
P2
P3
P4
P
P1
Z
Q
P5
P7
P6
9
Procedure

• CONVEX INCLUSION
• 1. Given a query point Z, determine by binary
  search the wedge in which it lies.
• 2. Determine that point Z lies between the rays
  defined by Pi and Pi1.
• 3.Once Pi and Pi1 are found, then Z is internal
  if and only if ( Pi Pi1 Z ) is a left turn .

10
Division Wedges for Convex Inclusion

• The inclusion question for a convex N-gon can
  be answered in O( log N ) time, given O( N )
  space and O( N ) preprocessing time.

11
A star-shaped polygon
P3
P2
Z
P4
P1
Q
P5
P
P7
P6
12
A star-shaped polygon

• The kernel Q can be founded in O( N ) time.
• The inclusion question for an N-vertex
  star-shaped polygon can be answered in O( log N )
  time and O( N ) storage, after O( N )
  preprocessing time.

13
Point Location in a planar subdivision

• The slab method
• The chain method
• The triangulation refinement method

14
The slab method

• Draw vertical lines through all vertices of the
  subdivision, as in figure. This partitions the
  plane into vertical slabs.

15
The slab method

• All edges intersecting a slab
• Have no endpoint in the slab
• Dont cross each other

16
The slab method

• Sort slabs in ascending x-coordinates.
• Store the segments crossing the interval ordered
  from top to bottom.
• Give a query point z.
• Use binary search to determine the slab
• by x-coordinates.
• Use binary search to determine the face
• by point-line discrimination.

17
Worst Case in Storage
18
The slab method

• Point-location in an N-vertex planar subdivision
  can be effected in O( log N )time using O( N2 )
  storages.

19
The Chain Method

• Definition
• A chain C ( u1, , up ) is a planar
  straight-line graph (PSLG) with vertex set u1,
  , up and edge set (ui, ui1) i 1, ,
  p-1

U1
U3
U2
U5
U6
U4
U8
U7
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The Monotone Chain

• Definition
• A chain C ( u1, , up ) is said to be
  monotone with respect to a straight line L if a
  line orthogonal to L intersects C in exactly one
  point.
• The orthogonal projections L(u1) , , L(up)
  of the vertices of C on L are ordered as L(u1) ,
  , L(up) .

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Where does the query point lie?
U2
U1
P
U4
U3
U5
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A query point lies ?

• The projection of P on L can be located with a
  binary search in a unique interval ( L(ui),
  L(ui1) )
• Determine on which side of the line containing
  uiui1 the query point lies.

P
U1
U2
L(u1)
U4
U3
L(u2)
L(P)
U5
L(u3)
L(u4)
L(u5)
L
23
The Chain Method

• Suppose there is a set C C1, , Cr of a
  polygon, we can apply the bisection to find the
  region query point lies.
• PS. C1, , Cr respect to y-axis.

C2
C1
C3
C6
C4
C5
24
The Chain Method

• If there are r chains in C and the longest chain
  has p vertices, then the search worst-case time
  is O( log p log r )

25
PSLG Transform into Chains

• 1. Regularize the PSLG.
• 2. Assign weights on the graph using
  weight-balancing algorithm.
• 3. Construct chains with respect to y-axis by
  traversing the graph.

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Definition Regular Graph

• Let G be a PSLG with vertex set v1, ..., vN ,
  where the vertices are indexed so that i lt j if
  and only if either y(vi) lt y(vj) or, if y(vi)
  y(vj), then x(vi) gt x(vj). A vertex vj is said to
  be regular if there are integers i lt j lt k such
  that ( vi, vj ) and ( vj, vk ) are edges of G.
• Graph G is said to be regular if each vj is
  regular for 1 lt j lt N.
• With the exception of the two extreme vertices v1
  and vN.

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Regularize Nonregular Vertices (remove cusps)

• Definition cusp
• Sweep-line to remove cusps
• Connect to the upper end point of the lower
  neighbor
• The connection does not cross any edge of G.

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Example 1 Remove cusps
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Example 2 Remove cusps
30
Remove cusps

• Regularization of an N-vertex PSLG is
  accomplished in time O(N log N), due to the
  possible initial sorting of vertex ordinates and
  to location of N vertices in the status structure
  , each at O( log N) cost.

31
Weight Assignment of Edges

• All edges satisfy
• 1. Each edge has positive weight
• 2. For each Vj( j ?1, N ),
• Win(Vj) Wout(Vj)

Vin( Vi ) IN(v)
Vout( Vi ) OUT(v)
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WEIGHT-BALANCING REGULAR PSLG

• 1. Initialization
• Begin for each edge e do W(e) 1
• 2. First pass
• for( i 2 i lt N-1 i )
• Win(Vi) sum of weight of incoming edges of Vi
• if (Win(Vi) gt Vout(Vi))
• d1 leftmost outgoing edge of Vi
• W(d1) Win(Vi) Vout(Vi) 1

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• 3. Second pass
• For (i N-1 i gt 2 i--)
• Wout(Vi) sum of weight of outgoing edges of Vi
• if (Wout(Vi) gt Win(Vi))
• d2 leftmost incoming edge of Vi
• W(d2) Wout(Vi) Win(Vi) W(d2)

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Example 1 1. Initialization
1
1
1
1
1
1
1
35
2. The first pass
1
X 2
1
1
1
1
1
1
36
3. The second pass
2
1
1
1
1
1
1
2 X
37
Construct chains
2
X 1
X 0
1
X 0
1
0 X
1
X 0
1
0 X
1
X 0
2
1 X
0 X
38
PSLG and Monotone Chains
39
Example 2 1. Initialization
40
2. The first pass
X 2
3 X
X 2
41
3. The second pass
3 X
42
Construct chains
Start from top traverse the leftmost possible
branch
2
X 1
X 0
1
0 X
3
2 X
1 X
0 X
1
0 X
1
X 0
0 X
1
1
2
0 X
X1
X0
X 0
1
1
X 0
1
1
X 0
X 0
0 X
1
0 X
1
3
2 X
1 X
0 X
1
X 0
43
PSLG and Monotone Chains
44
The chain method

• Time complexity O( log2 N )
• Space complexity O( N )
• Preprocessing time ( Regularization ) O( N log N
  )