line segment intersection - kit...calculate all line segment intersections compute regions 3 dr....

Dr. Tamara Mchedlidze · Dr. Darren Strash · Computational Geometry Lecture Line Segment Intersection

Tamara Mchedlidze · Darren Strash

Computational Geometry · Lecture

INSTITUT FUR THEORETISCHE INFORMATIK · FAKULTAT FUR INFORMATIK

Line Segment Intersection

26.10.2015

1


Aside: Organizational Items

2


Overlaying Map Layers

+

=

Example: Given two different map layers whose intersection isof interest.

Land use

Precipitation

Map combining themes

3


Overlaying Map Layers

+

=

Example: Given two different map layers whose intersection isof interest.

Land use

Precipitation

Map combining themes

Regions are polygonsPolygons are line segmentsCalculate all line segment intersectionsCompute regions

3


Problem Formulation

Given: Set S = s1, . . . , sn of line segments in the plane

Output: all intersections of two or more line segmentsfor each intersection, the line segments involved.

4


Problem Formulation



Def: Line segments are closed

4


Problem Formulation



Def: Line segments are closed

Discussion:– How can you solve this problemnaively?– Is this already optimal?– Are their better approaches?

4


The Sweep-Line Method: An Example

5



Events

5



5



sweep line

5



5


Data Structures

1.) Event Queue Qdefine p ≺ q ⇔def. yp > yq ∨ (yp = yq ∧ xp < xq)

p q`

6


Data Structures


p q`

Store events by ≺ in a balanced binary search tree

→ e.g., AVL tree, red-black tree, . . .

6


Data Structures


p q`



Operations insert, delete and nextEvent in O(log |Q|) time

6


Data Structures


p q`




2.) Sweep-Line Status T `

Stores ` cut lines ordered from left to right

6


Data Structures


p q`






Required operations insert, delete, findNeighbor

6


Data Structures


p q`






Required operations insert, delete, findNeighbor

This is also a balanced binary search tree with line segments stored inthe leaves!

6


Algorithm

FindIntersections(S)

Input: Set S of line segmentsOutput: Set of all intersection points and the line segments

involvedQ ← ∅; T ← ∅foreach s ∈ S doQ.insert(upperEndPoint(s))Q.insert(lowerEndPoint(s))

while Q 6= ∅ dop← Q.nextEvent()Q.deleteEvent(p)handleEvent(p)

7


Algorithm





What happenswith duplicates?

7


Algorithm





What happenswith duplicates?

This is the core of thealgorithm!

7


Algorithm

handleEvent(p)

U(p)← Line segments with p as upper endpointL(p)← Line segments with p as lower endpointC(p)← Line segments with p as interior pointif |U(p) ∪ L(p) ∪ C(p)| ≥ 2 then

return p and U(p) ∪ L(p) ∪ C(p)

remove L(p) ∪ C(p) from Tadd U(p) ∪ C(p) to Tif U(p) ∪ C(p) = ∅ then //sl and sr, neighbors of p in TQ ← check if sl and sr intersect below p

else //s′ and s′′ leftmost and rightmost line segment in U(p) ∪ C(p)Q ← check if sl and s′ intersect below pQ ← check if sr and s′′ intersect below p

8


Algorithm

handleEvent(p)





Stored with p in Q

8


Algorithm

handleEvent(p)





Stored with p in Q

Neighbors in T

8


Algorithm

handleEvent(p)





Stored with p in Q

Neighbors in T

Remove and insertreverses order in C(p)

8


What Happens Exactly?

s1

s2

s3

s4

s5s6

s7 s6 s5 s4 s1

s3 s7

s1

s5

s6 s4

s3

T

p

S

9



s1

s2

s3

s4

s5s6

s7 s6 s5 s4 s1

s3 s7

s1

s5

s6 s4

s3

T

p

U(p) = s2L(p) =C(p) =

S

9



s1

s2

s3

s4

s5s6

s7 s6 s5 s4 s1

s3 s7

s1

s5

s6 s4

s3

T

p

U(p) = s2L(p) = s4, s5C(p) = s1, s3

S

9



s1

s2

s3

s4

s5s6

s7 s6 s5 s4 s1

s3 s7

s1

s5

s6 s4

s3

T

p

U(p) = s2L(p) = s4, s5C(p) = s1, s3

Report (p, s1, s2, s3, s4, s5)

S

9



s1

s2

s3

s4

s5s6

s7 s6 s5 s4 s1

s3 s7

s1

s5

s6 s4

s3

T

p

U(p) = s2L(p) = s4, s5C(p) = s1, s3 s6

s7

s2

s3

s6

s1

s3

s2 s1

Delete L(p) ∪ C(p); add U(p) ∪ C(p)

S

9



s1

s2

s3

s4

s5s6

s7 s6 s5 s4 s1

s3 s7

s1

s5

s6 s4

s3

T

p

U(p) = s2L(p) = s4, s5C(p) = s1, s3 s6

s7

s2

s3

s6

s1

s3

s2 s1

Add event p′ = s1 × s7 in Q s′

s′′ sr

sl

p′

S

9


Correctness

Lemma 1:Algorithm FindIntersections finds all intersectionpoints and the line segments involved

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Correctness


Proof:Induction on the number of events processed.

Let p be an intersection point and all intersection points q ≺ pare already correctly computed.

Case 1: p is a line segment endpointp was inserted in QU(p) stores pL(p) and C(p) are in T

10


Correctness


Proof:Induction on the number of events processed.

Let p be an intersection point and all intersection points q ≺ pare already correctly computed.

Case 1: p is a line segment endpointp was inserted in QU(p) stores pL(p) and C(p) are in T

Case 2: p is not a line segment endpoint

Consider why p must be in Q!10


Running-Time Analysis


Input: Set S of line segmentsOutput: Set of all intersections with their line segmentsQ ← ∅; T ← ∅foreach s ∈ S doQ.insert(upperEndPoint(s))Q.insert(lowerEndPoint(s))


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O(1)

11






O(1)

O(n log n)

11






O(1)

O(n log n)

O(log |Q|)

11






O(1)

O(n log n)

O(log |Q|) ?

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Lemma 2:Algorithm FindIntersections has running timeO(n log n + I log n), where I is the number ofintersection points.

handleEvent(p)





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Summary

Thm 1:Let S be a set of n line segments in the plane. Thenwe can compute intersections in S together with theinvolved line segments in O((n + I) log n) time andO(?) space.

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Summary


Proof:Correctness XRunning time XSpace

13


Summary



Consider how much space thedata structures need!

13


Summary


– T has at most n elements– Q has at most O(n + I) elements– reduction of Q to O(n) space: an exercise

Consider how much space thedata structures need!

Thm 1:Let S be a set of n line segments in the plane. Thenwe can compute intersections in S together with theinvolved line segments in O((n + I) log n) time andO(n) space.

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Discussion

Is the Sweep-Line Algorithm always better than the naive one?

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Discussion


No, because if I ∈ Ω(n2) then the algorithm has running timeO(n2 log n).

14


Discussion



Can we do better?

14


Discussion



Can we do better?

Yes, in Θ(n log n + I) time and Θ(n) space [Balaban, 1995].

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Discussion



Can we do better?


How does this solve the map overlay problem?

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Discussion



Can we do better?


How does this solve the map overlay problem?

Using an appropriate data structure (doubly-connected edgelist) forplanar graphs we can compute in O((n + I) log n) time the overlay oftwo maps.(Details in Ch. 2.3 of the book)

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line segment intersection - kit...calculate all line segment intersections compute regions 3 dr....

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