UNC Chapel Hill M. C. Lin

Line Segment Intersection

Chapter 2 of the Textbook

Driving Applications–Map overlap problems– 3D Polyhedral Morphing

UNC Chapel Hill M. C. Lin

Thematic Map Overlay

GIS systems split each map into several layers

Each layer is called a thematic map, storing one type of information

Find overlay of several maps to locate interesting junctions

UNC Chapel Hill M. C. Lin

Transform to a Geometric Problem

Curves can be approximated by small (line) segments

Each thematic map can be viewed as a collection of line segments

Finding the overlay of two networks => computing all intersection points between the line segments of two sets

UNC Chapel Hill M. C. Lin

Let’s Be More Serious & Precise...

Segments closed or open?– Take a look at the original problem =>

should be closed

To simplify further, make 2 sets into 1– But, how do we identify the really

interesting one? Filter them out by checking if they are from the same set.

UNC Chapel Hill M. C. Lin

Problem Analysis

Brute Force Approach: O(n2) Desiderata: output(intersection) sensitive – Segments that are close together are the

candidates for intersection

x

y

UNC Chapel Hill M. C. Lin

Plane Sweep Algorithm

Status of l: the set of segments intersecting l Event points: where updates are required

l : sweep line

event point

UNC Chapel Hill M. C. Lin

Plane Sweep Algorithm (cont)

At a event point: update the status of the sweep line & perform intersection testsUpper: a new segment is added to the status of l

and it’s tested against the rest

Lower: it’s deleted from the status of l

=> Only testing pairs of segments for which there is a horizontal line intersects both segments.

=> But, this is not good enough. It may still be inefficient, O(n2) for some cases. (ex) a set of segments all intersect with x-axis.

UNC Chapel Hill M. C. Lin

Plane Sweep Algorithm (cont)

To include the idea of being close in the horizontal direction, only test segments that are adjacent in the horizontal direction --

Only test each with ones to its left and right New “status”: ordered sequence of segments New “event points”: endpoints and intersects

l

Sj Sk SlSm

UNC Chapel Hill M. C. Lin

Nasty Cases (Degeneracies)

Horizontal lines

Overlapping line segments

Multiple line segments intersect at one single point

UNC Chapel Hill M. C. Lin

Plane-Sweep Algorithm (Recap)

Move a horizontal sweep line l downwards

Halt l at event points (end pts & intersects)

While l moves, maintain ordered sequence of segments intersected by it

When l halts at an event point, the sequence of segments changes, status of l needs to be updated and to detect intersections depending on type of events

UNC Chapel Hill M. C. Lin

Event Types

NOTE: only intersection points below l are important, assuming all intersection points above l have been computed correctly.

Upper end point: If Si and Sk are adjacent on l, a new upper end point of Sj appears, check Sj with Si and Sk for intersections

Intersection point of 2 lines: Change their order. Each gets (at most) 1 new neighbor

Lower end point: Its two neighbors are now adjacent and must be tested for intersection below the sweep line l

UNC Chapel Hill M. C. Lin

Data Structure

Type: Event queue that stores events

Operations– remove next event and return it to be treated– among 2 events with the same y-coordinate, the

one with smaller x-coordinate is returned

(left-to-right priority order)– allow for insertions & check if it is already there– allow 2++ event points to coincide

(ex) two upper end points coincide

UNC Chapel Hill M. C. Lin

Implementing Event Queue

Define an order on event points, according to which they will be handled

Store the event points in a balanced binary search tree T according to their orders– both fetching & insertion takes O(log m) time, where m is

the number of events

Maintain the status of l using T– the left-to-right order of segments on the line l <=> the

left-to-right order of leaves in T– segments in internal nodes guide search– each update and search takes O(log m)

UNC Chapel Hill M. C. Lin

Status Structure, T

lSjSk Sl SmSi

Si

Sj Sk

Sl Sm

Si

Sj

Sk

SlT

UNC Chapel Hill M. C. Lin

FindIntersections (S)

Input : a set S of line segments in a plane Output : a set of intersections and their

associated line segments in S

1. Initialize Q. Insert the end points into Q

with their corresponding segments

2. Initialize an empty status structure T

3. While Q is not empty

4. Do Find next event point p in Q & delete it

5. HandleEventPoint (p)

UNC Chapel Hill M. C. Lin

Handling Changes in Status

l

S4

S1

S3 S8S7

S5

S2

S1S3

UNC Chapel Hill M. C. Lin

Handling Changes in Status

S7 S3

S1 S8

S3

S2

S1

S2

S7

T

S1

S3 S8

S5

S4

S1

S3

S7 S5S4

S7

T

S2

l

S4

S1

S3 S8S7S5

S1S3

UNC Chapel Hill M. C. Lin

HandleEventPoint (p)

1. Let U(p) be set of segments whose upper end point is p

2. Search in T for set S(p) of all segments that contains p;

they are adjacent in T. Let L(p) S(p) be the set of

segments whose lower endpts in p and C(p) S(p) be

the set of segments that contains p in its interior

3. If L(p) U(p) C(p) contains more than 1 segment

4. then Report p as an intersect with L(p), U(p) and C(p)

5. Delete segments in L(p) C(p) from T

6. Insert segments in U(p) C(p) into T. Order segments

in T according to their order on sweep line just below p.

A horizontal one comes last among all containing p.

UNC Chapel Hill M. C. Lin

HandleEventPoint (p)

7. (Deleting & re-inserting segments of C(p) reverses order)

8. If U(p) C(p) = 0

9. then Let sl and sr be the left/right neighbors of p in T

10. FindNewEvent(sl , sr , p)

11. else Let s’ be the left most segment of U(p) C(p) in T

12. Let sl be the left neighbor of s’ in T

13. FindNewEvent(sl , s’, p)

14. Find s’’ be rightmost segment of U(p) C(p) in T

15. Let sr be the right neighbor of s’’ in T

16. FindNewEvent(s’’, sr , p)

UNC Chapel Hill M. C. Lin

FindNewEvent (sl, sr, p)

1. If sl and sr intersect below the sweep line, or on it and to the right of current point p, and the intersection is not yet present as an event in Q

2. Then Insert the intersection point as an event into Q

UNC Chapel Hill M. C. Lin

Algorithm Analysis

Let S be a set of n segments in a plane

All intersections in S can be reported in– O(n log n + I log n) time– O(n) space– where I is the number of intersection points

UNC Chapel Hill M. C. Lin

Doubly-Connected Edge List

3 records: vertices, faces and “half-edges”

Vertex: coordinates(v) & a ptr to a half-edge

Face: – OuterComponent(f): bounding edges

– InnerComponent(f): edges on the boundary of holes contained in the face, f

Edge: a ptr to Origin(e), a ptr to a twin-edge, ptrs to Next(e) & Prev(e) edges and its left IncidentFace(e)

UNC Chapel Hill M. C. Lin

Computing Overlay of Subdivisions

Let S1, S2 be two planar subdivisions of complexity n1 and n2 respectively; and let n = n1 + n2

Overlay of S1 and S2 can be constructed in O(n log n + k log n) time, where k is the complexity of overlay

UNC Chapel Hill M. C. Lin

Boolean Operations

Let P1, P2 be two polygons with n1 and n2 vertices respectively; and let n = n1 + n2

Their boolean operations (intersection, union, and difference) can each be computed in O(n log n + k log n) time, where k is the complexity of the output