Computational Geometry:Computational Geometry:Intersection SearchIntersection Search

Joseph S. B. MitchellStony Brook University

Intersection SearchIntersection Search

InputInput: A set : A set SS of geometric objects of geometric objects (segments, polygons, disks, solid models, (segments, polygons, disks, solid models, etc)etc)

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Intersection SearchIntersection Search

Versions of the problem:Versions of the problem:• DETECTDETECT: Answer yes/no: Are there any : Answer yes/no: Are there any

intersections among the objects intersections among the objects SS??

(if “yes”, then we may insist on returning a (if “yes”, then we may insist on returning a witness)witness)

• REPORTREPORT: Output all pairs of objects that intersect: Output all pairs of objects that intersect• COMPUTECOMPUTE: Compute the common intersection of all : Compute the common intersection of all

objectsobjects• COUNTCOUNT: How many pairs intersect?: How many pairs intersect?• QUERYQUERY: Preprocess : Preprocess SS to support fast queries of the to support fast queries of the

form “Does object form “Does object QQ intersect any member of intersect any member of SS?” ?” (or “Report all intersections with (or “Report all intersections with QQ”)”)

33May also want to insert/delete in S, or allow objects of S to move.

Warm Up: 1D ProblemWarm Up: 1D Problem

Given n segments (intervals) on a Given n segments (intervals) on a lineline

DETECT: O(n log n), based on sortingDETECT: O(n log n), based on sorting REPORT: O(k+n log n), based on REPORT: O(k+n log n), based on

sorting, then marching through, left sorting, then marching through, left to rightto right

Lower bound to DETECT: Lower bound to DETECT: : : (n log n) (n log n) , from ELEMENT UNIQUENESS, from ELEMENT UNIQUENESS

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Element UniquenessInput: {x1, x2, …,xn }Are they distinct? (yes/no)(n log n)

Intersection Search: SegmentsIntersection Search: Segments Segment intersection: Given a set Segment intersection: Given a set SS

of of nn line segments in the plane, line segments in the plane, determine:determine:• Does some pair intersect? (Does some pair intersect? (DETECTDETECT))• Compute all points of intersection Compute all points of intersection

((REPORTREPORT))

Naïve: O(n2)CG: O(n log n) DETECT, O(k+n log n) REPORT

Lower Bound to DETECT: (n log n) , from ELEMENT UNIQUENESS

Element UniquenessInput: {x1, x2, …,xn }Are they distinct? (yes/no)(n log n)

Primitive ComputationPrimitive Computation

Does segment Does segment abab intersect segment intersect segment cdcd ? ?• Types of “intersect”Types of “intersect”

• Test using “Left” tests (sign of a cross Test using “Left” tests (sign of a cross product (determinant), which product (determinant), which determines the orientation of 3 points) determines the orientation of 3 points)

• Time O(1) (“constant”)Time O(1) (“constant”)66

a

bc

Left( a, b, c ) = TRUE ab ac > 0c is left of ab

Bentley-Ottmann SweepBentley-Ottmann Sweep

Main ideaMain idea: Process : Process eventsevents in order of in order of discovery as a horizontal line, discovery as a horizontal line, LL, , “sweeps” over the scene, from top to “sweeps” over the scene, from top to bottombottom

Two data structuresTwo data structures::• SLS: Sweep Line Status: left-to-right SLS: Sweep Line Status: left-to-right

ordering of the segments intersecting ordering of the segments intersecting LL• EQ: Event QueueEQ: Event Queue

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L

Two data structures:Two data structures:• SLSSLS: Sweep Line Status: left-to-right ordering of : Sweep Line Status: left-to-right ordering of

the segments intersecting the segments intersecting LL• EQEQ: Event Queue: Event Queue

Initialize: Initialize: • SLS = SLS = • EQ = sorted list of segment endpointsEQ = sorted list of segment endpoints

How to store:How to store:• SLS: balanced binary search treeSLS: balanced binary search tree (dictionary, support (dictionary, support O(log n) O(log n) insert, delete, search)insert, delete, search)

• EQ: priority queue (e.g., heap)EQ: priority queue (e.g., heap)

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O(n log n)

L

Hit top endpt of Hit top endpt of ee

Hit bottom endpt of Hit bottom endpt of ee

Hit crossing point Hit crossing point e e f f (only needed in REPORT) (only needed in REPORT)

Event HandlingEvent Handling

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O(log n)

L

a

b

e

Find segs a and b left/right of e in SLS. Insert e into SLS. Test(a,e), Test(b,e) and insert crossings (if any) in EQ

a

b

e

L

Find segs a and b left/right of e in SLS. Delete e from SLS. Test(a,b), and insert crossing (if any) in EQ

a b

L

ef

O(log n)

O(log n)

Exchange e, f in SLS. Test(a,f), Test(b,e) and insert crossings (if any) in EQ

Algorithm AnalysisAlgorithm Analysis InvariantsInvariants of algorithm: of algorithm:

• SLS is correctly ordered. SLS is correctly ordered. • Test(Test(a,ba,b) for intersection is done whenever segments ) for intersection is done whenever segments a a and and

b b become adjacent in the SLS orderbecome adjacent in the SLS order Discovered crossings are inserted into EQ Discovered crossings are inserted into EQ

(for REPORT; for DETECT, stop at first detected crossing)(for REPORT; for DETECT, stop at first detected crossing) Claim:Claim: All crossings are discovered All crossings are discovered Time: Time: O(O(n log nn log n) to DETECT ) to DETECT (O((O(nn) events @ O() events @ O(log nlog n) )) ) Time: Time: O((O((n+kn+k)) log n log n) to REPORT ) to REPORT (O((O(n+kn+k) events @ ) events @

O(O(log nlog n) )) ) Example AppletExample Applet

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k = # crossings = output size

Space: Working MemorySpace: Working Memory

Basic (original) version of B-O sweep: Basic (original) version of B-O sweep: Naively, the EQ has size at most Naively, the EQ has size at most O(n+k), since we store the SLS (O(n)) O(n+k), since we store the SLS (O(n)) and the EQ (size 2n+k )and the EQ (size 2n+k )

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Better bounds on size of EQ: O(n log2 n) [Pach and Sharir]

Space: Working MemorySpace: Working Memory

Modified B-O Sweep: Modified B-O Sweep: • Any time 2 segs STOP being adjacent in Any time 2 segs STOP being adjacent in

SLS, remove from EQ the corresponding SLS, remove from EQ the corresponding crossing point (if any); we will re-insert crossing point (if any); we will re-insert the crossing point again (at least once) the crossing point again (at least once) before the actual crossing event.before the actual crossing event.

• Note: Now the EQ has at most n-1 Note: Now the EQ has at most n-1 crossing events (and at most 2n crossing events (and at most 2n endpoint events)endpoint events)

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Optimal REPORT algorithms exist:Optimal REPORT algorithms exist:• O(O(k + n log nk + n log n), O(), O(k+nk+n) space (working memory)) space (working memory)

[Chazelle-Edelsbrunner][Chazelle-Edelsbrunner]

• O(O(k + n log nk + n log n), O(), O(nn) space ) space [Balaban][Balaban]

Special Case: REPORT for horiz/vert segsSpecial Case: REPORT for horiz/vert segs• Bentley-Ottmann sweep: O(Bentley-Ottmann sweep: O(k + n log nk + n log n) (optimal!)) (optimal!)

Special case: Simplicity testingSpecial case: Simplicity testing• O(O(nn), from Chazelle triangulation), from Chazelle triangulation

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(Complex)

L

All crossings happen when L hits a horizontal segment, si ; just locate (O( log n)) the endpoint and walk along horizontal segment in SLS to report all ki vertical segments crossed along si

Sweeping applies also toSweeping applies also to• UnionsUnions• IntersectionsIntersections• ArrangementsArrangements

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Practical Methods, 3DPractical Methods, 3D

Uniform gridUniform grid

QuadtreesQuadtrees

Bounding volume hierarchiesBounding volume hierarchies1919

With each pixel, store a list of objects intersecting it.Do brute force on pixel-by-pixel basis.

See Samet books, SAND website

Bounding Volume Bounding Volume HierarchiesHierarchies

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S

Bounding volume

Input: Set S of objects.

QuickCD: Collision DetectionQuickCD: Collision Detection

The 2-Box Cover ProblemThe 2-Box Cover Problem

•Find “smallest” (tightest fitting) pair of bounding boxes

•Motivation:Motivation: Best outer approximationBest outer approximation Bounding volume hierarchiesBounding volume hierarchies

Bounding Volume HierarchyBounding Volume Hierarchy

BV-tree: BV-tree: Level 0Level 0 k-dopsk-dops

BV-tree: Level 1BV-tree: Level 1

26-dops26-dops

14-dops14-dops6-dops6-dops

18-dops18-dops

BV-tree: Level 2BV-tree: Level 2

BV-tree: Level 5BV-tree: Level 5

BV-tree: Level 8BV-tree: Level 8