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Chapter 2 Line-Segment Intersection or: Map Overlay Alexander Wolﬀ TU/e Geometric Algorithms (2IL55) Line-Segment Intersection Application ? Line-Segment Intersection Application: map overlay (GIS) Line-Segment Intersection Application: map overlay (GIS) Deﬁnition: Line-Segment Intersection Application: map overlay (GIS) Deﬁnition: Is an intersection? Line-Segment Intersection Application: map overlay (GIS) Deﬁnition: Is an intersection? Answer: Check application! Line-Segment Intersection Application: map overlay (GIS) Deﬁnition: Is an intersection? Answer: Check application! Problem: Given a set S of n closed non-overlapping line segments in the plane, compute: – all points where at least two segments intersect and – for each such point report all segments that contain it. Line-Segment Intersection Application: map overlay (GIS) Deﬁnition: Is an intersection? Answer: Check application! yes! Problem: Given a set S of n closed non-overlapping line segments in the plane, compute: – all points where at least two segments intersect and – for each such point report all segments that contain it. Data structures for a top-to-bottom sweep 1) event (-point) queue Q Data structures for a top-to-bottom sweep 1) event (-point) queue Q p q ⇔def. Data structures for a top-to-bottom sweep 1) event (-point) queue Q p q ⇔def. yp > yq Data structures for a top-to-bottom sweep 1) event (-point) queue Q p q ⇔def. yp > yq or (yp = yq and xp < xq ) Data structures for a top-to-bottom sweep 1) event (-point) queue Q p q ⇔def. yp > yq or (yp = yq and xp < xq ) p q Data structures for a top-to-bottom sweep 1) event (-point) queue Q p q ⇔def. yp > yq or (yp = yq and xp < xq ) p q Data structures for a top-to-bottom sweep 1) event (-point) queue Q p q ⇔def. yp > yq or (yp = yq and xp < xq ) p q Data structures for a top-to-bottom sweep 1) event (-point) queue Q p q ⇔def. yp > yq or (yp = yq and xp < xq ) p q Store event pts in balanced binary search tree acc. to Data structures for a top-to-bottom sweep 1) event (-point) queue Q p q ⇔def. yp > yq or (yp = yq and xp < xq ) p q Store event pts in balanced binary search tree acc. to ⇒ nextEvent() and del/insEvent() take O(log |Q|) time Data structures for a top-to-bottom sweep 1) event (-point) queue Q p q ⇔def. yp > yq or (yp = yq and xp < xq ) p q Store event pts in balanced binary search tree acc. to ⇒ nextEvent() and del/insEvent() take O(log |Q|) time 2) (sweep-line) status T Data structures for a top-to-bottom sweep 1) event (-point) queue Q p q ⇔def. yp > yq or (yp = yq and xp < xq ) p q Store event pts in balanced binary search tree acc. to ⇒ nextEvent() and del/insEvent() take O(log |Q|) time 2) (sweep-line) status T Data structures for a top-to-bottom sweep 1) event (-point) queue Q p q ⇔def. yp > yq or (yp = yq and xp < xq ) p q Store event pts in balanced binary search tree acc. to ⇒ nextEvent() and del/insEvent() take O(log |Q|) time 2) (sweep-line) status T Store the segments intersected by in left-to-right order. Data structures for a top-to-bottom sweep 1) event (-point) queue Q p q ⇔def. yp > yq or (yp = yq and xp < xq ) p q Store event pts in balanced binary search tree acc. to ⇒ nextEvent() and del/insEvent() take O(log |Q|) time 2) (sweep-line) status T Store the segments intersected by in left-to-right order. How? Data structures for a top-to-bottom sweep 1) event (-point) queue Q p q ⇔def. yp > yq or (yp = yq and xp < xq ) p q Store event pts in balanced binary search tree acc. to ⇒ nextEvent() and del/insEvent() take O(log |Q|) time 2) (sweep-line) status T Store the segments intersected by in left-to-right order. How? In a balanced binary search tree! Sweep-line algorithm ﬁndIntersections(S) Input: set S of n non-overlapping closed line segments Output: set I of intersection pts for each p ∈ I every s ∈ S with p ∈ s Sweep-line algorithm ﬁndIntersections(S) Input: set S of n non-overlapping closed line segments Output: set I of intersection pts for each p ∈ I every s ∈ S with p ∈ s Q ← ∅; T ← ∅ foreach s ∈ S do Q.insert(upperEndPt(s)) Q.insert(lowerEndPt(s)) Sweep-line algorithm ﬁndIntersections(S) Input: set S of n non-overlapping closed line segments Output: set I of intersection pts for each p ∈ I every s ∈ S with p ∈ s Q ← ∅; T ← ∅ foreach s ∈ S do Q.insert(upperEndPt(s)) Q.insert(lowerEndPt(s)) while Q = ∅ do p ← Q.nextEvent() Q.deleteEvent(p) handleEvent(p) Correctness and running time Lemma: ﬁndIntersections() correctly computes all inter- section points & the segments that contain them. Correctness and running time Lemma: ﬁndIntersections() correctly computes all inter- section points & the segments that contain them. Lemma: ﬁndIntersections() ﬁnds I intersection points among n non-overlapping line segments in O((n + I) log n) time. Correctness and running time Lemma: ﬁndIntersections() correctly computes all inter- section points & the segments that contain them. Lemma: ﬁndIntersections() ﬁnds I intersection points among n non-overlapping line segments in O((n + I) log n) time. Theorem: We can report all I intersection points among n non-overlapping line segments in the plane and report the segments involved in the intersections in O((n + I) log n) time and O(n) space. Doubly-Connected Edge List Motivation: – data structure for planar subdivisions – map overlay (e.g., rivers and streets) – “walk” on the map, from region to region Doubly-Connected Edge List Motivation: – data structure for planar subdivisions – map overlay (e.g., rivers and streets) – “walk” on the map, from region to region graph nodes arcs abstract adjacency information Doubly-Connected Edge List Motivation: – data structure for planar subdivisions – map overlay (e.g., rivers and streets) – “walk” on the map, from region to region graph subdivision nodes vertices arcs edges abstract adjacency information Doubly-Connected Edge List Motivation: – data structure for planar subdivisions – map overlay (e.g., rivers and streets) – “walk” on the map, from region to region graph subdivision geometry nodes vertices points in R2 arcs edges line segments (or: circular arcs...) abstract adjacency information Doubly-Connected Edge List Motivation: – data structure for planar subdivisions – map overlay (e.g., rivers and streets) – “walk” on the map, from region to region graph subdivision geometry nodes vertices points in R2 incident arcs edges line segments (or: circular arcs...) abstract adjacency faces information Doubly-Connected Edge List Motivation: – data structure for planar subdivisions – map overlay (e.g., rivers and streets) – “walk” on the map, from region to region graph subdivision geometry nodes vertices points in R2 incident arcs edges line segments (or: circular arcs...) abstract adjacency faces information maximal connected subsets of the plane without vertices and edges Doubly-Connected Edge List Motivation: – data structure for planar subdivisions – map overlay (e.g., rivers and streets) – “walk” on the map, from region to region graph subdivision geometry nodes vertices points in R2 adjacent incident arcs edges line segments (or: circular arcs...) abstract adjacency faces information maximal connected subsets of the plane without vertices and edges Doubly-Connected Edge List Motivation: – data structure for planar subdivisions – map overlay (e.g., rivers and streets) – “walk” on the map, from region to region gy graph subdivision geometry lo po to nodes vertices points in R2 adjacent incident arcs edges line segments (or: circular arcs...) abstract adjacency faces information maximal connected subsets of the plane without vertices and edges Doubly-Connected Edge List Motivation: – data structure for planar subdivisions – map overlay (e.g., rivers and streets) – “walk” on the map, from region to region gy graph subdivision geometry lo po to nodes vertices points in R2 adjacent incident arcs edges line segments (or: circular arcs...) abstract adjacency faces information maximal connected subsets of the plane without vertices and edges complexity of a subdiv. := #vertices + #edges + #faces DCEL: Notation and an Example twin(e) origin(e) e next(e) prev(e) face(e) DCEL: Notation and an Example twin(e) Note: Half-edges are oriented origin(e) such that incident face e lies locally to the left. next(e) prev(e) face(e) DCEL: Notation and an Example twin(e) Note: Half-edges are oriented origin(e) such that incident face e lies locally to the left. next(e) prev(e) face(e) v1 e11 v2 f2 e21 e41 f1 v3 e32 v4 e31 DCEL: Notation and an Example twin(e) Note: Half-edges are oriented origin(e) such that incident face e lies locally to the left. next(e) prev(e) face(e) v1 e11 v2 f2 e21 edge origin twin face next prev e41 e11 v1 e22 f1 e21 e41 f1 v3 e12 v2 e11 f2 e42 e22 e32 e21 v2 e22 f1 e31 e11 v4 e31 ... DCEL: Notation and an Example twin(e) vertex coordinates edge origin(e) v1 (0, 3) e11 e v2 (2, 3) e21 next(e) v3 (2, 1) e31 prev(e) v4 (1, 0) e22 face(e) v1 e11 v2 f2 e21 edge origin twin face next prev e41 e11 v1 e22 f1 e21 e41 f1 v3 e12 v2 e11 f2 e42 e22 e32 e21 v2 e22 f1 e31 e11 v4 e31 ... DCEL: Notation and an Example twin(e) vertex coordinates edge origin(e) v1 (0, 3) e11 e v2 (2, 3) e21 next(e) v3 (2, 1) e31 prev(e) v4 (1, 0) e22 face(e) face outerComp. innerComp. e11 f1 nil e11 v1 v2 f2 e22 nil f2 e21 edge origin twin face next prev e41 e11 v1 e22 f1 e21 e41 f1 v3 e12 v2 e11 f2 e42 e22 e32 e21 v2 e22 f1 e31 e11 v4 e31 ... Map Overlay Input: subdivisions S1 , S2 as DCELs S2 S1 Map Overlay Input: subdivisions S1 , S2 as DCELs Output: DCEL D D S2 S1 Map Overlay Input: subdivisions S1 , S2 as DCELs Output: DCEL D with V (D) = V (S1 ) ∪ V (S2 ) ∪ {e1 ∩ e2 | ei ∈ E(Si )} E(D) = conn. comp. of E(S1 ) ∪ E(S2 ) \ V (D) D S2 S1 Map Overlay Input: subdivisions S1 , S2 as DCELs Output: DCEL D with V (D) = V (S1 ) ∪ V (S2 ) ∪ {e1 ∩ e2 | ei ∈ E(Si )} E(D) = conn. comp. of E(S1 ) ∪ E(S2 ) \ V (D) D S2 S1 Map Overlay Input: subdivisions S1 , S2 as DCELs Output: DCEL D with interpreted as line segments V (D) = V (S1 ) ∪ V (S2 ) ∪ {e1 ∩ e2 | ei ∈ E(Si )} E(D) = conn. comp. of E(S1 ) ∪ E(S2 ) \ V (D) D S2 S1 Map Overlay Input: subdivisions S1 , S2 as DCELs Output: DCEL D with interpreted as line segments V (D) = V (S1 ) ∪ V (S2 ) ∪ {e1 ∩ e2 | ei ∈ E(Si )} E(D) = conn. comp. of E(S1 ) ∪ E(S2 ) \ V (D) D S2 S1 Map Overlay Input: subdivisions S1 , S2 as DCELs Output: DCEL D with interpreted as line segments V (D) = V (S1 ) ∪ V (S2 ) ∪ {e1 ∩ e2 | ei ∈ E(Si )} E(D) = conn. comp. of E(S1 ) ∪ E(S2 ) \ V (D) Why do we need the big union? — Think and/or ask! D S2 S1 Map Overlay Input: subdivisions S1 , S2 as DCELs Output: DCEL D with interpreted as line segments V (D) = V (S1 ) ∪ V (S2 ) ∪ {e1 ∩ e2 | ei ∈ E(Si )} E(D) = conn. comp. of E(S1 ) ∪ E(S2 ) \ V (D) Map Overlay Input: subdivisions S1 , S2 as DCELs Output: DCEL D with interpreted as line segments V (D) = V (S1 ) ∪ V (S2 ) ∪ {e1 ∩ e2 | ei ∈ E(Si )} E(D) = conn. comp. of E(S1 ) ∪ E(S2 ) \ V (D) Idea: “recycle” ﬁndIntersections() Map Overlay Input: subdivisions S1 , S2 as DCELs Output: DCEL D with interpreted as line segments V (D) = V (S1 ) ∪ V (S2 ) ∪ {e1 ∩ e2 | ei ∈ E(Si )} E(D) = conn. comp. of E(S1 ) ∪ E(S2 ) \ V (D) Idea: “recycle” ﬁndIntersections() problem: input = set of segm. Map Overlay Input: subdivisions S1 , S2 as DCELs Output: DCEL D with interpreted as line segments V (D) = V (S1 ) ∪ V (S2 ) ∪ {e1 ∩ e2 | ei ∈ E(Si )} E(D) = conn. comp. of E(S1 ) ∪ E(S2 ) \ V (D) Idea: “recycle” ﬁndIntersections() problem: input = set of segm. Changes: Map Overlay Input: subdivisions S1 , S2 as DCELs Output: DCEL D with interpreted as line segments V (D) = V (S1 ) ∪ V (S2 ) ∪ {e1 ∩ e2 | ei ∈ E(Si )} E(D) = conn. comp. of E(S1 ) ∪ E(S2 ) \ V (D) Idea: “recycle” ﬁndIntersections() problem: input = set of segm. Changes: without ﬁnding/treating faces [book, p. 36–38] Map Overlay Input: subdivisions S1 , S2 as DCELs Output: DCEL D with interpreted as line segments V (D) = V (S1 ) ∪ V (S2 ) ∪ {e1 ∩ e2 | ei ∈ E(Si )} E(D) = conn. comp. of E(S1 ) ∪ E(S2 ) \ V (D) Idea: “recycle” ﬁndIntersections() problem: input = set of segm. Changes: without ﬁnding/treating faces [book, p. 36–38] self-study Map Overlay Input: subdivisions S1 , S2 as DCELs Output: DCEL D with interpreted as line segments V (D) = V (S1 ) ∪ V (S2 ) ∪ {e1 ∩ e2 | ei ∈ E(Si )} E(D) = conn. comp. of E(S1 ) ∪ E(S2 ) \ V (D) Idea: “recycle” ﬁndIntersections() problem: input = set of segm. Changes: without ﬁnding/treating faces [book, p. 36–38] • if all segments incident to an event pt belong to one subdivision, keep going! Map Overlay Input: subdivisions S1 , S2 as DCELs Output: DCEL D with interpreted as line segments V (D) = V (S1 ) ∪ V (S2 ) ∪ {e1 ∩ e2 | ei ∈ E(Si )} E(D) = conn. comp. of E(S1 ) ∪ E(S2 ) \ V (D) Idea: “recycle” ﬁndIntersections() problem: input = set of segm. Changes: without ﬁnding/treating faces [book, p. 36–38] • if all segments incident to an event pt belong to one subdivision, keep going! • otherwise “stitch” DCELs together Map Overlay Input: subdivisions S1 , S2 as DCELs Output: DCEL D with interpreted as line segments V (D) = V (S1 ) ∪ V (S2 ) ∪ {e1 ∩ e2 | ei ∈ E(Si )} E(D) = conn. comp. of E(S1 ) ∪ E(S2 ) \ V (D) Idea: “recycle” ﬁndIntersections() problem: input = set of segm. Changes: without ﬁnding/treating faces [book, p. 36–38] • if all segments incident to an event pt belong to one subdivision, keep going! • otherwise “stitch” DCELs together. Cases, e.g., Map Overlay Input: subdivisions S1 , S2 as DCELs Output: DCEL D with interpreted as line segments V (D) = V (S1 ) ∪ V (S2 ) ∪ {e1 ∩ e2 | ei ∈ E(Si )} E(D) = conn. comp. of E(S1 ) ∪ E(S2 ) \ V (D) Idea: “recycle” ﬁndIntersections() problem: input = set of segm. Changes: without ﬁnding/treating faces [book, p. 36–38] • if all segments incident to an event pt belong to one subdivision, keep going! • otherwise “stitch” DCELs together. Cases, e.g., – edge of S1 passes through vertex of S2 – edges of S1 and edges of S2 intersect