Point Location Geometry Motivation Point Location and Trapezoidal - PowerPoint PPT Presentation

Yazd Univ. Computational Point Location Geometry Motivation Point Location and Trapezoidal Maps A Randomized Incremental 1397-2 Algorithm Search Structure 1 / 38

Motivation: Point location in a map Point location query: Given a map and a query point q Yazd Univ. specified by its coordinates, find the region of the map containing q . Computational Geometry 5 ◦ 6 ′ Motivation Point Location and Trapezoidal Maps A Randomized Incremental Algorithm Search Structure 52 ◦ 3 ′ 2 / 38

Motivation: Point location in a map Store the map electronically, and let the computer do the point location for you. Yazd Univ. Preprocess the maps, and to store them in a data structure that makes it possible to answer point Computational Geometry location queries fast. Motivation 5 ◦ 6 ′ Point Location and Trapezoidal Maps A Randomized Incremental Algorithm Search Structure 52 ◦ 3 ′ 3 / 38

Motivation: Point location in a map Store the map electronically, and let the computer do the point location for you. Yazd Univ. Preprocess the maps, and to store them in a data structure that makes it possible to answer point Computational Geometry location queries fast. Motivation 5 ◦ 6 ′ Point Location and Trapezoidal Maps A Randomized Incremental Algorithm Search Structure 52 ◦ 3 ′ 3 / 38

Trapezoidal Maps: Store the map electronically, and let the computer do the point location for you. Preprocess the maps, and to store them in a data Yazd Univ. structure that makes it possible to answer point Computational location queries fast. Geometry 5 ◦ 6 ′ Motivation Point Location and Trapezoidal Maps A Randomized Incremental Algorithm Search Structure 52 ◦ 3 ′ 4 / 38

Trapezoidal Maps: Store the map electronically, and let the computer do the point location for you. Preprocess the maps, and to store them in a data Yazd Univ. structure that makes it possible to answer point Computational location queries fast. Geometry 5 ◦ 6 ′ Motivation Point Location and Trapezoidal Maps A Randomized Incremental Algorithm Search Structure 52 ◦ 3 ′ 4 / 38

Trapezoidal Maps: Given a planar subdivision S with n edges, store S in such a way that we can answer queries of the following type: Given a query point q , report the face Yazd Univ. f of S that contains q . Computational Geometry Motivation Point Location and Trapezoidal Maps A Randomized Incremental Algorithm Search Structure 5 / 38

Trapezoidal Maps: Find the slab containing the query point: O (log n ) time. Yazd Univ. Query the slab: O (log n ) time. Computational Geometry Motivation Point Location and Trapezoidal Maps A Randomized Incremental Algorithm Search Structure 6 / 38

Trapezoidal Maps: Find the slab containing the query point: O (log n ) time. Yazd Univ. Query the slab: O (log n ) time. Computational Geometry Motivation Point Location and Trapezoidal Maps A Randomized Incremental Algorithm Search Structure 6 / 38

Trapezoidal Maps: What about the storage requirements? Yazd Univ. Computational Geometry Motivation Point Location and Trapezoidal Maps A Randomized Incremental Algorithm Search Structure 7 / 38

Trapezoidal Maps: What about the storage requirements? Yazd Univ. Computational Geometry Motivation n Point Location and 4 Trapezoidal Maps A Randomized Incremental Algorithm Search Structure n slabs 4 7 / 38

Trapezoidal Maps: What about the storage requirements? O ( n 2 ) Yazd Univ. Computational Geometry Motivation n Point Location and 4 Trapezoidal Maps A Randomized Incremental Algorithm Search Structure n slabs 4 7 / 38

Trapezoidal Maps: Two Simplifications: Yazd Univ. Bounding Box. No two distinct points on a vertical line (general Computational Geometry position). Motivation Point Location and Trapezoidal Maps A Randomized Incremental Algorithm Search Structure 8 / 38

Trapezoidal Maps: Two Simplifications: Yazd Univ. Bounding Box. No two distinct points on a vertical line (general Computational Geometry position). Motivation Point Location and Trapezoidal Maps A Randomized Incremental Algorithm Search Structure 8 / 38

Trapezoidal Maps: Trapezoidal Map: Yazd Univ. drawing two vertical extensions from every endpoint Sides of a face Computational Geometry Motivation Point Location and Trapezoidal Maps A Randomized Incremental Algorithm Search Structure R 9 / 38

Trapezoidal Maps: Trapezoidal Map: Yazd Univ. drawing two vertical extensions from every endpoint Sides of a face Computational Geometry Motivation Point Location and Trapezoidal Maps A Randomized Incremental Algorithm Search Structure sides R 9 / 38

Trapezoidal Maps: Lemma 6.1 Each face in a trapezoidal map of a set S of line segments in general Yazd Univ. position has one or two vertical sides and exactly two non-vertical sides. Computational Geometry Proof. Each face is convex: all angles are at most 180 ◦ Motivation Because of the convexity, each face has at most 2 Point Location and Trapezoidal Maps vertical sides. A Randomized Incremental If there is more than two non-vertical sides, then at Algorithm least two of them must meet at a vertex which Search Structure contradict the definition of a face. Since a face is bounded, it cannot have less than two non-vertical sides and that it must have at least one vertical side. 10 / 38

Trapezoidal Maps: Lemma 6.1 Each face in a trapezoidal map of a set S of line segments in general Yazd Univ. position has one or two vertical sides and exactly two non-vertical sides. Computational Geometry Proof. Each face is convex: all angles are at most 180 ◦ Motivation Because of the convexity, each face has at most 2 Point Location and Trapezoidal Maps vertical sides. A Randomized Incremental If there is more than two non-vertical sides, then at Algorithm least two of them must meet at a vertex which Search Structure contradict the definition of a face. Since a face is bounded, it cannot have less than two non-vertical sides and that it must have at least one vertical side. 10 / 38

Trapezoidal Maps: Lemma 6.1 Each face in a trapezoidal map of a set S of line segments in general Yazd Univ. position has one or two vertical sides and exactly two non-vertical sides. Computational Geometry Proof. Each face is convex: all angles are at most 180 ◦ Motivation Because of the convexity, each face has at most 2 Point Location and Trapezoidal Maps vertical sides. A Randomized Incremental If there is more than two non-vertical sides, then at Algorithm least two of them must meet at a vertex which Search Structure contradict the definition of a face. Since a face is bounded, it cannot have less than two non-vertical sides and that it must have at least one vertical side. 10 / 38

Trapezoidal Maps: Lemma 6.1 Each face in a trapezoidal map of a set S of line segments in general Yazd Univ. position has one or two vertical sides and exactly two non-vertical sides. Computational Geometry Proof. Each face is convex: all angles are at most 180 ◦ Motivation Because of the convexity, each face has at most 2 Point Location and Trapezoidal Maps vertical sides. A Randomized Incremental If there is more than two non-vertical sides, then at Algorithm least two of them must meet at a vertex which Search Structure contradict the definition of a face. Since a face is bounded, it cannot have less than two non-vertical sides and that it must have at least one vertical side. 10 / 38

Trapezoidal Maps: Lemma 6.1 Each face in a trapezoidal map of a set S of line segments in general Yazd Univ. position has one or two vertical sides and exactly two non-vertical sides. Computational Geometry top (∆) Motivation Point Location and Trapezoidal Maps A Randomized ∆ Incremental Algorithm Search Structure bottom (∆) 11 / 38

Trapezoidal Maps: Cases for the left side: It degenerates to a point, which is the common left endpoint of top ( ∆ ) and (a) bottom ( ∆ ) . Yazd Univ. It is the lower vertical extension of the left endpoint of top ( ∆ ) that abuts (b) on bottom ( ∆ ) . Computational Geometry It is the upper vertical extension of the left endpoint of bottom ( ∆ ) that (c) abuts on top ( ∆ ) . (d) It consists of the upper and lower extension of the right endpoint p of a third Motivation segment s . These extensions abut on top ( ∆ ) and bottom ( ∆ ) , respectively. Point Location and It is the left edge of R . This case occurs for a single trapezoid of T ( S ) only, (e) Trapezoidal Maps namely the unique leftmost trapezoid of T ( S ) . A Randomized Incremental The first four cases are illustrated in Figure 6.4. The five cases for the right Algorithm leftp (∆) Search Structure leftp (∆) leftp (∆) top (∆) top (∆) top (∆) top (∆) bottom (∆) s bottom (∆) bottom (∆) bottom (∆) leftp (∆) (a) (b) (c) (d) 12 / 38