Arrangements of Lines C omputational Geometry By Samaneh shafi naderi Arrangements of lines L n . L (arrangement) L A(L) . Chapter 8- arrangements and duality Computational Geometry arrangements of lines Theorem 8.4 Constructing Arrangements Algorithm Zone Theorem(8.5) Levels and Discrepancy : Complexity : # vertices + # edges + # faces Arrangements of lines Chapter 8- arrangements and duality Computational Geometry arrangements of lines Theorem 8.4 Constructing Arrangements Algorithm Zone Theorem(8.5) Levels and Discrepancy Dual planePrimary plane Simple Arrangements . Chapter 8- arrangements and duality Computational Geometry arrangements of lines Theorem 8.4 Constructing Arrangements Algorithm Zone Theorem(8.5) Levels and Discrepancy simple Not simple Theorem 8.4 L n A(L) . 1 ) A(L) n(n-1)/2 2 ) A(L) n 3 ) A(L) n/2+n/2+1 A(L) Chapter 8- arrangements and duality Computational Geometry arrangements of lines Theorem 8.4 Constructing Arrangements Algorithm Zone Theorem(8.5) Levels and Discrepancy : L O(n) . Constructing Arrangements DCEL . :DCEL . : bounding box B(L) Chapter 8- arrangements and duality Computational Geometry arrangements of lines Theorem 8.4 Constructing Arrangements Algorithm Zone Theorem(8.5) Levels and Discrepancy DCEL 1) Plane sweep algorithm 2) Incremental algorithm Chapter 8- arrangements and duality Computational Geometry arrangements of lines Theorem 8.4 Constructing Arrangements Algorithm Zone Theorem(8.5) Levels and Discrepancy Plane sweep O(nlog n) . . Chapter 8- arrangements and duality Computational Geometry arrangements of lines Theorem 8.4 Constructing Arrangements Algorithm Zone Theorem(8.5) Levels and Discrepancy Incremental ( ) arrangements DCEL . DCEL 1 ) B(L) A(L) DCEL . 2 ) arrangement DCEL Chapter 8- arrangements and duality Computational Geometry arrangements of lines Theorem 8.4 Constructing Arrangements Algorithm Zone Theorem(8.5) Levels and Discrepancy B(L) B(L) O(n) . . Chapter 8- arrangements and duality Computational Geometry arrangements of lines Theorem 8.4 Constructing Arrangements Algorithm Zone Theorem(8.5) Levels and Discrepancy e B(L) L i A i- 1 . Chapter 8- arrangements and duality Computational Geometry arrangements of lines Theorem 8.4 Constructing Arrangements Algorithm Zone Theorem(8.5) Levels and Discrepancy Chapter 8- arrangements and duality Computational Geometry arrangements of lines Theorem 8.4 Constructing Arrangements Algorithm Zone Theorem(8.5) Levels and Discrepancy : 1 ) f . 2 ) f e f . e Chapter 8- arrangements and duality Computational Geometry arrangements of lines Theorem 8.4 Constructing Arrangements Algorithm Zone Theorem(8.5) Levels and Discrepancy O(n 2 ) O(n) ? zone l : A(L) L A(L) l . Zone: . Chapter 8- arrangements and duality Computational Geometry arrangements of lines Theorem 8.4 Constructing Arrangements Algorithm Zone Theorem(8.5) Levels and Discrepancy Zone Theorem L i A(L) zone L i . zone . Chapter 8- arrangements and duality Computational Geometry arrangements of lines Theorem 8.4 Constructing Arrangements Algorithm Zone Theorem(8.5) Levels and Discrepancy zone: zone m O(m) . Left bounding edge : . Right bounding edge: . Chapter 8- arrangements and duality Computational Geometry arrangements of lines Theorem 8.4 Constructing Arrangements Algorithm Zone Theorem(8.5) Levels and Discrepancy Proof of Zone theorem : L m . l x . L . ( ) : left-bounding edge zone l 5m . right-bounding edge . Chapter 8- arrangements and duality Computational Geometry arrangements of lines Theorem 8.4 Constructing Arrangements Algorithm Zone Theorem(8.5) Levels and Discrepancy Proof of Zone theorem : M=1 . ( 5>1) m-1 L .(5(m-1) < 5m) . Left bounding = X 5(m-1) + X < 5m Chapter 8- arrangements and duality Computational Geometry arrangements of lines Theorem 8.4 Constructing Arrangements Algorithm Zone Theorem(8.5) Levels and Discrepancy L1L1 Proof of Zone theorem : l m L . X=3 5(m-1)+3 < 5m Chapter 8- arrangements and duality Computational Geometry arrangements of lines Theorem 8.4 Constructing Arrangements Algorithm Zone Theorem(8.5) Levels and Discrepancy lmlm split Proof of Zone theorem l 1 Zone . Chapter 8- arrangements and duality Computational Geometry arrangements of lines Theorem 8.4 Constructing Arrangements Algorithm Zone Theorem(8.5) Levels and Discrepancy Proof of Zone theorem : l m l 2 L . X=5 5(m-1)+5