Problem definition Approach Dynamic program Summary Pseudo-Convex Decomposition of a Simple Polygon Stefan Gerdjikov and Alexander Wolff Fakultät für Informatik Karlsruhe University Stefan Gerdjikov and Alexander Wolff 1 19 Pseudo-Convex Decomposition Problem definition Approach Dynamic program Summary Outline 1 Problem definition 2 Approach Concave geodesics Characterization of pseudo-triangles 3 Dynamic program Computing pwij Computing cwij 4 Summary Stefan Gerdjikov and Alexander Wolff 2 19 Pseudo-Convex Decomposition Problem definition Approach Dynamic program Summary Outline 1 Problem definition 2 Approach Concave geodesics Characterization of pseudo-triangles 3 Dynamic program Computing pwij Computing cwij 4 Summary Stefan Gerdjikov and Alexander Wolff 3 19 Pseudo-Convex Decomposition 2 A pseudo-triangle is a simple polygon with 3 convex angles. 3 A pseudo-convex decomposition of a simple polygon Problem definition Approach Dynamic program Summary Preliminaries Definition 1 A convex angle is an angle ≤ 180◦. Stefan Gerdjikov and Alexander Wolff 4 19 Pseudo-Convex Decomposition 3 A pseudo-convex decomposition of a simple polygon Problem definition Approach Dynamic program Summary Preliminaries Definition 1 A convex angle is an angle ≤ 180◦. 2 A pseudo-triangle is a simple polygon with 3 convex angles. Stefan Gerdjikov and Alexander Wolff 4 19 Pseudo-Convex Decomposition 3 A pseudo-convex decomposition of a simple polygon Problem definition Approach Dynamic program Summary Preliminaries Definition 1 A convex angle is an angle ≤ 180◦. 2 A pseudo-triangle is a simple polygon with 3 convex angles. Stefan Gerdjikov and Alexander Wolff 4 19 Pseudo-Convex Decomposition Problem definition Approach Dynamic program Summary Preliminaries Definition 1 A convex angle is an angle ≤ 180◦. 2 A pseudo-triangle is a simple polygon with 3 convex angles. 3 A pseudo-convex decomposition of a simple polygon Stefan Gerdjikov and Alexander Wolff 4 19 Pseudo-Convex Decomposition Problem definition Approach Dynamic program Summary Preliminaries Definition 1 A convex angle is an angle ≤ 180◦. 2 A pseudo-triangle is a simple polygon with 3 convex angles. 3 A pseudo-convex decomposition of a simple polygon Stefan Gerdjikov and Alexander Wolff 4 19 Pseudo-Convex Decomposition Output: A pseudo-convex decomposition of P with the minimum number m of polygons. −→ Problem definition Approach Dynamic program Summary Problem Input: Simple polygon P = A0A1 ... An−1 in the plane. Stefan Gerdjikov and Alexander Wolff 5 19 Pseudo-Convex Decomposition Problem definition Approach Dynamic program Summary Problem Input: Simple polygon P = A0A1 ... An−1 in the plane. Output: A pseudo-convex decomposition of P with the minimum number m of polygons. −→ Stefan Gerdjikov and Alexander Wolff 5 19 Pseudo-Convex Decomposition −→ Problem definition Approach Dynamic program Summary Motivation Pseudo-convex decomposition: fewer polygons than a convex decomposition! Pseudo-triangles have important applications in rigidity theory. Hope: ? Pseudo-convex decomposition of simple polygons ⇒ approximation for pseudo-convex decomposition of point sets. Stefan Gerdjikov and Alexander Wolff 6 19 Pseudo-Convex Decomposition −→ Problem definition Approach Dynamic program Summary Motivation Pseudo-convex decomposition: fewer polygons than a convex decomposition! Pseudo-triangles have important applications in rigidity theory. Hope: ? Pseudo-convex decomposition of simple polygons ⇒ approximation for pseudo-convex decomposition of point sets. Stefan Gerdjikov and Alexander Wolff 6 19 Pseudo-Convex Decomposition −→ Problem definition Approach Dynamic program Summary Motivation Pseudo-convex decomposition: fewer polygons than a convex decomposition! Pseudo-triangles have important applications in rigidity theory. Hope: ? Pseudo-convex decomposition of simple polygons ⇒ approximation for pseudo-convex decomposition of point sets. Stefan Gerdjikov and Alexander Wolff 6 19 Pseudo-Convex Decomposition Problem definition Approach Dynamic program Summary Motivation Pseudo-convex decomposition: fewer polygons than a convex decomposition! Pseudo-triangles have important applications in rigidity theory. Hope: ? Pseudo-convex decomposition of simple polygons ⇒ approximation for pseudo-convex decomposition of point sets. −→ Stefan Gerdjikov and Alexander Wolff 6 19 Pseudo-Convex Decomposition Problem definition Approach Concave geodesics Dynamic program Characterization of pseudo-triangles Summary Outline 1 Problem definition 2 Approach Concave geodesics Characterization of pseudo-triangles 3 Dynamic program Computing pwij Computing cwij 4 Summary Stefan Gerdjikov and Alexander Wolff 7 19 Pseudo-Convex Decomposition 2 Each dij defines a simple polygon Pij = Ai Ai+1 ... Aj . 3 wij the minimum number of polygons in a pseudo-convex decomposition of Pij . Problem definition Approach Concave geodesics Dynamic program Characterization of pseudo-triangles Summary Diagonals and induced subpolygons Definition 1 Diagonals dij = Ai Aj , i < j and Aj is visible from Ai . Ai Aj dij Stefan Gerdjikov and Alexander Wolff 8 19 Pseudo-Convex Decomposition 3 wij the minimum number of polygons in a pseudo-convex decomposition of Pij . Problem definition Approach Concave geodesics Dynamic program Characterization of pseudo-triangles Summary Diagonals and induced subpolygons Definition 1 Diagonals dij = Ai Aj , i < j and Aj is visible from Ai . 2 Each dij defines a simple polygon Pij = Ai Ai+1 ... Aj . Pij Ai Aj dij Stefan Gerdjikov and Alexander Wolff 8 19 Pseudo-Convex Decomposition wij = 2 Problem definition Approach Concave geodesics Dynamic program Characterization of pseudo-triangles Summary Diagonals and induced subpolygons Definition 1 Diagonals dij = Ai Aj , i < j and Aj is visible from Ai . 2 Each dij defines a simple polygon Pij = Ai Ai+1 ... Aj . 3 wij the minimum number of polygons in a pseudo-convex decomposition of Pij . Pij Ai Aj dij Stefan Gerdjikov and Alexander Wolff 8 19 Pseudo-Convex Decomposition Problem definition Approach Concave geodesics Dynamic program Characterization of pseudo-triangles Summary Diagonals and induced subpolygons Definition 1 Diagonals dij = Ai Aj , i < j and Aj is visible from Ai . 2 Each dij defines a simple polygon Pij = Ai Ai+1 ... Aj . 3 wij the minimum number of polygons in a pseudo-convex decomposition of Pij . Pij Ai Aj dij wij = 2 Stefan Gerdjikov and Alexander Wolff 8 19 Pseudo-Convex Decomposition → dynamic programming . w0,n−1 = m Problem definition Approach Concave geodesics Dynamic program Characterization of pseudo-triangles Summary Diagonals and induced subpolygons Definition 1 Diagonals dij = Ai Aj , i < j and Aj is visible from Ai . 2 Each dij defines a simple polygon Pij = Ai Ai+1 ... Aj . 3 wij the minimum number of polygons in a pseudo-convex decomposition of Pij . wi,i+1 = 0 Stefan Gerdjikov and Alexander Wolff 8 19 Pseudo-Convex Decomposition → dynamic programming Problem definition Approach Concave geodesics Dynamic program Characterization of pseudo-triangles Summary Diagonals and induced subpolygons Definition 1 Diagonals dij = Ai Aj , i < j and Aj is visible from Ai . 2 Each dij defines a simple polygon Pij = Ai Ai+1 ... Aj . 3 wij the minimum number of polygons in a pseudo-convex decomposition of Pij . wi,i+1 = 0 . w0,n−1 = m Stefan Gerdjikov and Alexander Wolff 8 19 Pseudo-Convex Decomposition Problem definition Approach Concave geodesics Dynamic program Characterization of pseudo-triangles Summary Diagonals and induced subpolygons Definition 1 Diagonals dij = Ai Aj , i < j and Aj is visible from Ai . 2 Each dij defines a simple polygon Pij = Ai Ai+1 ... Aj . 3 wij the minimum number of polygons in a pseudo-convex decomposition of Pij . wi,i+1 = 0 . → dynamic programming w0,n−1 = m Stefan Gerdjikov and Alexander Wolff 8 19 Pseudo-Convex Decomposition min. #polygons in pseudo-convex pwij = pseudo-triangle decomposition of Pij if dij bounds a Problem definition Approach Concave geodesics Dynamic program Characterization of pseudo-triangles Summary Split the problem! wij = min. #polygons in pseudo-convex decomposition of Pij . Stefan Gerdjikov and Alexander Wolff 9 19 Pseudo-Convex Decomposition Problem definition Approach Concave geodesics Dynamic program Characterization of pseudo-triangles Summary Split the problem! wij = min. #polygons in pseudo-convex decomposition of Pij . min. #polygons in pseudo-convex pwij = pseudo-triangle decomposition of Pij if dij bounds a Pij Pij dij dij Stefan Gerdjikov and Alexander Wolff 9 19 Pseudo-Convex Decomposition Problem definition Approach Concave geodesics Dynamic program Characterization of pseudo-triangles Summary Split the problem! wij = min. #polygons in pseudo-convex decomposition of Pij . pwij min. #polygons in pseudo-convex pseudo-triangle = cwij decomposition of Pij if dij bounds a convex polygon Pij Pij dij dij diagonal-convex decomposition Stefan Gerdjikov and Alexander Wolff 9 19 Pseudo-Convex Decomposition Problem definition Approach Concave geodesics Dynamic program Characterization of pseudo-triangles Summary Split the problem! wij = min. #polygons in pseudo-convex decomposition of Pij . pwij min. #polygons in pseudo-convex pseudo-triangle = cwij decomposition of Pij if dij bounds a convex polygon wij = min(pwij, cwij) Compute wij , cwij and pwij in increasing order of j − i. Stefan Gerdjikov and Alexander Wolff 9 19 Pseudo-Convex Decomposition B1 = Ai and Bm = Aj . For each k < m: + Bk+1 = last vertex on P (Bk , Aj ) visible from Bk . B1B2 ... Bm is a convex, anticlockwise oriented polygon. Problem definition Approach Concave geodesics