Pseudo-Convex Decomposition of Simple Polygons∗

Pseudo-Convex Decomposition of Simple Polygons∗

EWCG 2006, Delphi, March 27–29, 2006 Pseudo-Convex Decomposition of Simple Polygons∗ Stefan Gerdjikov† Alexander Wolff‡ Abstract e.g. for point location or ray shooting. For decom- positions of simple polygons the same terms as for We extend a dynamic-programming algorithm of Keil decompositions of point sets apply. A decomposition and Snoeyink for finding a minimum convex decompo- is called minimum if it consists of the minimum num- sition of a simple polygon to the case when both con- ber of regions. vex polygons and pseudo-triangles are allowed. Our In this paper we give an algorithm for comput- algorithm determines a minimum pseudo-convex de- ing minimum pseudo-convex decompositions of simple composition of a simple polygon in O(n3) time where polygons. Given a simple polygon we use the same ap- n is the number of the vertices of the polygon. In this proach as Gudmundsson and Levcopoulos [4] to deter- way we obtain a well-structured decomposition with mine all geodesics in the polygon which can be sides fewer polygons, especially if the original polygon has of a pseudo-triangle and present a simple way to check long chains of concave vertices. whether three such geodesics form a pseudo-triangle. We use dynamic programming to solve proper sub- problems which then can be combined to obtain a 1 Introduction global solution. The resulting algorithm runs in O(n3) 2 Pseudo-triangles are simple polygons with exactly time and uses O(n ) space. three convex angles, i.e. interior angles of less than Our algorithm is based on a general technique for 180◦. Recently they have emerged to have geometri- decomposing a simple polygon into polygons of a cer- cal properties of interest for rigidity theory and ray- tain type proposed by Keil [5]. The technique is based shooting problems [2]. This is why pseudo-triangles on optimally decomposing subpolygons each of which is obtained from the original by drawing a single di- have been considered in relation with the decomposi- 3 tion problem of a set of points. It is defined as follows. agonal. This idea yields an O(n log n)-time algo- rithm for the convex decomposition problem [5]. Keil Given a set S of n points in the plane, decompose and Snoeyink [6] improve Keil’s result by giving an the convex hull of S into polygons of a given type O(min(nr2, r4))-time algorithm, where r is the num- such that the vertices of the polygons are in S and ber of reflex vertices of the polygon. each point in S is a vertex of at least one of the polygons. The decomposition is called convex if only convex polygons are allowed, pseudo-triangulation if 2 Characterization of Pseudo-Triangles only pseudo-triangles are allowed, and pseudo-convex + − if both pseudo-triangles and convex polygons can be We use P (Ai, Aj ) and P (Ai, Aj ) to denote the used. Convex decompositions have been considered paths on the boundary ∂P from a vertex Ai to a ver- by Fevens et al. [3]. Streinu [7] shows that the min- tex Aj of P in clockwise and anticlockwise direction, imum number of edges needed to obtain a pseudo- respectively. With vis(Ai) we denote the list of all triangulation is 2n−3 and thus, by Euler, the number vertices of P which are visible from Ai in clockwise of pseudo-triangles is n−2, which does not depend on order starting with Ai+1. Unless stated otherwise, the the structure of the point set but only on its size. This vertices of a polygon will be given in clockwise order. motivates research on the problem of enumerating all minimum pseudo-triangulations [2]. Aichholzer et al. Definition 1 Let P = A0A1 ...An−1 be a simple [1] study pseudo-convex decompositions. They show polygon. A path p = B1B2 ...Bm from Ai to Aj that each minimum pseudo-convex decomposition of is a concave geodesic with respect to the polygon P if a set of n points consists of less than 7n/10 polygons. it satisfies the following three conditions, see Fig. 1: A related problem is the decomposition of simple (G1) B1 = A and B = A . polygons into convex polygons or pseudo-triangles, i m j ∗ (G2) For each k<m it holds that Bk+1 is the last Work supported by grant WO 758/4-2 of the German Re- + search Foundation (DFG). vertex on P (Bk, Aj ) which is visible from Bk. †st [email protected] ‡Fakult¨at f¨ur Informatik, Universit¨at Karlsruhe, (G3) B1B2 ...Bm is a convex, anticlockwise oriented P.O. Box 6980, D-76128 Karlsruhe, Germany. WWW: polygon. i11www.ira.uka.de/people/awolff 13 22nd European Workshop on Computational Geometry, 2006 A P k Bm−1 B2 π 2 π3 Aj = Bm Ai = B1 π1 Aj Ai Figure 1: The geodesic B1B2 ...Bm from Ai to Aj is concave with respect to the simple polygon P . Figure 3: Testing whether three concave geodesics π1, π2, and π3 define a pseudo-triangle Remark 1 If B1B2 ...Bm is a concave geodesic from B1 to Bm with respect to a simple polygon P then + be such that Ai ∈ T (Aj , Ak) violates the construc- B2 ...Bm is a concave geodesic from B2 to Bm with tion proposed in property (G2). Let As be the vertex respect to P . + on T (Aj , Ak) after Ai and let k ≥ t>s be such For our further considerations we will need the fol- that At is visible from Ai. It is clear that s>i. lowing fact [6]: Due to Fact 1 we obtain that the edges AiAi+1, AiAs and AiAt appear in clockwise order around Ai. In + Fact 1 Let Ai be a vertex of P = A0A1 ...An−1. particular, because of the convexity of T (Ai, Ak)Ai, + Then the cyclic order of the line segments AiAj with AiAt intersects T (Ai, Ak) only in Ai and AiAs is + AiAj ⊆ P around Ai is the same as the order of their contained in the polygon P (Ai, At)Ai. However, Ak + other endpoints along ∂P . lies outside this polygon and thus T (Ai, Ak) leaves + P (Ai, At)Ai in some point x which does not belong + The following lemma states the relationship be- to AiAt, see Fig 2. Therefore T (Ai, Ak) leaves P . tween the concave geodesics in a simple polygon and Contradiction. the pseudo-triangles that can participate in a decom- position of the polygon. Next we establish the converse relation. Namely three concave geodesics determine a pseudo-triangle. Lemma 1 If a pseudo-triangle T is contained in a simple polygon P = A0A1 ...An−1 with convex ver- Lemma 2 Let P = A0A1 ...An−1 be a simple poly- tices at Aj , Ak and Al, j<k<l, then the paths + + + gon. Further let i<j<k and π1 = Ai ...Aj , π2 = T (Aj , Ak), T (Ak, Al) and T (Al, Aj ) are concave A ...A and π3 = A ...A be concave geodesics geodesics with respect to P . j k k i with respect to P . If the triangle AiAj Ak is clock- wise oriented, then the polygon π1π2π3 is a pseudo- Proof. (Sketch) Due to symmetry it suffices to prove triangle. + that T (Aj , Ak) is a concave geodesic. Proper- ties (G1) and (G3) of a concave geodesic obviously hold. Thus we have to verify only property (G2). Proof. (Idea) See Fig. 3. Using that, say π1 and π2 + First note that T (Aj , Ak) contains only vertices have only one common vertex, one can show that they + of P that lie on P (Aj , Ak), for otherwise T wouldn’t have no other common points. Then the orientation + be simple. Now assume that T (Aj , Ak) does not of the triangle AiAj Ak together with property (G2) satisfy property (G2), see Fig. 2. Let k >i ≥ j from Definition 1 provide that in fact π1 is contained in the triangle AiAj Ak. Similar considerations for A the paths π2 and π3 show that π1π2π3 is a pseudo- l triangle. A t 3 Algorithm A s Ai x We use the same approach for finding a minimum pseudo-convex decomposition of a simple polygon as Keil and Snoeyink [6] for finding the minimum con- Ak Aj vex decomposition of a polygon. Namely we consider smaller simple polygons which are obtained from the Figure 2: Each pseudo-triangle consists of three con- original polygon by drawing a single diagonal. For cave geodesics that connect its convex vertices. The each such polygon we make assumptions in what sort + arcs denote the boundary of P (Aj , At) and the solid of polygon the diagonal can be included. In case the + lines denote the edges of T (Aj , Ak). diagonal is a part of a convex polygon we use the 14 EWCG 2006, Delphi, March 27–29, 2006 algorithm of Keil and Snoeyink [6]. In case the di- convex polygon C = Aj ...Ak ∪ T , where Aj ...Ak is agonal is part of a pseudo-triangle we proceed as fol- a smaller convex polygon. In the former case, an op- lows. Assume we have a precomputed list L of all timal decomposition of Pij consists of wik + wkj +1 concave geodesics w.r.t. P . Then we can filter L to polygons. In the latter case the decomposition of Pij find all pseudo-triangles that contain the diagonal as is the union of two pseudo-convex decompositions: an edge. For each such pseudo-triangle T we compute (i) that of Pik and (ii) that of Pkj under the con- 0 the size of an optimal decomposition that contains T .

View Full Text

Details

  • File Type
    pdf
  • Upload Time
    -
  • Content Languages
    English
  • Upload User
    Anonymous/Not logged-in
  • File Pages
    4 Page
  • File Size
    -

Download

Channel Download Status
Express Download Enable

Copyright

We respect the copyrights and intellectual property rights of all users. All uploaded documents are either original works of the uploader or authorized works of the rightful owners.

  • Not to be reproduced or distributed without explicit permission.
  • Not used for commercial purposes outside of approved use cases.
  • Not used to infringe on the rights of the original creators.
  • If you believe any content infringes your copyright, please contact us immediately.

Support

For help with questions, suggestions, or problems, please contact us