Lecture 4 (CGAL) Panos Giannopoulos, Dror Atariah AG TI WS 2012/13

Transcription

1 Lecture 4 (CGAL) Panos Giannopoulos Dror Atariah AG TI WS 2012/13

2 Exercise 3?

3 Outline Note: Random points Delaunay Triangulations and Voronoi diagrams Exercise 4 Project ideas FU Berlin Lecture 4 (CGAL) WS 2012/13 3

4 Note: Random points Choosing random points in a convex polygon FU Berlin Lecture 4 (CGAL) WS 2012/13 4

5 t f e e n e t r r n r y e k l s. Triangulation of a point set P Collection of triangles convex hull boundary triangle corners are the points of P union of triangles = convex hull of P any two triangles are either disjoint or intersect in a corner or edge Note: the number triangles is the same any triangulation of P Delaunay Triangulation: max min angle (avoids skinny triangles ) FU Berlin Lecture 4 (CGAL) WS 2012/13 5

6 We can model a piece of the earth s surface as a terrain. A terrain is a Delaunay Triangulation 2-dimensional surface in 3-dimensional space with a special property: every vertical line intersects it in a point if it intersects it at all. In other words it is the graph of a function f : A R2 R that assigns a height f (p) to every point p in the domain A of the terrain. (The earth is round so on a global scale terrains defined in this manner are not a good model of the earth. But (Some) motivation: (Polyhedral) Terrains on a more local scale terrains provide a fairly good model.) A terrain can be visualized with a perspective drawing like the one in Figure 9.1 or with contour lines lines of equal height like on a topographic map. Figure 9.1 A perspective view of a te Of course we don t know the height of every point on earth; we only know it where we ve measured it. This means that when we talk about some terrain we only know the value of the function f at a finite set P A of sample points. From the height of the sample points we somehow have to approximate the height FU Berlin Lecture 4 (CGAL) WS 2012/13 at the other points in the domain. A naive approach assigns to every p A the 6

7 Delaunay Triangulation ook very natural. Therefore our approach for approximating a terrain ows. We first determine a triangulation of P: a planar subdivision ounded faces are triangles and whose vertices are the points of P. (We hat the sample points are such that we can make the triangles cover Terrain from sample points ain of the terrain.) We then lift each sample point to its correct height mapping every triangle in the triangulation to a triangle in 3-space..2 illustrates this. What we get is a polyhedral terrain the graph of a us function that is piecewise linear. We can use the polyhedral terrain proximation of the original terrain. Approximation of a real terrain (earth s surface) measure height at sample points P approximate the height of other points (real terrain) triangulate P lift each sample point to its correct height question remains: how do we triangulate the set of sample points? In his can be done in many different ways. But which triangulation is the ropriate one for our purpose namely to approximate a terrain? There nitive answer to this question. We do not know the original terrain we w its height at the sample points. Since we have no other information eight at FU theberlin sample points is the correct height for any triangulation all Lecture 4 (CGAL) WS 2012/13 7

8 3 g e only know its height at the sample points. Since we have no other information and the height at the sample points is the correct height for any triangulation all Delaunay Triangulation triangulations of P seem equally good. Nevertheless some triangulations look more natural than others. For example have a look at Figure 9.3 which shows two triangulations the same point set. From the heights of the sample points Terrain fromofsample points: which triangulation? we get the impression that the sample points were taken from a mountain ridge. Triangulation (a) reflects this intuition. Triangulation (b) however where one single edge has been flipped has introduced a narrow valley cutting through flip orridge. notintuitively to flip? this looks wrong. Can we turn this intuition into thetomountain a criterion that tells us that triangulation (a) is better than triangulation (b)? q height = (a) q height = (b) Skinny triangles (-> small angles) give results The problem with triangulation (b) is that the height of thecounter-intuitive point q is deter(e.g. cutting a mountain ridge into a narrow valey) FU Berlin Lecture 4 (CGAL) WS 2012/13 8

9 Delaunay Triangulation Delaunay Triangulation: max min angle (over all triangles) over all triangulations Note: there is only a finite number of different triangulations FU Berlin Lecture 4 (CGAL) WS 2012/13 9

10 Now consider an edge e = pi p j of a triangulation T of P. If e is not an edge Triangulation oflocally the unbounded face is incident to two triangles we candelaunay increase theitsmallest angle i p j pk and pi p j pl. If these Sectionp9.1 two triangles form a convex quadrilateral we can obtain a newoftriangulation observation immediately follows from the TRIANGULATIONS PLANAR POINT (Given some initial triangulation) " T by removing pi p j from T and inserting psets k pl instead. We call this operation Flip illegal edges to increase min angleof(if possible) an edge flip. The only difference in the angle-vector T and T " are the six pl ation with an illegal edge e. Let T! be the! pj pping e. Then A(Tα ) > A(T). α3 2 α5 pl edge flip α1 angles αα... α α!... α! to compute the pi 9.4Instead al. weαcan use the simple criterion dge of this criterion follows pk from Thales s ness α4" α2" pi α1" α3" α6" pj α5" pk angles α1... α6 in A(T) which are replaced by α1"... α6" in A(T " ). Figure 9.4 illustrates this. We call the edge e = pi p j an illegal edge if nt to triangles pi p j pk and pi p j pl and let C min α < min αi". The edge pi p j is illegal if and only ifi the 1!i!6 1!i!6 thermore if the points Illegal edge pcriterion: i p j pk pl form e on a common circle then exactly one of metric in pk and pl : pl lies inside the circle s inside circle FU the Berlin Lecture through 4 (CGAL) WSp 2012/13 i p j pl. When pl pj pi pk illegal 10

11 ch noi ne no Voronoi diagram ( dual of DT) Given a set P of n points (sites) in the plane are is tes We of ere noi p j. en of ase use lel Voronoi diagram V(P): subdivision of the plane into n cells (one for each site of P) such that a point q lies in the cell corresponding to a site pi P iff distance(q pi ) < distance(q pi ) pk e p pj FU Berlin Lecturei 4 (CGAL) WS 2012/13 11

12 Voronoi diagram ( dual of DT) How are the cells formed? Intersection of (open) half-planes formed by bisectors between sites FU Berlin Lecture 4 (CGAL) WS 2012/13 12

13 Voronoi diagram ( dual of DT) Section 7.2 Characterization of DIAGRAM edges and COMPUTING THE VORONOI vertices of a Voronoi diagram a point is a vertex of V(P) iff its largest empty circle contains three or more sites on its boundary a bisector between two sites defines an edge iff there is a point on the bisector such that its largest empty circle contains both sites on its boundary but no other site The Voronoi diagram can be computed in O(n log n) time FU Berlin Lecture 4 (CGAL) WS 2012/13 13

14 .2 Delaunay Triangulations and Voronoi diagrams The Delaunay Triangulation What s the connection? et P be a set of n points or sites as we shall sometimes call them in the lane. Recall from Chapter 7 that the Voronoi diagram of P is the subdivision f the plane into n regions one for each site in P such that the region of a te p P contains all points in the plane for which p is the closest site. The Dual of V(P) Voronoi diagram of Pgraph is denoted G by Vor(P). The region of a site p is called G a node for every Voronoi cell (equiv. site) an arc between two nodes if the corr. cells share and edge Vor(P) he Voronoi cell of p; it is denoted by V(p). In this section we will study the ual graph of the Voronoi diagram. This graph G has a node for every Voronoi ell equivalently for every site and it has an arc between two nodes if the orresponding cells share an edge. Note that this means that G has an arc for very edge of Vor(P). As you can see in Figure 9.5 there is a one-to-one orrespondence between the bounded faces of G and the vertices of Vor(P). FU Berlin Lecture 4 (CGAL) WS 2012/13 14

15 edge of V(pi ) and V(p j ).) contained in V(pi ) fine ti j todelaunay be the triangle whose vertices are pi p j and the Voronoi center of Ci j. diagrams Triangulations and eevoronoi cell of oftip; it is denoted we in will the that the edge pi toby thev(p). centerin ofthis Ci j issection contained V(pstudy j connecting i ); pi milar observation holds for p j. Now let graph pk pl be another edgeforofevery DG(P) al graph of the Voronoi diagram. This G has a node Voronoi define the circle Cfor the site and triangle tkl it similar the between way Ci j and were if the ll equivalently every has antoarc twoti jnodes kl and fined. rresponding cells share an edge. Note that this means that G has an arc for Ci j Suppose contradiction thatcan pi psee Both is pk aand pl j and k pl intersect. As you inpfigure 9.5 there one-to-one ery edge for of avor(p). st lie outside Cibetween lie outside that pk pofl must j and so they j. This rrespondence the also bounded facestiof G andimplies the vertices Vor(P). pj ersect one of the edges of ti j incident to the center of Ci j. Similarly pi p j st intersect one of the edges of tkl incident to the center of Ckl. It follows contained in V(p j ) t one of the edges of ti j incident to the center of Ci j must intersect one of the ges of tkl incident to the center of Ckl. But this contradicts that these edges contained in disjoint Voronoi cells. What s the connection? Delaunay graph DG(P) Sraight-line embedding of the dual graph G It s a plane graph It has a face for every vetex of V(P) The Delaunay graph of P is an embedding of the dual graph of the Voronoi gram. As observed earlier it has a face for every vertex of Vor(P). The edges und a face correspond to the Voronoi edges incident to the corresponding onoi vertex. In particular if a vertex v of Vor(P) is a vertex of the Voronoi s for the sites p1 p2 p3... pk then the corresponding face f in DG(P) has p2 p3... pk as its vertices. Theorem 7.4(i) tells us that in this situation the nts p1 p2 p3... pk lie on a circle around v so we not only know that f is a on but even that it is convex. If the points of P are distributed at random the chance that four points pen to lie on a circle is very small. We will in this chapter say that a set points is in general position if it contains no four points on a circle. If P is Consider the straight-line embedding of G where the node corresponding general position then all vertices of the Voronoi diagram have degree three the Voronoi cell V(p) is the point p and the arc connecting the nodes of consequently all bounded faces of DG(P) are triangles. This explains why p) and V(q)called is thethe segment pq see Figureof 9.6. call this (P) is often Delaunay triangulation P. We We shall be aembedding bit more the elaunay of DG(P) P and we denote it by DG(P). name sounds eful andgraph will call the Delaunay graph of P. (Although We define athe Delaunay ench Delaunay graphs have nothing withedges the French painter. They ngulation to be any triangulation obtainedtobydo adding to the Delaunay ph. Since all faces of DG(P) are convex obtaining such a triangulation The Delaunay triangulation is any triangulation obtained from DG(P) by adding edges to it (if needed) f v If P contains no 4 points on a cirlce then all vertices of V(P) have degree 3 -> all 197 (bounded) faces of DG(P) are triangles FU Berlin Lecture 4 (CGAL) WS 2012/13 15

16 Delaunay Triangulations and Voronoi diagrams Both Delaunay triangulation and Voronoi diagram of an n point set can be computed in O(n log n) time FU Berlin Lecture 4 (CGAL) WS 2012/13 16

17 Exercise 4 Read CGAL 2D Triangulations (Ch ) Compute the Delaunay triangulation T and its dual Voronoi diagram V of your favorite n point set P (2d). Draw both using Qt. Given a point p (not necessarily in P) find the triangle of T and the cell of V that p lies in. Given two points s and t (not necessarily in P) find a shortest path (measured in number of triangles) from s to t in T. Given two points s and t compute the set of triangles of T interected by the straight-line segment st. FU Berlin Lecture 4 (CGAL) WS 2012/13 17

18 Project ideas (so far...) Porting Arrangments package demo (currently in Qt3) to Qt4 Random point generator on a circle Convex hull (2d 3d) in O(nlogh) time (T. Chan) Triangulating simple polygons (non-convex) in O(n) time (Amato et al.) Shape matching: minimizing the Hausdorff distance of two point sets under translation Guarding art galleries guarding terrains (visibility polygons regions) Shortest paths among obstacles (visibility graphs) Shortest paths in terrains Motion planning in robotics: can a robot reach its distination? (minkowski sums visibility-voronoi complex) Surface reconstruction from point sets Binding CGAL to other (scripting) languages.. Look at the packages of CGAL to find something that you like.. FU Berlin Lecture 4 (CGAL) WS 2012/13 18