Lecture 3 (CGAL) Panos Giannopoulos, Dror Atariah AG TI WS 2012/13

Transcription

1 Lecture 3 (CGAL) Panos Giannopoulos Dror Atariah AG TI WS 2012/13

2 Exercises?

3 Outline Convex Hull in 3d Polyhedra and the Half Edge (or DCEL) data structure Exercise 3 3

4 stated above each λi is non-negative and the sum of convex combination. In Chapter 1 we defined the con as the smallest convex set containing the points or m section of all convex sets containing the points. One of a set of points is exactly the set of all possible Convex hull of n points is a hull convex polytope: the points. We can therefore test whether a mixture base mixtures computing the convex hull of their r n by vertices (points) a planar polygon is equal to the number Section 11.1 checking whether the point representing the mixture 2n 4 facets true; the number of edges of a polytope THE COMPLEXITY CONVEX HULLS What if there areof more than two interesting com es. But fortunately the difference cannot IN 3-SPACE 3nhave 6said edges Well what we above remains true; we just h ing theorem on the number of edges and higher dimension. More precisely if we want to take d a facet of a convex polytope is defined we have to represent a mixture by a point in d-dimen edge nts on its boundary. A facet of a convex hull of the points representing the base mixtures wh on. An edge of a convex polytope is an represents the set of all possible mixtures. Convex Hull in 3d Convex hulls in particular convex hulls in 3-dimens ytope with n vertices. The number of various applications. For instance they are used to sp vertex umber of facets of P is at most 2n 4. in computer animation. Suppose that we want to chec If the answer to this question is nega facet es for a connected planar graph with n and P2 intersect. the following strategy pays off. Approximate the ob ng relation holds:!1 and P!2 that contain the originals. If we want to c P!1 and P!2 intersect; + n f = 2. intersect we first check whether P we need to perform the supposedly more costly te 4

5 Convex Hull in 3d Theorem The convex hull of a set of n points in R3 can be computed in O(n log n) time. Here: O(n2 )-time algorithm 5

6 Convex Hull in 3d Input: P ( R3 P = n Output: CH(P) 1. Choose 4 points that are NOT coplanar. (Their convex hull is a tetrahedron.) Let P4 = {p1... p4 } be these points. C CH(P4 ). 2. Let Pr = {p1... pr } for r 1. (p5... pn are the remaining points.) For r 5 to n Add pr to CH(Pr 1 ). (construct CH(Pr ) from CH(Pr 1 ).) C CH(Pr ). 3. Return C. 6

7 of P using a randomized incremental algorithm following the paradigm we have met before in Chapters 4 6 and 9. Convex Hull in 3d The incremental construction starts by choosing four points in P that do not lie in a common plane so that their convex hull is a tetrahedron. This can be done as follows. Let p1 and p2 be two points in P. We walk through the set P until we find a point p3 that does not lie on the line through p1 and p2. We continue searching P until we find a point p4 that does not lie in the plane through p1 p2 and p3. (If we cannot find four such points r 1then all points in P lie in a plane. In this case we can use the planar convex hull algorithm of Chapter 1 to compute the convex hull.) Next we compute a random permutation p5... pn of the remaining points. We will consider the points one by one in this random order maintaining the convex hull as we go. For an integer r! 1 let Pr := {p1... pr }. In a generic step of the algorithm r 1 we have to add the point pr to the convex hull of Pr 1 that is we have to transform CH(Pr 1 ) into CH(Pr ). There are two cases. Add pr to CH(Pr 1 ): visible region (Pr 1 ) 1 ). We re 11.2 btained ection. ane h f pletely ce f is er side 1. If pr lies inside (or on the boudary of) CH(P then CH(Pr ) = CH(Pr 1 ). ) 2. If pr lies outside CH(Pr 1 ) find11.2 visible region of pr on CH(P ) (test every facet) Section COMPUTING CONVEXregion HULLS IN of visible facets enclosed by the horizon). (connected 3- SPACE If pr lies inside CH(Pr 1 ) or on its boundary then CH(Pr ) = CH(Pr 1 ) and there is nothing to be done. Horizon Visibility hf p f q horizon pr Figure 11.2 f is visible fromthep horizon of a polytope H(Pr 1 ) but not from q hat can 3. Replace visible region by facets (triangles) connecting pr standing to Now suppose that pr lies outside CH(P r 1 ). Imagine that you are 246 at pr and that you are looking at CH(Pr 1 ). You will be able to see some must be it s horizon. facets 7

8 we store the convex hull in the form of a doubly-connected edge list a data structure developed Convex HullininChapter 3d 2 for storing planar subdivisions. The only difference is that vertices will now be 3-dimensional points. We will keep the convention that thevisible half-edgesregion are directed that the ones boundingconnecting any face 3. Replace bysuch facets (triangles) pr to form ait s counterclockwise horizon: cycle when seen from the outside of the polytope. pr CH(Pr 1 ) pr facets: we have to check Figure whether 11.3 we ha Chapter 11 CH(P ) Adding r happens if pr lies in the plane of aapoint facetoofth CONVEX HULLS from pr by our definition of visibility ab to the addition pr to new the convex hull. We have their a doubly-connected 4.Back Note: Mergeofany triangle with co-planar faces of and we willinto adda triangles connecting pr edge list representing CH(P ) which we have to transform doublyr 1 CH(P ) (if any). r 1 Those triangles are coplanar w connected edge list for CH(Pr ). Suppose that wehorizon. knew all facets of CH(P ) r 1 f r to remove all the information stored visible from pr. Then it would be peasy with f into one facet. for these facets from the doubly-connected edge list compute the new facets connecting pr to the horizon and store the information for the new facets in the Intime theindiscussion so far we have ignored doubly-connected edge list. All this will take linear the total complexity CH(Pr 1 ) that are visible from pr. Of of the facets that disappear. There is one subtlety we should take care of every after thefacet. addition of thesuch new a test takes const FU Berlin Lecture 3 (CGAL) WS 2012/13 8 Since

9 Proof. Recall that Euler s formula states for a connected planar graph with n nodes ne arcs and n f faces the following relation holds: facet Polyhedra and Half Edge (or DCEL) data structure n ne + n f = 2. Since we can interpret the boundary of a convex polytope as a planar graph see The boundary of a holds 3d convex polytope can beand interpreted Figure 11.1 the same relation for the numbers of vertices edges planar graph: facets in a convex polytope. (In fact Euler s formula was originally stated in as a Figure 11.1 A cube interpreted as a planar g note that one facet maps to the unbounded face of the graph terms of polytopes not convex in terms of hull planarin graphs.) Every face We store the the form of aof the graph corresponding to P has at least three arcs and every arc is incident to two faces Doubly-Connected Edge List so we have 2ne! 3n f. Plugging this into Euler s formula we get (as for planar subdivisions) n + n f 2! 3n f /2 so n f " 2n 4. Applying Euler s formula once more we see that ne " 3n 6. For the special case that every facet is a triangle the case of a simplicial polytope the bounds on the number of edges and facets of an n-vertex polytope 9

10 Chapter 2 The half-edge record of a half-edge!e a pointer Twin(!e) to its twin half-e the face that it bounds. We don t ne because it is equal to Origin(Twin IncidentFace(!e) lies to the left of! destination. The half-edge record al geometric and to the next and previous edge on th topological information Next(! e) is the unique half-edge on th stored in records the destination of!e as its origin and (vertex face half-edge) boundary of IncidentFace(!e) that ha Polyhedra and Half Edge (or DCEL) data structure LINE SEGMENT INTERSECTION Doubly-Connected Edge List: Origin(!e) Twin(!e)!e Next(!e) Prev(!e) IncidentFace(!e) two oriented half-edges A constant information is use for anamount edge of (origin distination) require more storage since the list Inner incident as there are holes in the face of aface. Because a lies to its left from half-edge all InnerComponents( f ) lists toge storage is linear in the complexity of the counterclockwise connected edge list a simple subdivis traversal of afor face corresponding to an edge ei are labeled! Vertex v1 Coordinates (0 4) IncidentEd 10!e11

11 by facets Polyhedra and Half Edge (or DCEL) data structure epresent For a cube it looks like: dary of a herefore t a data The only keep the any face tope. 11

12 3D Polyhedral Surfaces in CGAL CGAL::Polyhedron_3<PolyhedronTraits_3> for storing the convex hull (Uses a DCEL no need for low lever operations from the user): template < class PolyhedronTraits_3 class PolyhedronItems_3 = CGAL::Polyhedron_items_3 template < class T class I> class HalfedgeDS = CGAL::HalfedgeDS_default class Alloc = CGAL_ALLOCATOR(int)> class Polyhedron_3; Use a Kernel as PolyhedronTraits_3 : typedef CGAL::Simple_cartesian<double> Kernel; typedef CGAL::Polyhedron_3<Kernel> Polyhedron; 12

13 CGAL::Polyhedron_3<PolyhedronTraits_3> Container (It provides all sorts of iterators): typedef typedef typedef typedef CGAL::Simple_cartesian<double> Kernel; Kernel::Point_3 Point_3; CGAL::Polyhedron_3<Kernel> Polyhedron; Polyhedron::Vertex_iterator Vertex_iterator; int main() { Point_3 p( ); Point_3 q( ); Point_3 r( ); Point_3 s( ); Polyhedron P; P.make_tetrahedron( p q r s); CGAL::set_ascii_mode( std::cout); for ( Vertex_iterator v = P.vertices_begin(); v!= P.vertices_end(); ++v) std::cout << v >point() << std::endl; return 0; } It provides handles (e.g. Polyhedron::Halfedge_handle ) as trivial lightweight iterators 13

14 CGAL::Polyhedron_3<PolyhedronTraits_3> (It supports plane information by default) E.g. compute plane equation of a plane of a facet: struct Plane_equation { template <class Facet> typename Facet::Plane_3 operator()( Facet& f) { typename Facet::Halfedge_handle h = f.halfedge(); typedef typename Facet::Plane_3 Plane; return Plane( h >vertex() >point() h >next() >vertex() >point() h >next() >next() >vertex() >point()); } }; 14

15 CGAL::Polyhedron_3<PolyhedronTraits_3>... typedef Kernel::Plane_3 Plane_3; typedef CGAL::Polyhedron_3<Kernel> Polyhedron; int main() { Point_3 p( 1 0 0); Point_3 q( 0 1 0); Point_3 r( 0 0 1); Point_3 s( 0 0 0); Polyhedron P; P.make_tetrahedron( p q r s); std::transform( P.facets_begin() P.facets_end() P.planes_begin() Plane_equation()); CGAL::set_pretty_mode( std::cout); std::copy( P.planes_begin() P.planes_end() std::ostream_iterator<plane_3>( std::cout "\n")); return 0; } 15

16 CGAL::Polyhedron_3<PolyhedronTraits_3> It supports I/O from/to.off file format: template <class PolyhedronTraits_3> ostream& ostream& out << CGAL::Polyhedron_3<PolyhedronTraits_3> P template <class PolyhedronTraits_3> istream& istream& in >> CGAL::Polyhedron_3<PolyhedronTraits_3>& P.OFF format 16

17 Exercise 3 Implement the O(n2 )-time algorithm for computing the convex hull of a point set in R3. Use CGAL::Polyhedron_3<PolyhedronTraits_3>. Use predicates from the Kernel to check visibility. Construct the horizon by performing a depth first search. Possible functions for updating the current convex hull: erase_facet (...) erase_connected_component (...) add_vertex_and_facet_to_border (...) add_facet_to_border (...) Visualize the steps of the algorithm (partial results) with a.off viewer (e.g. Polyhedron_3 CGAL Demo). 17