Convex Hulls in 3-spaceProblem StatementComplexitySlide 4AlgorithmInitializationInserting Points into CHCase 2: Pr outside CH(Pr-1)VisibilityRethinking the HorizonCH(Pr-1)  CH(Pr)Algorithm So Far…How to Find Visible RegionConflict ListsConflict Graph GInitializing GUpdating GDetermining New ConflictsSlide 19What About the Other Facets?Final AlgorithmFine PointAnalysisExpected Number of Facets CreatedExpected Number of New FacetsExpected Degree of prExpected Number of Created FacetsRunning TimeTotal Time to Find New ConflictsRandomized Insertion OrderSlide 31Convex Hulls in Dual SpaceHalf-Plane IntersectionSlide 34Voronoi Diagrams RevisitedVoronoi DiagramsSimplified CaseDemoDelaunay Triangulations from CHHigher Dimensional Convex HullsSlide 41October 7, 2003 Lecture 10: Convex Hulls in 3D 1 / 41Convex Hullsin 3-space(slides mostly by Jason C. Yang)October 7, 2003 Lecture 10: Convex Hulls in 3D 2 / 41Problem Statement•Given P: set of n points in 3D•Return:–Convex hull of P: CH(P), i.e. smallest polyhedron s.t. all elements of P on or inthe interior of CH(P).October 7, 2003 Lecture 10: Convex Hulls in 3D 3 / 41•Complexity of CH for n points in 3D is O(n)•..because the number of edges of a convex polytope with n vertices is at most 3n-6 and the number of facets is at most 2n-4•..because the graph defined by vertices and edges of a convex polytope is planar•Euler’s formula: n – ne + nf = 2ComplexityOctober 7, 2003 Lecture 10: Convex Hulls in 3D 4 / 41Complexity•Each face has at least 3 arcs•Each arc incident to two faces2ne  3nf•Using Eulernf  2n-4 ne  3n-6October 7, 2003 Lecture 10: Convex Hulls in 3D 5 / 41Algorithm•Randomized incremental algorithm•Steps:–Initialize the algorithm–Loop over remaining pointsAdd pr to the convex hull of Pr-1 to transform CH(Pr-1) to CH(Pr)[for integer r1, let Pr:={p1,…,pr} ]October 7, 2003 Lecture 10: Convex Hulls in 3D 6 / 41Initialization•Need a CH to start with•Build a tetrahedron using 4 points in P–Start with two distinct points in P, say, p1 and p2–Walk through P to find p3 that does not lie on the line through p1 and p2–Find p4 that does not lie on the plane through p1, p2, p3–Special case: No such points exist? Planar case!•Compute random permutation p5,…,pn of the remaining pointsOctober 7, 2003 Lecture 10: Convex Hulls in 3D 7 / 41Inserting Points into CH•Add pr to the convex hull of Pr-1 to transform CH(Pr-1) to CH(Pr)•Two Cases:1) Pr is inside or on the boundary of CH(Pr-1)–Simple: CH(Pr) = CH(Pr-1)2) Pr is outside of CH(Pr-1) – the hard caseOctober 7, 2003 Lecture 10: Convex Hulls in 3D 8 / 41Case 2: Pr outside CH(Pr-1)•Determine horizo n of pr on CH(Pr-1)–Closed curve of edges enclosing the visible region of pr on CH(Pr-1)October 7, 2003 Lecture 10: Convex Hulls in 3D 9 / 41Visibility•Consider plane hf containing a facet f of CH(Pr-1)•f is visible from a point p if that point lies in the open half-space on the other side of hfOctober 7, 2003 Lecture 10: Convex Hulls in 3D 10 / 41Rethinking the Horizon–Boundary of polygon obtained from projecting CH(Pr-1) onto a plane with pr as the center of projectionOctober 7, 2003 Lecture 10: Convex Hulls in 3D 11 / 41CH(Pr-1)  CH(Pr)•Remove visible facets from CH(Pr-1)•Found horizon: Closed curve of edges of CH(Pr-1)•Form CH(Pr) by connecting each horizon edge to pr to create a new triangular facetOctober 7, 2003 Lecture 10: Convex Hulls in 3D 12 / 41Algorithm So Far…•Initialization–Form tetrahedron CH(P4) from 4 points in P–Compute random permutation of remaining pts.•For each remaining point in P–pr is point to be inserted–If pr is outside CH(Pr-1) then•Determine visible region•Find horizon and remove visible facets•Add new facets by connecting each horizon edge to prHow do we determine the visible region?October 7, 2003 Lecture 10: Convex Hulls in 3D 13 / 41How to Find Visible Region•Naïve approach:–Test every facet with respect to pr–O(n2) work•Trick is to work ahead:Maintain information to aid in determining visible facets.October 7, 2003 Lecture 10: Convex Hulls in 3D 14 / 41Conflict Lists•For each facet f maintainPconflict(f) {pr+1, …, pn}containing points to be inserted that can see f•For each pt, where t > r, maintain Fconflict(pt) containing facets of CH(Pr) visible from pt•p and f are in conflict because they cannot coexist on the same convex hullOctober 7, 2003 Lecture 10: Convex Hulls in 3D 15 / 41Conflict Graph G•Bipartite graph –pts not yet inserted–facets on CH(Pr)•Arc for every point-facet conflict•Conflict sets for a point or facet can be returned in linear timeAt any step of our algorithm, we know all conflicts between the remaining points and facets on the current CHOctober 7, 2003 Lecture 10: Convex Hulls in 3D 16 / 41Initializing G •Initialize G with CH(P4) in linear time•Walk through P5-n to determine which facet each point can seef1f2p6p5p7p6p7p5f2f1GOctober 7, 2003 Lecture 10: Convex Hulls in 3D 17 / 41Updating G•Discard visible facets from pr by removing neighbors of pr in G•Remove pr from G•Determine new conflictsp6p7p5f2f1Gf1f2f3f3p6p7p5f2f1Gp6p7p5f2f1Gp5p7p6October 7, 2003 Lecture 10: Convex Hulls in 3D 18 / 41Determining New Conflicts•If pt can see new f, it can see edge e of f.•e on horizon of pr, so e was already in and visible from pt in CH(Pr-1)•If pt sees e, it saw either f1 or f2 in CH(Pr-1)•Pt was in Pconflict(f1) or Pconflict(f2) in CH(Pr-1)ptOctober 7, 2003 Lecture 10: Convex Hulls in 3D 19 / 41Determining New Conflicts•Conflict list of f can be found by testing the points in the conflict lists of f1 and f2 incident to the horizon edge e in CH(Pr-1)ptOctober 7, 2003 Lecture 10: Convex Hulls in 3D 20 / 41What About the Other Facets?•Pconflict(f) for any f unaffected by pr remains unchangedpt•Pconflict(f) for any f unaffected by pr remains unchanged•Deleted facets not on horizon already accounted forOctober 7, 2003 Lecture 10: Convex Hulls in 3D 21 / 41Final Algorithm•Initialize CH(P4) and G•For each remaining point–Determine visible facets for pr by checking G–Remove Fconflict(pr) from CH–Find horizon and add new facets to CH and G–Update G for new facets by testing the points in existing conflict lists for facets in CH(Pr-1) incident to e on the new facets–Delete pr and Fconflict(pr) from GOctober 7, 2003 Lecture 10: Convex Hulls in 3D 22 / 41Fine Point•Coplanar