Convex Hulls in 3D. Definitions. Size of a Convex Hull in 3D. Collision detection: The smallest volume polyhedral that contains a set of points in 3D.

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1 Convex Hulls in 3D SMD156 Lecture 11 Convex Hulls in Higher Dimensions Voronoi diagrams 3D Incremental Algorithm Collision detection: 1 2 Definitions The smallest volume polyhedral that contains a set of points in 3D.! Facets! Edges! Vertices "Generalizes to higher dimensions.# Size of a Convex Hull in 3D "Theorem Corollary 11.2#! A convex hull in 3D with n vertices has at most 3n$6 edges and 2n$4 facets. 1. It is %planar% 2. Euler&s formula 3 4

2 Computing a Convex Hull in 3D Horizon Incremental Algorithm! Add points one at a time and maintain a current convex hull.! Use DCEL:s to represent the convex hull. Start with four "4# points that form a pyramid. Then add new facets for each point considered that lie outside the current hull.! "and delete old facets that end up inside.# 5 6 Adding One Point 7 Deleting/Adding Facets The horizon can be found by traversing the "surface of the# convex hull.! Each edge of the horizon bounds two faces of which just one is visible. A point is!isibl" to another point if the line segment that is bounded by the points doesn&t intersect the interior of the convex hull. A face/edge is visible if all its points are visible. Remove visible faces.! Easy since, by construction, each facet is oriented counterclockwis" when seen from the outside of the convex hull. Insert new triangular facets for each edge in the horizon unless 8

3 Worst-Case Analysis the new face would be coplanar with an existing face %behind% the horizon, in which case the two are merged. Each time a linear number "in the size of the current convex hull# of new facets replace as many old facets. Total time: O"n 2 # 9 10 How to Find the Visible Facets Conflict Graph As on previous slide: Brute$force. Clever trick: Use a conflict graph.! Bipartite.! Nodes for each point not yet considered.! Nodes for each facet in the current convex hull.! Arcs between points and facets that are visible to each other. Maintain the con'ict graph during the computation

4 Using the Conflict Graph The facets visible to a point are its neighbors in the graph. Likewise: The points visible from a facet are its neighbors in the graph. The con'ict sets of points and facets can be extracted in time linear in their sizes. Adding a Point Using the con'ict graph, we get the horizon and all visible facets. Remove all visible facets, but keep those next to the horizon in a separate data structure for a while.! Update the con'ict graph accordingly When adding a new point! and new facets! to the convex hull, we add new nodes to the con'ict graph as well. The crucial point is how to construct their con'ict lists Adding a Point (cont.) New Faces lie Next to the Horizon Consider adding a new face f. 1. f is coplanar with f 1 "%behind% the horizon# Con'ictList"f# = Con'ictList"f 1 #. 2. f is not coplanar with f 1 Con'ictList"f# = Visible points of Con'ictList"f 1 # and Con'ictList"f 2 #. f 15 16

5 Convex Hulls in 3D and Delaunay Triangulations in 2D

6 Conclusion The Delaunay triangulation/voronoi diagram of a set of n points "x i, y i # in the plan can be computed by computing the convex hull in 3D of the points "x i, y i, x i2 +y i2 #.! Extract the lower hull. Conclusion "Lemma 11.4#! Given a set of n points in 3D, CONVEXHULL computes a convex hull in O"n log n# expected time. Comment:! There is also a deterministic algorithm by Preparata and Hong that runs in O"n log n# time, but that one is more complicated