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SMD156Lecture 16

Convex Hulls inHigher Dimensions

Projects andSurveys

3D

Voronoidiagrams Incremetal

Algorithm

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Convex Hulls in 3D

• Collision detection:

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Definitions

• The smallest volumepolyhedra that contains a setof points in 3D.# Facets

# Edges

# Vertices

• !Generalizes to higherdimensions."

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Size of a Convex Hull in 3D

• !Theorem 11.1 + 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

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Computing a Convex Hull in 3D

• 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 pointconsidered that lie outside the current hull.# !and delete old facets that end up inside."

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Horizon

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Adding One Point

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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 visible to another point if the line

segment that is bounded by the points doesn%tintersect the interior of the convex hull.

• A face/edge is visible if all its points arevisible.

• Remove visible faces.# Easy since, by construction, each facet is oriented

counterclock-wise when seen from the outside ofthe convex hull.

• Insert new triangular facets for each edge inthe horizon unless…

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• … the new face would be coplanar with anexisting face $behind$ the horizon, in whichcase the two are merged.

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• Each time a linear number !in the size of thecurrent convex hull" of new facets replace asmany old facets.

• Total time: O!n2"

Worst-Case Analysis

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• 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 toeach other.

• Maintain the conflict graph during thecomputation.

How to Find the Visible Facets

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Conflict Graph

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Using the Conflict Graph

• The facets visible to a point are its neighborsin the graph.

• Likewise: The points visible from a facet areits neighbors in the graph.

• The conflict sets of points and facets can beextracted in time linear in their sizes.

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Adding a Point

• Using the conflict graph, we get the horizon and allvisible facets.

• Remove all visible facets, but keep those next to thehorizon in a separate data structure for a while.# Update the conflict graph accordingly

• When adding a new point # and new facets # to theconvex hull , we add new nodes to the conflict graphas well.

• The crucial point is how to construct their conflictlists.

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Adding a Point (cont.)

• Consider adding a new face f.1. f is coplanar with f1 !$behind$ the horizon"

• ConflictList!f" = ConflictList!f1".

2. f is not coplanar with f1

• ConflictList!f" = Visible points of ConflictList!f1" andConflictList!f2".

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f

New Faces lie Next to the Horizon

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Convex Hulls in 3D and Delaunay

Triangulations in 2D

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Conclusion

• The Delaunay triangulation/Voronoi diagramof a set of n points !xi, yi" in the plan can becomputed by computing the convex hull in3D of the points !xi, yi, xi

2+yi2".

# Extract the lower hull.

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Conclusion

• !Lemma 11.4"# Given a set of n points in 3D, CONVEXHULL

computes a convex hull in O!n log n" expectedtime.

• 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.

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