Voronoi Diagram (Supplemental)

The Universal Spatial Data Structure (Franz Aurenhammer)

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Outline

Voronoi and DelaunayFacility location problemNearest neighborFortune’s algorithm revisitedGeneralized Voronoi diagrams

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Voronoi Diagram

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Dual: Delaunay Triangulation

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Facility Location Problems

Determine a location to minimize the distance to its furthest customerMinimum enclosing circle

Determine a location whose distance to nearest store is as large as possibleLargest empty circle

ip

jp

kp

q

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Facility Location (version 2)

Seek location for new grocery store, whose distance to nearest store is as large as possible — center of largest empty circleOne restriction: center in convex hull of the sites

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Facility Location (cont)

Center in hull: p must be coincident with a voronoi vertex

Center on hull: p must lie on a voronoi edge

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Largest Empty Circle

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Nearest Neighbor Search

A special case of point-location problem where every face in the subdivision is monotoneUse chain method to get O(log n) time complexity for query

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Cluster Analysis

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Closest Pairs

In collision detection, two closest sites are in greatest danger of collisionNaïve approach: (n2)

Each site and its closest pair share an edge check all Voronoi edges O(n)Furthest pair cannot be derived directly from the diagram

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Motion Planning (translational)

Collision avoidance:

stay away from obstacle

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Fortune’s Algorithm Revisited

ConesIdeaH/W implementation

The curve of intersection of two cones projects to a line.

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45 deg Cone

distance=heightsite

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Cone (cont)

intersection of cone equal-distance point

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Cone (cont)When viewed from –Z, we got colored V-cells

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Nearest Distance Function

Viewed from here[less than]

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Furthest Distance Function

Viewed from here[greater than]

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Fortune’s Algorithm (Cont)

Cone slicing

Cone cut up by sweep plane and L are sweeping toward the right.

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Fortune’s Algorithm (Cont)

Viewed from z = -, The heavy curve is the parabolic front.

How the 2D algorithm and

the 3D cones are related…

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Generalized Voronoi Diagram

V(points), Euclidean distance V(points, lines, curves, …)Distance function: Euclidean, weighted, farthest

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Brute Force Method

Record ID of the closest site to each sample

point

Coarsepoint-sampling

result

Finerpoint-sampling

result

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Graphics Hardware Acceleration

Our 2-part discrete Voronoidiagram representation

Distance

Depth Buffer

Site IDs

Color Buffer

Simply rasterize the cones using gra

phics hardware

Haeberli90, Woo97

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Algorithm

Associate each primitive with the corresponding distance meshRender each distance mesh with depth test onVoronoi edges: found by continuation methods

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Ex: Voronoi diagram between a point and a line

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Distance Meshes

linecurve

polygon

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Applications (Mosaic)

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Hausner01, siggraph

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Medial Axis Computation

Medial axes as part of Voronoi diagram

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Piano Mover: Real-time Motion Planning (static and dynamic)

Plan motion of piano through 100K triangle model

Distance buffer of floorplan used as potential field

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Variety of Voronoi Diagram

(regular) Voronoi diagram

Furthest distance Voronoi diagram

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Minimum Enclosing Circle

Center of MEC is at the vertex of furthest site Voronoi diagram