Other Voronoi/Delaunay Structures

Overview

• Alpha hulls (a subset of Delaunay graph)• Extension of Voronoi Diagrams

Convex Hull

• What is it good for?– The bounding region of a point set

• Not so good for describing shapes

Convex Hull

• Subtractive definition– Taking away all empty half

planes– Edge 𝑝𝑝𝑖𝑖 ,𝑝𝑝𝑗𝑗 lies on the hull

if it lies on the boundary of an empty half plane

Alpha Hull

• Subtractive definition– Taking away all empty half

planes circles of radius α– Edge 𝑝𝑝𝑖𝑖 ,𝑝𝑝𝑗𝑗 lies on the hull

if it lies on the boundary of an empty half planecircle.

Alpha Hull

α=0

α=∞

(α controls the level of details)

Alpha Hull

• Alpha hull is a subset of the Delaunay graph– Each hull edge has an empty circle– Let 𝛼𝛼𝑚𝑚𝑖𝑖𝑚𝑚(𝑝𝑝𝑖𝑖 ,𝑝𝑝𝑗𝑗), 𝛼𝛼𝑚𝑚𝑎𝑎𝑎𝑎(𝑝𝑝𝑖𝑖 ,𝑝𝑝𝑗𝑗) be the minimum

and maximum radius of all empty circles of edge 𝑝𝑝𝑖𝑖 ,𝑝𝑝𝑗𝑗. The edge is on the hull if

𝛼𝛼𝑚𝑚𝑖𝑖𝑚𝑚 𝑝𝑝𝑖𝑖 ,𝑝𝑝𝑗𝑗 < α < 𝛼𝛼𝑚𝑚𝑎𝑎𝑎𝑎(𝑝𝑝𝑖𝑖 ,𝑝𝑝𝑗𝑗)

Alpha Hull

𝛼𝛼𝑚𝑚𝑖𝑖𝑚𝑚

𝛼𝛼𝑚𝑚𝑎𝑎𝑎𝑎𝑝𝑝𝑖𝑖

𝑝𝑝𝑗𝑗

𝑝𝑝𝑖𝑖

𝑝𝑝𝑗𝑗𝛼𝛼𝑚𝑚𝑖𝑖𝑚𝑚

𝛼𝛼𝑚𝑚𝑎𝑎𝑎𝑎

Computing Alpha Hull

• Compute the Voronoi Diagram of point set• For each Voronoi edge

– Compute 𝛼𝛼𝑚𝑚𝑖𝑖𝑚𝑚,𝛼𝛼𝑚𝑚𝑎𝑎𝑎𝑎– If α is in range, output the dual Delaunay edge.

• O(n log n)– Subsequent computation of alpha hulls with

different α takes only O(n) (or faster…)

Alpha Hull

• Interior of alpha hull is a subset of the Delaunay triangulation– An element (point, edge, face) of Delaunay

triangulation is on or inside α-Hull if the radius of its smallest empty circle is smaller than α

Alpha Hull in 3D

α=0

α=∞

Voronoi Diagram

• A finite set of point sites 𝑝𝑝𝑖𝑖

• Euclidean distance 𝑑𝑑:𝑑𝑑 𝑥𝑥,𝑝𝑝𝑖𝑖 = 𝑥𝑥 − 𝑝𝑝𝑖𝑖

• Voronoi diagram is the set of 𝑥𝑥 with multiple nearest sites

Voronoi Diagram

• A finite set of point sites 𝑝𝑝𝑖𝑖

• Euclidean distance 𝑑𝑑:𝑑𝑑 𝑥𝑥,𝑝𝑝𝑖𝑖 = 𝑥𝑥 − 𝑝𝑝𝑖𝑖

• Voronoi diagram is the set of 𝑥𝑥 with multiple nearest sites

Weighted Voronoi Diagram

• A finite set of point sites 𝑝𝑝𝑖𝑖 with weights 𝑟𝑟𝑖𝑖

• Additively weighted distance 𝑑𝑑:𝑑𝑑 𝑥𝑥, 𝑝𝑝𝑖𝑖 = 𝑥𝑥 − 𝑝𝑝𝑖𝑖 − 𝑟𝑟𝑖𝑖

• Voronoi diagram is the set of 𝑥𝑥 with multiple nearest sites

Weighted Voronoi Diagram

• 𝑑𝑑 𝑥𝑥, 𝑝𝑝𝑖𝑖 measures signed distance from x to a circle centered at 𝑝𝑝𝑖𝑖 with radius 𝑟𝑟𝑖𝑖

𝑝𝑝𝑖𝑖

𝑥𝑥𝑑𝑑𝑟𝑟𝑖𝑖

Weighted Voronoi Diagram

• The “bisector” of two sites in the weighted distance metric

A hyperbola(if one circle is not completely within another)

Does not exist

Cell of p1

Cell of p2

Cell of p1

Cell of p2 Cell of p2

Weighted Voronoi Diagram

• A weighted Voronoi cell – May be empty– May be non-convex – Always contains the site

Power Diagram

• A finite set of point sites 𝑝𝑝𝑖𝑖 with weights 𝑟𝑟𝑖𝑖

• Power distance 𝑑𝑑:𝑑𝑑 𝑥𝑥, 𝑝𝑝𝑖𝑖 = 𝑥𝑥 − 𝑝𝑝𝑖𝑖 2 − 𝑟𝑟𝑖𝑖2

• Voronoi diagram is the set of 𝑥𝑥 with multiple nearest sites

Power Diagram

• 𝑑𝑑 𝑥𝑥, 𝑝𝑝𝑖𝑖 measures:– 𝑥𝑥 outside circle 𝑝𝑝𝑖𝑖 , 𝑟𝑟𝑖𝑖 : squared length of tangent segment

from 𝑥𝑥 to the circle– 𝑥𝑥 inside circle 𝑝𝑝𝑖𝑖 , 𝑟𝑟𝑖𝑖 : negative squared length of half-

chord perpendicular to diameter at 𝑥𝑥

𝑝𝑝𝑖𝑖

𝑥𝑥𝑑𝑑

𝑝𝑝𝑖𝑖

𝑥𝑥−𝑑𝑑

Power Diagram

• The “bisector” of two sites in the power metric is always a straight line– Not always “between” the sites

Power Diagram

• A power cell – May be empty– Always convex– May not contain the site

Applet!

http://pages.cpsc.ucalgary.ca/%7Elaneb/Power/

Voronoi Diagram

• A finite set of point sites 𝑝𝑝𝑖𝑖

• Euclidean distance 𝑑𝑑:𝑑𝑑 𝑥𝑥,𝑝𝑝𝑖𝑖 = 𝑥𝑥 − 𝑝𝑝𝑖𝑖

• Voronoi diagram is the set of 𝑥𝑥 with multiple nearest sites

Voronoi Diagram of Segments

• A finite set of line segments 𝑙𝑙𝑖𝑖

• Euclidean distance 𝑑𝑑:𝑑𝑑 𝑥𝑥, 𝑙𝑙𝑖𝑖 = min𝑝𝑝∈𝑙𝑙𝑖𝑖

𝑥𝑥 − 𝑝𝑝

• Voronoi diagram is the set of 𝑥𝑥 with multiple nearest segments

Voronoi Diagram of Segments

• The “bisector” of two (disjoint) segments is made up of straight and parabolic pieces

Voronoi Diagram of Segments

• When the segments from a closed polygon, the diagram is known as medial axis

Medial Axis

• Captures shape and topology of objects

2D objects 3D objects

Voronoi Diagram

• A finite set of point sites 𝑝𝑝𝑖𝑖

• Euclidean distance 𝑑𝑑:𝑑𝑑 𝑥𝑥,𝑝𝑝𝑖𝑖 = 𝑥𝑥 − 𝑝𝑝𝑖𝑖

• Voronoi diagram is the set of 𝑥𝑥 with multiple nearest sites

Furthest-point Voronoi Diagram

• A finite set of point sites 𝑝𝑝𝑖𝑖

• Euclidean distance 𝑑𝑑:𝑑𝑑 𝑥𝑥,𝑝𝑝𝑖𝑖 = 𝑥𝑥 − 𝑝𝑝𝑖𝑖

• Voronoi diagram is the set of 𝑥𝑥 with multiple furthest sites– 𝑉𝑉𝑉𝑉𝑟𝑟 𝑝𝑝𝑖𝑖 = 𝑥𝑥 𝑑𝑑 𝑥𝑥, 𝑝𝑝𝑖𝑖 > 𝑑𝑑 𝑥𝑥, 𝑝𝑝𝑗𝑗 ,∀𝑖𝑖 ≠ 𝑗𝑗}

Furthest-point Voronoi Diagram

• A cell is also an intersection of half-planes defined by bisector lines– It uses the half-planes that do not contain the site

𝑝𝑝1 𝑝𝑝2

Cell of p1Cell of p2

Furthest-point Voronoi Diagram

• A cell – May be empty (if the site

is not on the convex hull)– Always convex– Never contains the site

Furthest-point Voronoi Diagram

• Can be used to find the smallest circle containing the set– The center of this circle

is on the furthest-point Voronoi diagram

Applet!

http://cgm.cs.mcgill.ca/%7Egodfried/teaching/projects97/belair/alpha.html

Other Voronoi/Delaunay StructuresOverviewConvex HullConvex HullAlpha HullAlpha HullAlpha HullAlpha HullComputing Alpha HullAlpha HullAlpha Hull in 3DVoronoi DiagramVoronoi DiagramWeighted Voronoi DiagramWeighted Voronoi DiagramWeighted Voronoi DiagramWeighted Voronoi DiagramPower DiagramPower DiagramPower DiagramPower DiagramVoronoi DiagramVoronoi Diagram of SegmentsVoronoi Diagram of SegmentsVoronoi Diagram of SegmentsMedial AxisVoronoi DiagramFurthest-point Voronoi DiagramFurthest-point Voronoi DiagramFurthest-point Voronoi DiagramFurthest-point Voronoi Diagram