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CS133 Computational Geometry Voronoi Diagram Delaunay Triangulation 1 Nearest Neighbor Problem Given a set of points and a query point , find the closest point to , , ,

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CS133

Computational Geometry

Voronoi Diagram Delaunay Triangulation

1

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Nearest Neighbor Problem

Given a set of points 𝑄 and a query point π‘Ÿ, find the closest point π‘ž ∈ 𝑄 to π‘Ÿ βˆ€π‘ž, 𝑠 ∈ 𝑄, 𝑒𝑗𝑑𝑒 π‘ž, π‘Ÿ ≀ 𝑒𝑗𝑑𝑒(𝑠, π‘Ÿ) Simple algorithm: Scan and find the minimum An efficient algorithm: Use a spatial index structure such as K-d tree What if we need to repeat this for every point in the space, i.e., an infinite number of points?

2

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Application: Cell Coverage

3

Voronoi Diagram

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Other Applications

Service coverage for hospitals, post offices, schools, … etc. Marketing: Find candidate locations for a new restaurant Routing: How an electric vehicle should travel while staying close to charging stations

4

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Voronoi Region

Given a set 𝑄 of points (also called sites), a Voronoi region (Voronoi face) of a site π‘žπ‘— ∈ 𝑄, π‘Š π‘žπ‘— is the set of points in the Euclidean space where π‘žπ‘— is (one of) the closest sites π‘Š π‘žπ‘— = 𝑦: π‘žπ‘— βˆ’ 𝑦 ≀ π‘žπ‘˜ βˆ’ 𝑦 βˆ€π‘žπ‘˜ ∈ 𝑄

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Voronoi Diagram

The Voronoi diagram is the set of points that belong to two or more Voronoi regions Voronoi diagram is a tessellation of the space into regions where each region contains all the points that are closest to one site

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VD of Two Points

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π‘ž1 π‘ž2 π‘Š π‘ž1 π‘Š π‘ž2

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VD of Three Points

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π‘ž1 π‘ž2 π‘Š π‘ž1 π‘Š π‘ž2 π‘ž3 π‘Š π‘ž3

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VD of Three Points

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π‘ž1 π‘ž2 π‘Š π‘ž1 π‘Š π‘ž2 π‘ž3 π‘Š π‘ž3

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Voronoi Region

A Voronoi region of a set π‘žπ‘— is the intersection

  • f all half spaces defined by the

perpendicular bisectors π‘Š π‘žπ‘— =Ϊπ‘˜β‰ π‘— 𝐼 π‘žπ‘—, π‘žπ‘˜

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VD of a Set of Points

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𝑄 π‘ŠπΈ 𝑄

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Mother Nature Loves VD

http://forum.woodenboat.com/showthread.php?112363-Voronoi-Diagrams-in-Nature

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Mother Nature Loves VD

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Mother Nature Loves VD

http://forum.woodenboat.com/showthread.php?112363-Voronoi-Diagrams-in-Nature

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Mother Nature Loves VD

Onion cells under the microscope

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Mother Nature Loves VD

http://forum.woodenboat.com/showthread.php?112363-Voronoi-Diagrams-in-Nature

A thin slice of carrot under the scope

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Mother Nature Loves VD

http://forum.woodenboat.com/showthread.php?112363-Voronoi-Diagrams-in-Nature

A dead maple leaf at 160X

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Mother Nature Loves VD

http://forum.woodenboat.com/showthread.php?112363-Voronoi-Diagrams-in-Nature

An oak leaf

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VD Properties

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Voronoi regions are convex Each Voronoi region contains a single site Voronoi regions (faces) can be unbounded Most intersection points connect three segments

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VD Properties

π‘Š π‘žπ‘— is unbounded iff π‘žπ‘— ∈ π’Ÿβ„‹ 𝑄 If a point 𝑦 is at the intersection of three or more Voronoi regions, say π‘Š π‘ž1 , π‘Š π‘ž2 , … , π‘Š π‘žπ‘™ , then 𝑦 is the center of a circle 𝐷 that have π‘ž1, … , π‘žπ‘™ at its boundary 𝐷 contains no other sites VD is unique

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Delaunay Triangulation (DT)

Delaunay triangulation is the straight-line dual

  • f the Voronoi diagram

Each site is a corner of at least one triangle Each two Voronoi regions that share an edge are connected with an edge in DT

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DT Properties

The edges of 𝐸 𝑄 do not intersect Is 𝐸 𝑄 unique?

Yes, if no four sites are co-circular

If π‘žπ‘— and π‘žπ‘˜ are the closest pair

  • f sites, they are connected with an edge in DT

If π‘žπ‘— and π‘žπ‘˜ are nearest neighbors, they are connected with an edge in DT The circumcircle of π‘žπ‘—, π‘žπ‘˜, and π‘žπ‘™ is empty ⟺ π‘žπ‘—, π‘žπ‘˜, π‘žπ‘™ is a triangle in DT

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DT is a Planar Graph

Since the edges in DT do not intersect, they form a planar graph

The number of edges/faces in a Delaunay Triangulation is linear in the number of vertices. The number of edges/vertices in a Voronoi Diagram is linear in the number of faces. The number of vertices/edges/faces in a Voronoi Diagram is linear in the number of sites.

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Theorem 7.3

For π‘œ β‰₯ 3, the number of vertices in the Voronoi diagram (π‘œπ‘€) of a set of π‘œ point sites in the plane is at most 2π‘œ βˆ’ 5, and the number of edges π‘œπ‘“ is at most 3π‘œ βˆ’ 6

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Proof

For any connected graph 𝐻 Euler’s rule: 𝑛𝑀 βˆ’ 𝑛𝑓 + 𝑛𝑔 = 2

𝑛𝑀: Number of vertices (nodes) 𝑛𝑓: Number of edges (arcs) 𝑛𝑔: Number of faces

π‘œπ‘€ + 1 βˆ’ π‘œπ‘“ + π‘œ = 2 Each edge connects two vertices The sum of degrees of vertices βˆ‘π‘’ 𝑀𝑗 = 2π‘œπ‘“ 𝑒 𝑀𝑗 β‰₯ 3

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Proof (cont’d)

3π‘œπ‘€ ≀ βˆ‘π‘’ 𝑀𝑗 3 π‘œπ‘€ + 1 ≀ 2π‘œπ‘“ π‘œπ‘€ + 1 ≀ 2

3 π‘œπ‘“

But: π‘œπ‘€ + 1 βˆ’ π‘œπ‘“ + π‘œ = 2 π‘œπ‘€ + 1 = 2 βˆ’ π‘œ + π‘œπ‘“ ≀ 2

3 π‘œπ‘“ 1 3 π‘œπ‘“ ≀ π‘œ βˆ’ 2

π‘œπ‘“ ≀ 3π‘œ βˆ’ 6 π‘œπ‘€ ≀ 2π‘œ βˆ’ 5

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DT Properties

The boundary of 𝐸 𝑄 is the convex hull of 𝑄

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π‘žπ‘— π‘žπ‘˜ π‘Š π‘žπ‘— π‘Š π‘žπ‘˜

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DT Properties

The boundary of 𝐸 𝑄 is the convex hull of 𝑄

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π‘žπ‘— π‘žπ‘˜ π‘Š π‘žπ‘— π‘Š π‘žπ‘˜

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DT Properties

The boundary of 𝐸 𝑄 is the convex hull of 𝑄

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π‘žπ‘— π‘žπ‘˜ π‘Š π‘žπ‘— π‘Š π‘žπ‘˜

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DT Properties

The boundary of 𝐸 𝑄 is the convex hull of 𝑄

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π‘žπ‘— π‘žπ‘˜ π‘Š π‘žπ‘— π‘Š π‘žπ‘˜

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DT Properties

The boundary of 𝐸 𝑄 is the convex hull of 𝑄

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π‘žπ‘— π‘žπ‘˜ π‘Š π‘žπ‘— π‘Š π‘žπ‘˜

∞

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DT Properties

If π‘žπ‘˜ is the nearest neighbor of π‘žπ‘— then π‘žπ‘—π‘žπ‘˜ is a Delaunay edge π‘žπ‘˜ is the nearest neighbor of π‘žπ‘— iff. the circle around π‘žπ‘— with radius π‘žπ‘— βˆ’ π‘žπ‘˜ is empty of

  • ther points.

β‡’The circle through π‘žπ‘— + π‘žπ‘˜ /2 with radius |π‘žπ‘— βˆ’ π‘žπ‘˜|/2 is empty of other points. β‡’ π‘žπ‘— + π‘žπ‘˜ /2 is on the Voronoi diagram. β‡’ π‘žπ‘— + π‘žπ‘˜ /2 is on a Voronoi edge.

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VD Plane Sweep

Scan the plane from top to bottom Compute the VD of the points above the sweep line Is it that simple?

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VD of a Line and a Point

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𝑧 = 1 2 𝑦 βˆ’ π‘žπ‘—π‘¦ 2 π‘žπ‘—π‘§ βˆ’ ℓ𝑧 + ℓ𝑧 + π‘žπ‘—π‘§

π‘žπ‘—π‘¦, π‘žπ‘—π‘§ ℓ𝑧

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VD of a Line and a n Points

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VD of a Line and a n Points

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Fortune’s Algorithm

As the line sweeps the plane, the algorithm maintains the VD of the set of points and the sweep line Since the sweep line is closer than any future point, it acts as a barrier that isolates the VD from all future points

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Fortune’s Algorithm in Action

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VD Properties

The VD part above the beach line (blue) is

  • final. Why?

This area is closer to some site than the beach line … closer to some site than any future site We already know the nearest site to those areas

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VD Properties

The beach line is 𝑦-monotone. Why?

Each parabola is 𝑦-monotone At each 𝑦-coordinate, the beach line takes one value which is the minimum of all the parabolas Therefore, it is 𝑦-monotone

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Figure Credits: http://www.cs.sfu.ca/~binay/813.2011/Fortune.pdf

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VD Properties

The breakpoints of the beach line lie on Voronoi edges of the final diagram

Each breakpoint is equidistant from two sites A breakpoint is as close to some site as to the sweep line The sweep line is (closer) to the blue sites than future sites

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Fortune’s Algorithm

Move the sweep line downwards and update the VD as the line moves When the line reaches βˆ’βˆž, we will have our final VD. (Because any point in the space is closer to some site than y = βˆ’βˆž) Note: We never create the beach line

  • explicitly. We only maintain enough

information that allows us to reconstruct parts

  • f it when we need them

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Beach Line Changes

How can the beach line change (topologically)

A new arc appears An existing arc is removed

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Site Event

When the sweep line hits a new site Where are the points that are equi-distant from the new site and the sweep line? A vertical line that crosses the new site

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Site Event

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Lemma: The only way in which a new arc can appear on the beach line is through a site event Proof by contradiction

Case 1: An existing arc π›Ύπ‘˜ breaks through the middle of an existing arc 𝛾𝑗 Case 2: An existing arc π›Ύπ‘˜ appears in between two arcs

Proof is in the book

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Circle (Vertex) Event

An existing arc shrinks into a point and disappears This happens when three (or more) sites become closer to a point than the sweep line shielding the point from the sweep line

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Circle (Vertex) Event

The sweep line will only go further down while the points stay This results in a vertex on the Voronoi Diagram Lemma: The only way in which an existing arc can disappear from the beach line is through a circle event

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Circle (Vertex) Event

A circle event happens between three adjacent arcs

  • f three different

sites A circle event is added at the lowest point of the circle and is associated with the point of the disappearing arc

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Circle event

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Plane Sweep Constructs

Sweep line status: The VD of the sites and the sweep line. In other words, the final part

  • f the VD + the beach line in non-decreasing

𝑦 order Event points:

Site event: A new site that adds a new arc to the

  • VD. 1-to-1 mapping to an input site

Circle event: The disappearance of an arc resulting in a vertex in VD. Can only be discovered along the way

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Sweep Line Status

The final part of VD is stored in the Doubly- Connected Edge List (DCEL) data structure The beach line is stored as a BST (𝜐) of arcs sorted by 𝑦

Leaves store arcs Internal nodes store the breakpoints as a pair of sites π‘žπ‘—, π‘žπ‘˜

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Sweep Line Status

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π‘ž1 π‘ž2 π‘ž3 𝛽1 𝛽2 𝛽3

π‘ž1, π‘ž2 π‘ž1, π‘ž2 π‘ž2, π‘ž3 π‘ž2, π‘ž3 π‘ž1 π‘ž2 π‘ž3

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Event Points

Stored in a priority queue 𝑅 as a max-heap

  • rdered by 𝑧

𝑅 is initialized with all sites

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Handle Site Event (𝒒𝒋)

If 𝜐 is empty, add the site to it and return Search in 𝜐 for the arc 𝛽 vertically above π‘žπ‘— If exists, delete a circle event linked with 𝛽 Split 𝛽 into two arcs and insert a new arc 𝛽𝑗 corresponding to π‘žπ‘— The new intersections are 𝛽, 𝛽𝑗 and 𝛽𝑗, 𝛽 Check the new triples of arcs and add their corresponding circle event to 𝑅

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Handle Site Event (𝒒𝒋)

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π‘ž1 π‘ž2 π‘ž3 𝛽1 𝛽2 𝛽3

π‘ž1, π‘ž2 π‘ž1, π‘ž2 π‘ž2, π‘ž3 π‘ž2, π‘ž3 π‘ž1 π‘ž2 π‘ž3

π‘ž4

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Handle Site Event (𝒒𝒋)

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π‘ž1 π‘ž2 π‘ž3 𝛽1 𝛽2 𝛽3

π‘ž1, π‘ž2 π‘ž1, π‘ž2 π‘ž2, π‘ž3 π‘ž2, π‘ž3 π‘ž1 π‘ž2 π‘ž3

π‘ž4

π‘ž4, π‘ž2 π‘ž2, π‘ž4

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Handle Site Event (𝒒𝒋)

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π‘ž1 π‘ž2 π‘ž3 𝛽1 𝛽2 𝛽3

π‘ž1, π‘ž2 π‘ž1, π‘ž2 π‘ž2, π‘ž3 π‘ž2, π‘ž3 π‘ž1 π‘ž3

π‘ž4

π‘ž4, π‘ž2 π‘ž2, π‘ž4 π‘ž2, π‘ž4 π‘ž4, π‘ž2 π‘ž4 π‘ž2 π‘ž2

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Handle Site Event (𝒒𝒋)

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π‘ž1 π‘ž2 π‘ž3 𝛽1 𝛽2 𝛽3

π‘ž1, π‘ž2 π‘ž1, π‘ž2 π‘ž2, π‘ž3 π‘ž2, π‘ž3 π‘ž1 π‘ž3

π‘ž4

π‘ž4, π‘ž2 π‘ž2, π‘ž4 π‘ž2, π‘ž4 π‘ž4, π‘ž2 π‘ž4 π‘ž2 π‘ž2 𝛽1𝛽2𝛽3 are no longer adjacent βž” Remove the circle event that corresponds to 𝛽2 𝛽1𝛽2𝛽4 are now adjacent βž” Create a new circle event for them Similarly, create a circle event for 𝛽4𝛽2𝛽3 No circle event for the triple 𝛽2𝛽4𝛽2 because they don’t correspond tot three different sites

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Creating a Circle Event

Given three sites π‘žπ‘—, π‘žπ‘˜, π‘žπ‘™ that have three adjacent arcs, we first compute the center of their circumcircle, i.e., the intersection of the two perpendicular bisectors to π‘žπ‘—π‘žπ‘˜ and π‘žπ‘˜π‘žπ‘™ Compute the bottom point of the circle as 𝑦𝑑, 𝑧𝑑 βˆ’ 𝑠 where

𝑦𝑑, 𝑧𝑑 are the coordinates of the circle center and 𝑠 is the circle radius

Associate the circle event with the middle site in the tree order

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Handle Circle Event (𝜹)

Delete the leaf 𝛿 that corresponds to the disappearing arc 𝛽𝑗 from 𝜐 Delete the two breakpoints that involve 𝛽𝑗 Insert a new break point Add the center of the circle event as a vertex in VD. This center is one side of two half- edges Check for any new circle events caused by the now adjacent triples of arcs Running time: 𝑃 π‘œ log π‘œ

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Circle Event (𝜹)

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π‘ž1 π‘ž2 π‘ž3 𝛽1 𝛽2 𝛽3

π‘ž1, π‘ž2 π‘ž2, π‘ž3 Circle event point π‘ž1, π‘ž2 π‘ž2, π‘ž3 π‘ž1 π‘ž2 π‘ž3

π‘ž4

𝜐

𝛽4

π‘ž3, π‘ž4 π‘ž3, π‘ž4 π‘ž4

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Circle Event (𝜹)

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π‘ž1 π‘ž2 π‘ž3 𝛽1 𝛽2 𝛽3

π‘ž1, π‘ž2 π‘ž2, π‘ž3 Circle event point π‘ž1, π‘ž2 π‘ž2, π‘ž3 π‘ž1 π‘ž2 π‘ž3

π‘ž4

𝜐

𝛽4

π‘ž3, π‘ž4 π‘ž3, π‘ž4 π‘ž4

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Circle Event (𝜹)

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π‘ž1 π‘ž2 π‘ž3 𝛽1 𝛽2 𝛽3

π‘ž1, π‘ž2 π‘ž2, π‘ž3 Circle event point π‘ž1, π‘ž3 π‘ž1 π‘ž3

π‘ž4

𝜐

𝛽4

π‘ž3, π‘ž4 π‘ž3, π‘ž4 π‘ž4 (π‘ž1, π‘ž3, π‘ž4) are now adjacent in the tree, create a corresponding circle event

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Delaunay Triangulation

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Delaunay Triangulation

A Delaunay triangulation can be defined as the (unique) triangulation in which the circumcircle of each triangle has no other sites NaΓ―ve algorithm:

Consider all possible triangles 𝑃 π‘œ3

Check if the circumcircle of the triangle is empty 𝑃 π‘œ

Running time 𝑃 π‘œ4

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Guibas and Stolfi’s Algorithm

A divide and conquer algorithm

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Algorithm Outline

DelaunayTriangulation(P)

If (|P| <= 3)

return TrivialDT(P)

Split P into P1 and P2 DT1 = DelaunayTriangulation(P1) DT2 = DelaunayTriangulation(P1) Merge(DT1, DT2)

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Split

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Pre-sort by x

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TrivialDT(P)

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P TrivialDT(P)

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Merge(P1, P2)

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Merge(P1, P2)

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Merge(P1, P2)

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Merge(P1, P2)

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Find the First LR edge

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Upper tangent of π’Ÿβ„‹ 𝑄

1 , π’Ÿβ„‹ 𝑄2 Base LR edge

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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New Base LR edge

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Rising Bubble

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Terminate

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Rising Bubble Implementation

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πœ„π‘† πœ„π‘€

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Rising Bubble Implementation

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πœ„π‘† πœ„π‘€

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Rising Bubble Implementation

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πœ„π‘† πœ„π‘€

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Rising Bubble Implementation

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πœ„π‘† πœ„π‘€

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Rising Bubble Implementation

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πœ„π‘† πœ„π‘€

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Rising Bubble Implementation

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Terrain Problem

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Terrain Problem

We would like to build a model for the Earth terrain We can measure the altitude at some points How to approximate the altitude for non- measured points?

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Nearest Neighbor

One possibility, approximate it to the nearest measured point Does not look natural

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Triangulation

Determine a triangulation Raise each point to its altitude Question: Which triangulation?

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SLIDE 127

Angle-optimal Triangulation

For a triangulation 𝒰 𝐡(𝒰): is the angle vector which consists of the angles 𝛽’s in sorted order 𝛽1 ≀ 𝛽2 ≀ β‹― ≀ π›½π‘œ We say that 𝐡 𝒰 > 𝐡(𝒰′) if 𝐡(𝒰) is lexicographically larger than 𝐡(𝒰′) 𝒰 is angle optimal if 𝐡 𝒰 β‰₯ 𝐡(𝒰′) for all triangulations 𝒰′

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𝛽1 𝛽2 𝛽3 𝛽4 𝛽5 𝛽6

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SLIDE 128

Edge Flip

The edge π‘žπ‘—π‘žπ‘˜ is illegal if min

1≀𝑗≀6 𝛽𝑗 < min 1≀𝑗≀6 𝛽𝑗 β€²

Flipping an edge increases the angle vector

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𝛽1 𝛽2 𝛽3 𝛽4 𝛽5 𝛽6 π‘žπ‘— π‘žπ‘˜ π‘žπ‘™ π‘žπ‘š 𝛽1

β€²

𝛽2

β€²

𝛽3

β€²

𝛽4

β€²

𝛽5

β€²

𝛽6

β€²

π‘žπ‘— π‘žπ‘˜ π‘žπ‘™ π‘žπ‘š Edge Flip

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SLIDE 129

Detect Illegal Edges

Thale’s Theorem 𝑏𝑐 is a chord in 𝐷 βˆ‘π‘π‘ π‘ > βˆ‘π‘π‘žπ‘ βˆ‘π‘π‘žπ‘ = βˆ‘π‘π‘Ÿπ‘ βˆ‘π‘π‘Ÿπ‘ > βˆ‘π‘π‘‘π‘

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𝑏 𝑐 𝑠 π‘ž π‘Ÿ 𝑑 𝐷

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SLIDE 130

Detect Illegal Edges

By Thale’s Theorem βˆ‘π‘žπ‘—π‘žπ‘˜π‘žπ‘™ < βˆ‘π‘žπ‘—π‘žπ‘šπ‘žπ‘™ βˆ‘π‘žπ‘˜π‘žπ‘—π‘žπ‘™ < βˆ‘π‘žπ‘˜π‘žπ‘šπ‘žπ‘™ An angle-optimal triangulation is equivalent to Delauany Triangulation

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π‘žπ‘— π‘žπ‘˜ π‘žπ‘™ π‘žπ‘š Illegal Edge

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SLIDE 131

Delaunay Triangulation

  • 1. Start with any valid triangulation
  • 2. If no illegal edges found, terminate
  • 3. Pick an illegal edge and flip it
  • 4. Go to 2
  • Does this algorithm terminate?
  • Running time: 𝑃 π‘œ2

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SLIDE 132

Incremental Algorithm

Given an existing Delaunay triangulation πΈπ‘ˆ(𝑄) We need to add a point π‘žπ‘— to πΈπ‘ˆ

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SLIDE 133

Incremental Algorithm

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Incremental Algorithm

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Incremental Algorithm

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Incremental Algorithm

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Incremental Algorithm

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π‘žβˆ’1 π‘žβˆ’2

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SLIDE 138

Incremental Algorithm

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SLIDE 139

Incremental Algorithm

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Insert

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Legalize Edge

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Correctness

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Correctness

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Incremental Algorithm

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Search Data Structure

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