Introduction to Voronoi Diagramsand Delaunay Triangulations

Solomon Boulos

Introduction to Voronoi Diagrams and Delaunay Triangulations – p.1

Voronoi DiagramsVoronoi region:V (pi) = {x ∈ R

n | ‖pi − x‖ ≤ ‖pj − x‖,∀j ≤ n}

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Delaunay TriangulationDual to Voronoi Diagram

Connect Vertices Across Common Line

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Circumcircle PropertyTi ∈ D(S) ⇐⇒ The circumcircle of Ti is empty

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General PositionWhat if you had points on the same circle?

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PerturbationWhat if we move a point just slightly?

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Circumcircle TestSimple determinant test∣

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x2 + y2 x y 1

x21+ y2

1x1 y1 1

x22+ y2

2x2 y2 1

x23+ y2

3x3 y3 1

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Equal 0 → On Circle

Less Than 0 → Outside Circle

Greater Than 0 → Inside Circle

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Symbolic PerturbationDeterminant == 0 is very sensitive

Sensitive enough for an ǫ

△(ǫ) =

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1 ξ2i−1 + ǫ2i−1 ξ2i + ǫ2i

1 ξ2j−1 + ǫ2j−1 ξ2j + ǫ2j

1 ξ2k−1 + ǫ2k−1 ξ2k + ǫ2k

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Further breaks up into sub-determinants

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Naive AlgorithmTake 3 points at random

See if Circumcircle is Empty (test against all points)

If not, add triangle

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Cirumcircle Test is localizedFor nicely distributed points the circumcircles are small

Suggests incremental algorithm

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Incremental AlgorithmAssume we start with a current delaunay triangulation

Choose a new vertex to add at random

Add new triangles, flip edges

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Example

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Example Continued

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Example Finished

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Complete AlgorithmFind the triangle abc containing the new point

Add new edges linking the new point to the vertices ofabc

Perform Swap Test until Stack is Empty

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The Swap TestIf edge is on the convex hull, skip

Otherwise check quad for circumcircle test

If test fails, add new edges

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Why does it work?You start with a Delaunay Triangulation

In each step, only a local neighborhood needs fixup

Each flip may break two more edges (ad and db)

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The Initial TriangulationUgliest Detail

Create a “big triangle”

Needs vertices that won’t be within the circumcircles

Remove vertices from final triangulation

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Total Running TimeTime to find containing triangle (happens n times)

There exist O(logn) methods for this

Time to perform insertion (happens n times)Time to perform a swap (O(1), but how many times?)

Total Time = O(nlogn + nu)

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Look at single flipReplaces link edge with new edge

Adds back two link edges (net increase 1)

Number Link Edges - 3 = Number of Flips

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Upper boundNumber Link Edges = Vertex Degree

3 ≤ Vertex Degree ≤ 6 for planar graphs

Max flips = 3

Total Running Time = O(nlogn)

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LimitationsEdge Flips don’t extend past 2D

Needs initial big triangle

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Determinants and OrientationA determinant is a signed volume∣

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x1 y1 1

x2 y2 1

x3 y3 1

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= ±‖(~v2 − ~v1) × (~v3 − ~v1)‖

Less than 0 → Clockwise

Greater than 0 → Counter-Clockwise

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Circumcircle as OrientationThe Circumcircle test is an orientation test

Let p′ = (p, ‖p‖2)

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Properties of the Space of SpheresPoints below the cutting plane are inside the circle

Also applies to triangular regions

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Using Convex HullsSimple Algorithm

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Advantages of Convex Hull ApproachMany good Convex Hull Algorithms (e.g. QuickHull)

Simple extension to arbitrary dimensions

No strange infinite triangle initialization

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Questions?

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