Meshing Formally introd by Voronoi in 1908 Meshing Delaunay 1934 - - PDF document

CS 101

Meshing

CS 101

Meshing

Introduction to Delaunay Triangulations and Voronoi Diagrams Part I

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Delaunay/Voronoi

An essential notion

Formally intro’d by Voronoi in 1908

Delaunay 1934

Applications galore

path planning in robotics crystallography minimum spanning tree travel salesman medial axis

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All About Proximity

We will see connections with:

nearest neighbors

by definition

convex hull (CH)

much less trivial

• ne of many wonderful properties…

circumcircles and triangles

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Basic Definitions

Let P = (p1, p2, …, pn) be n 2D points

called sites in this context

Partition the plane in n regions Vi

Vi = points closer to pi than any other

= Voro Vorono noi cell ell (caveat: can be unbounded)

Points with more than one nearest site

= Voro Vorono noi dia iagra gram

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Example

2 points

Points that are equidistant (bisector) CS101 - Meshing

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Example

circumcircle

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Voronoi cell

Definition using halfplanes

let be the halfplane:

boundary = bisector of the two sites

• is in the halfplane

Then we have: Consequence: convex cells

convex cells

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Voronoi Cell

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Voronoi Diagram

Once we have defined the cell

boundary made out of edges

Voronoi edges 2 closest sites

Voronoi edges linking vertices

Voronoi vertices multiple closest sites

remember: cell complex…

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Example

Four points are cocircular…

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Example

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Voronoi Example I

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Voronoi Example II

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Voronoi in Nature?

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Voronoi Properties

Too many to mention…

cell always convex if V vertex of degree 3, circumcenter if pj n.n. of pi they share a V-edge cell unbounded iff site on CH

proof? think about a point at infinity p on CH iff closest point from inf. point

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Some Applications

Motion Planning

what path should a robot follow to

stay as far as possible from sites?

Ambulance Dispatcher

which hospital is closest to scene?

Farthest point sampling

where to put a new Starbuck’s?

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Delaunay Triangulation

Definition:

DT is the (straight-line) dual of VD dual in the sense of:

V cell -> D vertex V edge -> D edge V vertex -> D triangle

careful:

what happen when 4 cocircular points?

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(Rough) Examples

An example from the sea…

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Realtime Demo

click here

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Example in 3D

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Properties

Too many…

“same” as Voronoi

e.g., if pj n.n. of pi (pi,pj) is a D-edge

important property (Empty Circle):

For every D-edge (pi,pj), there’s a circle through pi and pj that doesn’t contain ANY other vertex. The reverse holds too.

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Proof?

• If ab is in DT, V(a) and V(b) share an edge e.

Pick x on e, and trace a circle center on x through a and b. if there’s a site c in or on the circle, x would be in V(c) too! So no site inside.

• Suppose there’s an empty circle through a and

b centered on a point x. x is on bisector btw a and b (equidistant). Now wiggle x a bit while keeping the circle empty—along bisector. Clearly, x is on a V-edge, thus ab D-edge.

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Applications

Nearest neighbor search:

NNG ⊆ DT

i.e., if b n.n. of a, then ab is in DT

Medial Axis – generalization (grassfire)

Minimum Spanning Tree

MST is a subgraph of DT

suppose (a,b) edge of MST, and assume (a.b)

NOT Delaunay; prove contradiction (use empty circle…)

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Numerical Properties

In 2D: maximizes minimum angle

stronger: best lexicographic

sequence of triangle angles

great for numerics (as detailed later) Edelsbrunner ’87 ain’t true in 3D….

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Connection to CH

We already saw link

CH of sites goes along D-edges

But what if we think higher dim…

I.e., 2D plane and sites are

projections of something 3D

great property, valid in nD

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Lifting Map

Lift points (x, y) to paraboloid (x, y, x2 + y2)

placement of origin irrelevant

DT = CH of 3D points

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Proof

Equation of tangent plane at (a,b): Shift upwards by r2: Intersection with paraboloid? Now, imagine you have a face (p1,p2,p3) of 3D CH

• ther vertices are at least r2 away, right?

thus, they project OUTSIDE of the projected circle bingo: empty circle, thus the face maps to D-triangle.

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Same for Voronoi?

Almost… VD = proj. of tangent envelope

Before…