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MotivationTriangulating a polygon

Triangulating a polygon

Geometric Algorithms

Lecture 2: Triangulating a polygon

Geometric Algorithms Lecture 2: Triangulating a polygon


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Visibility in polygonsTriangulationProof of the Art gallery theorem

Polygons and visibility

Two points in a simple polygon can see each other if theirconnecting line segment is in the polygon



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Art gallery problem

Art Galley Problem: How many cameras are needed toguard a given art gallery so that every point is seen?



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Art gallery problem

In geometry terminology: How many points are needed in asimple polygon with n vertices so that every point in thepolygon is seen?

The optimization problem is computationally difficult

Art Galley Theorem: n/3 cameras are occasionallynecessary but always sufficient



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Triangulation, diagonal

Why are n/3 always enough?

Assume polygon P is triangulated: a

decomposition of P into disjointtriangles by a maximal set ofnon-intersecting diagonals

Diagonal of P: open line segment

that connects two vertices of P andfully lies in the interior of P



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A triangulation always exists

Lemma: A simple polygon with n verticescan always be triangulated, and alwayshave n 2 triangles

Proof: Induction on n. If n = 3, it istrivial

Assume n > 3. Consider the leftmostvertex v and its two neighbors u and w.Either uw is a diagonal (case 1), or part of

the boundary of P is in uvw (case 2)

Choose the vertex t in uvw farthest fromthe line through u and w, then vt must bea diagonal

v

v

u

u

w

w



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A triangulation always exists

In case 1, uw cuts the polygon into a triangle and a simplepolygon with n 1 vertices, and we apply induction

In case 2, vt cuts the polygon into two simple polygons withm and nm + 2 vertices 3 m n 1, and we also applyinduction

By induction, the two polygons can be triangulated usingm 2 and nm + 2 2 = nm triangles. So the originalpolygon is triangulated using m 2 + nm = n 2triangles



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A 3-coloring always exists

Observe: the dual of a triangulated simplepolygon is a tree

Lemma: The vertices of a triangulatedsimple polygon can always be 3-colored

Proof: Induction. True for a triangle

Every tree has a leaf. Remove the

corresponding triangle from thetriangulated polygon, color its vertices, addthe triangle back, and let the extra vertexhave the color different from its neighbors



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A 3-coloring always exists


Vi ibili i l


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n/3 cameras are enough

For a 3-colored, triangulated simple

polygon, one of the color classes is used byat most n/3 colors. Place the cameras atthese vertices

This argument is called

the pigeon-hole principle


Vi ibilit i l


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n/3 cameras are enough

Question: Why does the proof fail when the polygon hasholes?


Towards an efficient algorithm


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gPartitioning into monotone piecesTriangulating a monotone polygonTriangulating a simple polygon

Two-ears for triangulation

Using the two-ears theorem:(an ear consists of three consecutivevertices u,v,w where uw is a diagonal)

Find an ear, cut it off with a diagonal,triangulate the rest iteratively

Question: Why does every simple polygon

have an ear?

Question: How efficient is this algorithm?




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Partitioning into monotone piecesTriangulating a monotone polygonTriangulating a simple polygon

Overview

A simple polygon is y-monotone iff anyhorizontal line intersects it in a connectedset (or not at all)

Use plane sweep to partition the polygoninto y monotone polygons

Then triangulate each y-monotone polygon




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Monotone polygons

A y-monotone polygon has a top vertex, a

bottom vertex, and two y-monotone chainsbetween top and bottom as its boundary

Any simple polygon with one top vertexand one bottom vertex is y-monotone


M i iTowards an efficient algorithmP i i i i i


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Vertex types

What types of vertices does ageneral simple polygon have?

start

stop

split

merge

regular. . . imagining a sweep line going topto bottom

start

merge regular

split

end


M ti tiTowards an efficient algorithmP titi i i t t i


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Sweep ideas

Find diagonals from each mergevertex down, and from each split

vertex up

A simple polygon with no split ormerge vertices can have at most onestart and one stop vertex, so it is

y-monotone


MotivationTowards an efficient algorithmPartitioning into monotone pieces


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Sweep ideas

Find diagonals from each mergevertex down, and from each split

vertex up

A simple polygon with no split ormerge vertices can have at most onestart and one stop vertex, so it is

y-monotone




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Sweep ideas

explored

unexplored

explored

unexplored




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Sweep ideas

Where can a diagonal from asplit vertex go?

Perhaps the upper endpoint ofthe edge immediately left of

the merge vertex?




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Sweep ideas



the merge vertex?




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Triangulating a polygong p

Triangulating a monotone polygonTriangulating a simple polygon

Sweep ideas



the merge vertex?




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Triangulating a polygong p

Triangulating a monotone polygonTriangulating a simple polygon

Sweep ideas


Perhaps the last vertex passedin the same component?




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Triangulating a polygon Triangulating a monotone polygonTriangulating a simple polygon

Sweep ideas






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Sweep ideas




MotivationT i l i l

Towards an efficient algorithmPartitioning into monotone piecesT i l i l


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Status of sweep

The status is the set of edgesintersecting the sweep line thathave the polygon to theirright, sorted from left to right,and each with their helper: thelast vertex passed in that

component


MotivationT i l ti l

Towards an efficient algorithmPartitioning into monotone piecesT i l ti t l


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Helpers of edges

The helper for an edge e thathas the polygon right of it,

and a position of the sweepline, is the lowest vertex vabove the sweep line such thatthe horizontal line segmentconnecting e and v is inside

the polygon



Towards an efficient algorithmPartitioning into monotone piecesTriangulating a monotone polygon


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Status structure, event list

The status structure stores all edges that have the polygon tothe right, with their helper, sorted from left to right in theleaves of a balanced binary search tree

The events happen only at the vertices: sort them byy-coordinate and put them in a list (or array, or tree)





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Main algorithm





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Event handling

Start vertex v:

Insert the counterclockwiseincident edge in T with v as thehelper





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Event handling

End vertex v:Delete the clockwise incidentedge and its helper from T





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g g p yg g g p ygTriangulating a simple polygon

Event handling

Regular vertex v:

If the polygon is right of the twoincident edges, then replace theupper edge by the lower edge inT, and make v the helper

If the polygon is left of the twoincident edges, then find the

edge e directly left of v, andreplace its helper by v





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g g p yg g g p ygTriangulating a simple polygon

Event handling

Merge vertex v:

Remove the edge clockwisefrom v from T

Find the edge e directly left ofv, and replace its helper by v

e





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Triangulating a simple polygon

Event handling

Split vertex v:

Find the edge e directly left of

v, and choose as a diagonal theedge between its helper and v

Replace the helper of e by v

Insert the edge counterclockwise

from v in T, with v as its helper

e





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Efficiency

Sorting all events by y-coordinate takes O(n log n) time

Every event takes O(log n) time, because it only involvesquerying, inserting and deleting in T



Towards an efficient algorithmPartitioning into monotone piecesTriangulating a monotone polygonT i l ti i l l


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Degenerate cases

Question: Which degenerate cases arise in this algorithm?



Towards an efficient algorithmPartitioning into monotone piecesTriangulating a monotone polygonTriang lating a simple pol gon


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Representation

A simple polygon with somediagonals is a subdivision use a DCEL

Question: How manydiagonals may be chosen to

the same vertex?



Towards an efficient algorithmPartitioning into monotone piecesTriangulating a monotone polygonTriangulating a simple polygon


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More sweeping

With an upward sweep in eachsubpolygon, we can find a

diagonal down from everymerge vertex (which is a splitvertex for an upward sweep!)

This makes all subpolygons

y-monotone





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Result

Theorem: A simple polygon with n vertices can bepartitioned into y-monotone pieces in O(n log n) time





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Triangulating a monotone polygon

How to triangulate ay-monotone polygon?





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g g p p yg







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T i l i l


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T i l i l


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T i l i l


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T i l ti t l


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Th l ith


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The algorithm

Sort the vertices top-to-bottom by a merge of the twochains

Initialize a stack. Pop the first two vertices

Take the next vertex v, and triangulate as much aspossible, top-down, while popping the stack

Push v onto the stack





Result


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Result

Theorem: A simple polygon with n vertices can bepartitioned into y-monotone pieces in O(n log n) time

Theorem: A monotone polygon with n vertices can betriangulated O(n) time

Can we immediately conclude:

A simple polygon with n vertices can be triangulated

O(n log n) time ???





Result


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Result

We need to argue that all monotone polygons together thatwe will triangulate have O(n) vertices

Initially we had n edges. We add at most n 3 diagonals in

the sweeps. So all monotone polygons together have at most2n 3 edges, and therefore at most 2n 3 vertices

Hence we can conclude that triangulating all monotonepolygons together takes only O(n) time

Theorem: A simple polygon with n vertices can betriangulated O(n log n) time


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