AMS 345/CSE 355 Computational Geometry

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AMS 345/CSE 355 Computational Geometry

• Triangulation Algorithms

Joe Mitchell
Some figures ORourke Computational Geometry
in C Chap 2
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Triangulation

• Input Set S of n points
• Input Other shapes
• 3D Surfaces and solids (tetrahedralization)

Simple polygon
Planar Straight-Line Graph (PSLG)
Polygon with holes
Triangulation applet for simple polygons
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Triangulation Theory in 2D
Also with holes

• Thm A simple polygon has a triangulation.
• Lem An n-gon with n?4 has a diagonal.
• Thm Any triangulation of a simple n-gon has n-3
  diagonals, n-2 triangles.
• Thm The dual graph is a tree.
• Thm An n-gon with n?4 has ?2 ears.
• Thm The triangulation graph can be 3-colored.

But, NOT true in 3D!
Proofs Induction on n.
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Triangulating a Simple Polygon

• Simple ear-clipping methods O(n2 )
• Cases with simple O(n) algorithms
• Convex polygons (trivial!)
• Monotone polygons, monotone mountains
• General case (even with holes!)
• Sweep algorithm to decompose into
• monotone mountains
• O(n log n)
• Best theoretical results
• Simple polygons O(n) Chazelle90
• Polygons with h holes O(nh log1? h), ?(nh
  log h) BC
• Good practical method FIST Held, based on
  clever methods of ear clipping (worst-case O(n2 )
  )

fan
Not practical!
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Lower Bound ?(nh log h)

• ?(n) Have to read the data
• ?(h log h) from SORTING

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FIST Fast Industrial-Strength Triangulation
Based on ear clipping
Constrained Delaunay
Simple polygon
FIST
Works nicely also for highly degenerate and
crazy polygons
3D cycles
http//www.cosy.sbg.ac.at/held/projects/triang/tr
iang.html
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Ear-Clipping Triangulation
Input Simple polygon P
vi1
vi-1
vi
pq is a diagonal, cutting off a single triangle
(the ear)
Naive O(n3) Smarter Keep track of ear
tip status of each vi (initialize O(n2)
) Each ear clip requires O(1) ear tip tests ( _at_
O(n) per test ) Thus, O(n2) total, worst-case

• Ear-clipping applet

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Triangulate
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Ear-Clipping

• Lemma When clipping ear wth tip vi the only ear
  tip statuses that can change are at vi-1 and
  vi1

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(No Transcript)
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Example Triangulate
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Example Output
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W(n2 ) Examples Exist
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Faster Algorithm O(n log n)

• Input PSLG of size n can enclose by a big box B
• Step 1 Use sweep to decompose B into y-monotone
  mountains y-monotone polygons having one side
  (left/right) a single segment (the base) O(n
  log n)
• Step 2 Triangulate each y-monotone polygon (size
  ni ) in time O(ni), for total O(n)
• Overall O(n log n) to triangulate PSLG

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Monotone Polygons

• P is monotone in direction d

d
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Monotone Polygons

• P is monotone in direction d

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y-Monotone Polygon
d
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y-Monotone Polygon
d
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Monotone Mountains
d
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Triangulating a Monotone Mountain in O(n)

• Ear clipping is easy!
• Testing if vi-1 vi1 is a diagonal takes only
  O(1) time

t
vi-1
vi
vi1
vi is ear tip iff Left(vi1 , vi , vi-1 )
Just traverse vertices from top to bottom.
Test/clip ears. If an ear is clipped, re-test
the earity of the upper endpoint (vi-1 ) of the
diagonal just clipped.
monotone
b
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Triangulating a Monotone Mountain in O(n)
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Example
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Triangulation in O(n log n)

• (1) Plane sweep to get horizontal
  trapezoidalization

L
Fire bullets left/right from each vertex
SLS left-to-right ordering of segments crossed
by L (balanced binary tree) Events L hits a
vertex
Time O(n log n)
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Trapezoidalization
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Trapezoidalization
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Trapezoidalization
In each case We do O(1) updates to the SLS, each
taking time O(log n), since the SLS is stored in
a balanced binary search tree.
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Trapezoidalization
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Triangulation in O(n log n)

• (2) Join top vertex to bottom vertex in each
  trapezoid

Lemma Resulting pieces are monotone mountains
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Triangulation in O(n log n)

• (3) Triangulate each monotone mountain

Triangulate each, in time O(ni ), for total time
O(n)
Summary O(n log n) to triangulate n points or a
planar straight-line graph (PSLG)
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Minimum-Weight Triangulation

• MWT of a simple polygon O(n3), using dynamic
  programming
• MWT of a polygon with holes (or of a set of
  points in the plane) is NP-hard
• Min-Weight Steiner Triangulation allow extra
  Steiner points to be added
• Not known to exist
• 316-approximation known

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Convex Decomposition

• Partition simple polygon P into a small number of
  convex pieces

One way to do it Triangulate P
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Convex Decomposition

• Partition simple polygon P into a small number of
  convex pieces

Another way to do it Use diagonals to partition
P into convex polygons
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Convex Decomposition

• Partition simple polygon P into a small number of
  convex pieces

Another way to do it Allow Steiner points
(non-vertices) inside P. May get fewer pieces!
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Convex Decomposition

• Goal Partition P into a small number of convex
  pieces (convex polygons)
• A triangulation is one possible decomposition
  into convex pieces, but it may have many more
  pieces than necessary!
• Dynamic Programming algorithms yield optimal
  solutions for simple polygons (for both Steiner
  and non-Steiner versions), in roughly O(n3)
• Hertel-Mehlhorn algorithm 4-approximation in
  time O(n)

O(r2n log n) without Steiner Keil85 O(nr3)
allowing Steiner Chazelle80
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Convex Decomposition
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Convex Decomposition
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Optimal Convex Decomposition

• Allowing Steiner points

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Convex Decomposition
r6
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Convex Decomposition
One diagonal resolves the local nonconvexities
at 2 reflex vertices at once
r6
We need at least r/2 segments to resolve all r
reflex vertices
Results in at least ceil(r/2)1 pieces
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Hertel-Mehlhorn Algorithm

• Start with any triangulation of simple polygon P
  (time O(n), Chazelle)
• Remove inessential diagonals, in any order (time
  O(n), since we can test a diagonal locally in
  time O(1) to see if it is essential if we remove
  a diagonal, we only have to update the
  inessential flag of O(1) other diagonals)

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Hertel-Mehlhorn Algorithm

• Lemma 2.5.2 At the end of the algorithm, for
  each reflex vertex v, there can be at most 2
  diagonals essential for v

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Hertel-Mehlhorn Algorithm

• Theorem 2.5.3 The H-M algorithm yields a
  decomposition into at most 4OPT pieces, where
  OPT is the minimum possible number of convex
  pieces in a (Steiner) convex decomposition.

We say that the H-M Algorithm is a
4-approximation algorithm
OPEN Find a better factor than 4, which still
runs very efficiently (say, in O(n log n) or O(n)
time).
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Hertel-Mehlhorn Algorithm

• Proof At end, each diagonal is essential for
  some (reflex) vertex.
• By Lemma 2.5.2, there are at most 2r diagonals
  left (since each reflex vertex is responsible
  for at most 2 diagonals)
• Thus, the number, M, of pieces is 2r1 lt 2r4
  4OPT
• (Since, by Theorem 2.5.1, OPT ceil(r/2)1, so
  4OPT 4ceil(r/2)42r4)

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H-M Algorithm Example