Triangulation Supplemental

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Triangulation Supplemental

• From ORourke (Chs. 12)
• Fall 2002

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Contents

• Ear clipping algorithm
• Triangulating monotonic polygons
• Monotonic decomposition via trapezoidalization

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Ear Clipping

• Go around the polygon to check whether a diagonal
  can be drawn from (i) to (i2)
• Diagonal check
• No edge crossing AND inside P

What is the time complexity in the worst case?
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Validity of Diagonal

• No edge crossing
• Each potential diagonal (i, i2) need to check
  with ? edges

• In-Cone check
• local geometry

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Triangulating Monotonic Polygons

• Linear time algorithm O(n)
• Fact on monotonic polygons
• Def a vertex is called reflex if its internal
  angle is strictly greater than p
• Def cusp
• A reflex vertex whose adjacent vertices v- and v
  are either both above or below v.
• Lemma
• If a polygon P has no cusps, then it is monotone.

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Algorithm Ideas

• Cut off triangles from the top in a greedy
  fashion
• At each step, the first available triangle
  removed
• For each vertex v, connect v to all the vertices
  above it and visible via a diagonal, and remove
  the top portion of the polygon thereby
  triangulated
• Continue with the next vertex below v

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Case 1
Case 2a, 2b
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0,1,2
0,1,2,3
0,1,2,3,4
0,1,2,3,4
4,5
4,6
6,7
7,8
7,9
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Trapezoidalization

• Accomplish monotonic subdivision via horizontal
  trapezoidalization
• Supporting vertices the vertices through which
  the horizontal lines are drawn
• Assume P be a polygon with no two vertices on a
  horizontal line
• Each trapezoid has exactly two supporting
  vertices one on top, one on bottom
• Remove cusps by connecting the supporting vertex
  to the opposite vertex

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Pass through each v the maximal horizontal
segment s such that s ? P and s??P v
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Trapezoidalization via Plane Sweep

• Time complexity O(n log n)
• Maintain a balanced tree of edges

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Sweep Line Events
(, a, c, b, )
(, a, d, b, )
(, a, c, d, b, )
(, a, b, )
(, a, b, )
(, a, c, d, b, )
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Summary
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Homework

• Analyze the worst case complexity of ear clipping
  algorithm
• Triangulate (and keep track of the reflex chain)
  of the polygon on the next page
• Decompose the polygon on page 13 into monotonic
  pieces using trapezoidalization

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