1 Triangulation Supplemental From O’Rourke (Chs. 1&2) Fall 2005

2 Contents Ear clipping algorithm Triangulating monotonic polygons Monotonic decomposition via trapezoidalization

3 Ear Clipping Go around the polygon to check whether a diagonal can be drawn from (i-1) to (i+1) Diagonal check: –No edge crossing AND inside P What is the time complexity in the worst case?

5 Think… Can every simple polygon be triangulated? Does every simple polygon have at two ears (so that ear clipping algorithm can work)?

6 Validity of Diagonal No edge crossing –Each potential diagonal (i-1, i+1) need to check with ? edges “In-Cone” check: –local geometry

7 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  –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. Why?

8 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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10 Case 1: Case 2a, 2b:

11 Sort by y-coordinate reflex chain: {1,2} v = Triangulate Monotone Polygon

12 Case 2b: chain {1,2,3} v = 4 v = 3 chain: {1,2} [same side, non-convex]

13 Case 2b: chain {1,2,3,4} v = 5 v = 4 chain: {1,2,3} [same side, non-convex]

Case 1: diagonal (5,2) reflex chain {1,2,3,4} v = 5 chain: {1,2,3,4} [opposite side]

Case 1: diagonal (5,3) reflex chain {2,3,4} v = 5 chain: {2,3,4} [opposite side]

Case 1: diagonal (5,4) reflex chain {3,4} reflex chain {4,5} v = 6 v = 5 chain: {3,4} [opposite side]

Case 2b: reflex chain {4,5,6} v = 7 v = 6 chain: {4,5} [same side, non-convex]

Case 2a: diagonal (7,5) reflex chain {4,5,6} v = 7 chain: {4,5,6} [same side, convex]

Case 2a: diagonal (7,4) reflex chain {4,5} reflex chain {4,7} v = 8 v = 7 chain: {4,5} [same side, convex]

v = 8 chain: {4,7} [lowest vertex; stop]

{0,1,2} {0,1,2,3} {0,1,2,3,4} {4,5} {4,6} {6,7} {7,8} {7,9} {4,5} {6,7} {7,8} {9,10} {7,9} {4,6}

22 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 Assume no two points have same y coord.

23 The support line stops at the boundary

24 Trapezoidalization via Plane Sweep Time complexity: O(n log n) Maintain a balanced tree of edges

25 Sweep Line Events (…, a, c, b, …) (…, a, d, b, …) (…, a, c, d, b, …) (…, a, b, …) (…, a, c, d, b, …)

26 Summary

27 Review Questions 1.Analyze the worst case complexity of ear clipping algorithm 2.Analyze the time complexity of the algorithm for triangulating monotone polygon 3.Triangulate (and keep track of the reflex chain) of the polygon on the next page 4.Decompose the polygon on page 24 into monotonic pieces using trapezoidalization

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29 Monotone-ization If the polygon has no cusp, it is monotone and can be triangulated immediately. [seek an orientation where the polygon is monotone] Decompose into monotone pieces via horizontal trapezoidalization

30 Three Types of Event Points (…, a, c, b, …) (…, a, d, b, …) (…, a, c, d, b, …) (…, a, b, …) (…, a, c, d, b, …) Regular Upward cusp Downward cusp replace remove insert

31 Making Trapezoids Regular point: Extend toward material side until a boundary is reached Cusps: extend both sides until boundaries are reached

32 a b c d e f g h i j {bc} {aj} {ajih} {ajicdh}u.cusp {ajicdg} {acdg}d.cusp {acdefg}u.cusp {bcdefg} {bcde} {} Upward cusp: connect to support vertex above Downward cusp: connect to support vertex below [implementation]