Graham scan triangulation
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Graham Scan Triangulation. Reference. Summary. Time complexity: O(kn), k-1 is the number of concave vertices in the polygon Worst case: O(n 2 ), yet easy to implement Algorithm: Loop around polygon to cut ear Once an ear is cut, update the list of concave vertices An ear can be cut if:

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Graham Scan Triangulation

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Graham scan triangulation

Graham Scan Triangulation

Reference


Summary

Summary

  • Time complexity: O(kn), k-1 is the number of concave vertices in the polygon

  • Worst case: O(n2), yet easy to implement

  • Algorithm:

    • Loop around polygon to cut ear

    • Once an ear is cut, update the list of concave vertices

    • An ear can be cut if:

      • Ear is at a convex vertex

      • Triangle does not contain any concave vertex


Algorithm

Algorithm

Build diagonal; cut ear

Check concavity of two sides of the ear; update R if necessary

P: vertices in the polygon

R: concave vertices in P


Graham scan triangulation

P={0,1,2,3,4,5}

R={3,4}

2

i=2

IsAnEar(P,R,1) = F

{3 in ∆012}

i=3

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5

0


Graham scan triangulation

P={0,1,2,3,4,5}

R={3,4}

2

i=3

IsAnEar(P,R,2) = T

Diagonal(1, 3)

P -= {2}

Check concavity of 3 and 1:

R -= {3}

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3

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5

0


Graham scan triangulation

P={0,1,3,4,5}

R={4}

2

i = 3

IsAnEar(P,R,1) = T

{1 is convex  4 not in ∆013}

Diagonal(0, 3)

P -= {1}

Check concavity of 3 and 0:

{no-op}

PRED(3) = 0 i=SUCC(3)=4

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3

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5

0

Never cut the ear at p0


Graham scan triangulation

P={0,3,4,5}

R={4}

2

i = 4

IsAnEar(P,R,3) = T

{3 is convex  4 not in ∆034}

Diagonal(0, 4)

P -= {3}

Check concavity of 4 and 0:

R -= {4}

PRED(4) = 0 i=SUCC(4)=5

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3

4

5

0


Graham scan triangulation

P={0,4,5}

R={ }

2

i = 5

P is a triangle = T

i=SUCC(5) = 0

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3

4

5

0


Graham scan triangulation

P={0,4,5}

R={ }

2

i = 0

END Triangulate

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3

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5

0


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