1. Brute-Force Triangulation
1. Find a diagonal.
2. Triangulate the two resulting subpolygons recursively.
How to find a diagonal?
leftmost
vertex
case 2
case 1

2. Triangulating a Convex Polygon
Idea:
monotone
Decompose a simple polygon into convex pieces.
Triangulate the pieces.
as difficult
as triangulation

3. y-monotone Pieces Stragegy: walk always downward or
highest
vertex
Stragegy:
walk always
downward or
horizontal
Partition the polygon into monotone
pieces and then triangulate.
lowest
vertex

4. Turn Vertex Turn vertex is where the walk
from highest vertex to the lowest
vertex switches direction.
 Both adjacent edges are below.
The polygon interior lies above.
Choose a diagonal that goes up.

5. Five Types of Vertices 4 types of turn vertices
4 types of turn vertices
Regular vetex: the remaining vertices (no turn).
start vertices  end vertices split vertices  merge vertices

6. Local Non-Monotonicity
Proof
split
vertex
interior
exterior
exterior
interior
merge vertex

7. Partitioning into Monotone Pieces
 Add a downward diagonal at every merge vertex.
 Add an upward diagonal at every split vertex.
Use a downward plane sweep.
No new event point will be created except the vertices.
The event queue is implemented as priority queue (e.g., heap).

8. Removal of a Split Vertex
edge
helper

9. Removal of a Merge Vertex
Merge vertices can be handled the same way in an upward sweep
as split vertices in a downward sweep.
But why not all in the same downward sweep?
Check if the old helper is a merge vertex and add the diagonal if so.
(The diagonal is always added if the new helper is a split vertex).

10. Sweep-Line Status Implemented as a binary search tree.
Only edges to the left of the polygon interior are stored.
This is because we are only interested in edges to
the left of split and merge vertices.
Edges are stored in the left-to-right order.

11. DCEL Representation
Construct a doubly-connected edge list to represent the polygon.
Add in diagonals computed for split and merge vertices.
Edges in the status BST and corresponding ones in DCEL
cross-point each other.

12. The Algorithm

13. Handling Start & End Verticesv 5 v 3 e 5 v 4 e 4 v 6 e 3 e 6 v 1 v 14 e 2 v 7 v 2 v 9 v 8 e 7 e 1 e 8 e 15 e 14 e 9 v 12 v 15 v 10 e 10 e 11 e 13 e 12 v 11 v 13

14. Handling Split Vertex v v e v e v e e v e v v v v e e e e v e e v v vv 5 v 3 e 5 v 4 e 4 v 6 e 3 e 6 v 1 e 2 v 7 v 2 v 9 v 8 e 7 e 1 e 8 e 15 v 14 e 14 e 9 v 12 v 15 v 10 e 10 e 11 e 13 e 12 v 11 v 13

15. Handling Merge Vertex (1)v 5 v 3 e 5 v 4 e 4 v 6 e 3 e 6 v 1 e 2 v 7 v 2 v 9 v 8 e 7 e 1 e 8 e 15 v 14 e 14 e 9 v 12 v 15 v 10 e 10 e 11 e 13 e 12 v 11 v 13

16. Handling Merge Vertex (2)i

17. Handling Regular Vertices (1)v 5 v 3 e 5 v 4 e 4 v 6 e 3 e 6 v 1 e 2 v 7 v 2 v 9 v 8 e 7 e 1 e 8 e 15 v 14 e 14 e 9 v 12 v 15 v 10 e 10 e 11 e 13 e 12 v 11 v 13

18. Handling Regular Vertices (2)

19. Correctness Theorem The algorithm adds a set of non-intersecting
diagonals that partitions the polygon into
monotone pieces.
Proof The pieces that result from the partitioning contain no split
or merge vertices. Hence they are monotone by an earlier
lemma.
We need only prove that the added segments are diagonals
that intersect neither the polygon edges nor each other.
Establish the above claim for the handling of each of the
five type of vertices during the sweep. (Read the textbook
on how to do this for the case of a split vertex.)

20. Running Time on Partitioning

21. It is for clarity of presentation and will be easily removed later.
It is for clarity of presentation and will be easily removed later.
convex
vertex
lowest vertex on top.
funnel
Idea: add as many diagonals from the current
vertex handled to those on the stack as
possible.
current
vertex
Invariants of iteration:
reflex
vertices
One boundary of the funnel is a polygon edge.
The other boundary is a chain of reflex vertices
(with interior angles > ) plus one convex vertex
(the highest) at bottom of the stack.

22. Case 1: Next Vertex on Opposite Chain
This vertex must be the lower endpoint of the
single edge e bounding the chain.
Pop these vertices from the stack.
Add diagonals from the current vertex
to them (except the bottom one) as they
are popped.
current
vertex
Push the previous top of the stack and
the current vertex back onto the stack. j

23. Case 2: Next Vertex on the Same Chain
The vertices that can connect to the current
vertex are all on the stack.
Pop one vertex from the stack.
It shares an edge with the current vertex.
current
vertex
Pop other vertices from the stack as
long as they are visible from the current
vertex.
Draw a diagonal between each of them
and the current vertex.
Pushed the last popped vertex back onto
the stack followed by the current vertex.

24. The Triangulation Algorithm

25. An Example u u u u u Start: u u u u u u u u u u u u u u u u u u u u u1 u 2 u 3 u 2 1 u 4
Start: u 6 u 5 u 7 u 8 u 9 u 10 u 11 u 4 u 3 u 3 u 12 u 2 u 2 u 13 u 14 u 1 u 1 u 15 u 16 u 18 u 17 u 4 u 19 u 2 u 5 u 5 u 6 u 20 u 1 u 5 u 1

26. Example (cont’d) u u u u u u u u u u u u u u u u u u u u u u u u u u u1 u 2 u 3 u 4 u 7 u 6 u 6 u 5 u 6 u 5 u 7 u 8 u 9 u 10 u 11 u 8 u 7 u 9 u 12 u 7 u 6 u 8 u 13 u 14 u 15 u 16 u 18 u 17 u 11 u 10 u 10 u 9 u 19 u 20 u 10 u 8 u 8

27. Example (Cont’d) u u u u u u u u u u u u u u u u u u u u u u u u u u u1 u 2 u 3 u 4 u 15 u 6 u 5 u 10 u 7 u 8 u 9 u 10 u 11 u 16 u 16 u 15 u 17 u 15 u 12 u 13 u 14 u 10 u 10 u 15 u 16 u 18 u 17 u 18 u 17 u 19 u 18 u 19 u 20 u 17 u 10 u 17

28. Removal of Strict y-monotonicity
#pops  #pushes
Treat them from left to right.
The effect of this is equivalent to that of rotating the plane
slightly clockwise and then every vertex will have different
y coordinate.

29. Time Complexity of Triangulation
1. Partition a simple polygon into monotone pieces.
2. Triangulate each monotone piece.

30. Triangulation of a Planar Subdivision
The algorithm for splitting a polygon into monotone pieces
does not use the fact that the polygon was simple.
The plane sweep for decomposition of a polygon into monotone
pieces takes as input only edges that lie to the left of the interior.
This easily generalizes to a planar subdivision in a bounding box.