Triangulating a monotone polygon, introduction
The algorithm to triangulate a monotone polygon
depends on its monotonicity.
Developed in 1978 by Garey, Johnson, Preparata, and Tarjan,
it is described in both Preparata pp. 239241 (1985) and
Laszlo pp. 128135 (1996).
The former uses
y
monotone polygons, the latter uses
x
monotone.
Initialization
Sort the
N
vertices of monotone polygon
P
in order by decreasing
y
coordinate.
(Here
N
is the number of vertices of
P
, not
S
.)
The sort can be done in
O
(
N
) time, not
O
(
N
log
N
),
by merging the two monotone chains of
P
.
Let
u
1
,
u
2
, …,
u
N
be the sorted sequence of vertices,
so
y
(
u
1
) >
y
(
u
2
) > … >
y
(
u
N
).
Because of the regularization process and the monotonicity of
P
,
for every
u
i
1
≤
i
<
N
there exists
u
j
1 <
j
≤
N
such that edge
u
i
u
j
is an edge of
P
.
Triangulation of Monotone Polygon
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Description of the processing
The algorithm processes one vertex at a time in order of decreasing
y
coordinate, creating diagonals of polygon
P
.
Each diagonal bounds a triangle, and leaves a polygon
with one less side still to be triangulated.
Stack
The algorithm uses a stack to store vertices that have been visited
but not yet connected with a diagonal.
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 Summer '11
 Mukherjee
 monotone polygon

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