Triangulation%20of%20Monotone%20Polygon

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Triangulation of Monotone Polygon
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. 239-241 (1985) and Laszlo
pp. 128-135 (1996). The former uses y-monotone
polygons, the latter uses x-monotone. Initializat
ion 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 u1, u2, , uN be
the sorted sequence of vertices, so y(u1) gt y(u2)
gt gt y(uN). Because of the regularization
process and the monotonicity of P, for every ui 1
? i lt N there exists uj 1 lt j ? N such that edge
uiuj is an edge of P.
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ProximityConstrained triangulation
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. The stack content
is v1, v2, , vi, where v1 is the bottom and vi
the top of the stack. At any time during the
execution, there are two invariants 1. The
vertices v1, v2, , vi on the stack from a
chain on the boundary of P, where y(v1) gt y(v2)
gt gt y(vi). 2. If i ? 3, angle vjvj1vj2 ? ?
for 1 ? j ? i - 2.
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ProximityConstrained triangulation
Algorithm By adjacent we mean connected by an
edge in P. Recall that v1 is the bottom of the
stack, vi is the top. 1. Push u1 and u2 on the
stack. 2. j 3 / j is index of current vertex
/ 3. u uj 4. Case (i) u is adjacent to v1
but not vi. add diagonals uv2, uv3, ,
uvi. pop vi, vi-1, , v1 from stack. push vi,
u on stack. Case (ii) u is adjacent to vi but
not v1. while i gt 1 and angle uvivi-1 lt
? add diagonal uvi-1 pop vi from
stack endwhile push u Case (iii) u
adjacent to both v1 and vi. add diagonals uv2,
uv3, , uvi-1. exit 5. j j 1 Go to step 3.
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ProximityConstrained triangulation
Algorithm cases Case (i) u is adjacent to v1
but not vi.
v1
v2
v3
v4 vi top of stack
u
Case (ii) u is adjacent to vi but not v1.
v1
v2
v3
v4
v5 vi top of stack
u
Case (iii) u adjacent to both v1 and vi.
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ProximityConstrained triangulation
Example, 1
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ProximityConstrained triangulation
Example, 2
v1
v2
v3
8, Case (ii)
u
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ProximityConstrained triangulation
Example, 3
9, Case (iii)
10, final
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ProximityConstrained triangulation
Proof of correctness The correctness of the
algorithm depends on the fact that all the added
diagonals lie inside the polygon P. For details,
see Preparata pp. 240-241, or Laszlo pp.
134-135. Analysis of triangulating a monotone
polygon The initial sort (merge) requires O(N)
time. Each of the N vertices is visited and
placed on the stack exactly once, except when the
while fails in case (ii). This happens at most
once per vertex, so that time can be charged to
the current vertex. ? The algorithm requires O(N)
time to triangulate a monotone polygon, where N
is the number of vertices of the polygon.
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ProximityConstrained triangulation
S, E
(1) Inscribe S in minimum enclosing axis
parallel rectangle. O(N)
PSLG
rect(S), S, E
(2) Regularize rect(S), S, E. O(N log N)
Each region within regularized rect(S), S, E is a
monotone polygon.
Regularized rect(S), S, E
(3) Decompose regularized rect(S), S, E
into monotone polygons. O(N)
Monotone polygon
(4) Triangulate monotone polygons. O(N)
Triangulation of rect(S)
Overall comments O(N log N) regularization
dominates time. O(N) for triangulating all
monotone polygons (here N is the number of
vertices in S and rect(S). We know TRIANGULATION
has lower bound in ?(N log N). ? This algorithm
is optimal, ?(N log N).
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ProximityTriangulating a simple polygon
Definitions In a simple polygon, edges intersect
only at vertices, and non-adjacent edges do not
intersect. Three consecutive vertices a, b, c of
a polygon form an ear of the polygon if segment
ac is a diagonal b is the ear tip. Meisters
Two Ears Theorem. Every polygon with N ? 4
vertices has at least two nonoverlapping ears.
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ProximityTriangulating a simple polygon
Example, simple polygon with ears
c
b
ear
a
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ProximityTriangulating a simple polygon
Triangulation by otectomy See ORourke, pp.
39-46. Let P be a simple polygon, with vertices
p1, p2, , pN. 1. if N gt 3 2. for each
potential ear diagonal pipi2 3. if pipi2 is
a diagonal 4. add diagonal pipi2 5. recurse
on P - pi1 6. endif 7. endfor 8. endif Ana
lysis Step 2 is O(N), search around P. Step 3 is
a test for diagonal by checking for
intersections, O(N). Step 5 can occur at most
O(N) times. Overall time required is O(N3).
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ProximityTriangulating a simple polygon
Example
p12
p5
7
p11
2
8
p10
10
9
p13
p7
p15
3
p6
11
p9
p14
p8
12
13
p16
4
p4
14
5
p17
1
6
15
p2
p18
p3
p1
Numbers indicate sequence in which diagonals were
added.