Algorithms
NonLecture P: Polygon Triangulation
If triangles had a god, they would give him three sides.
— Charles Louis de Secondat Montesquie (1721)
Down with Euclid! Death to triangles!
— Jean Dieudonné (1959)
P Polygon Triangulation
P.1 Introduction
Recall from last time that a
polygon
is a region of the plane bounded by a cycle of straight edges joined
end to end. Given a polygon, we want to decompose it into triangles by adding
diagonals
: new line
segments between the vertices that don’t cross the boundary of the polygon. Because we want to keep
the number of triangles small, we don’t allow the diagonals to cross. We call this decomposition a
triangulation
of the polygon. Most polygons can have more than one triangulation; we don’t care which
one we compute.
Two triangulations of the same polygon.
Before we go any further, I encourage you to play around with some examples. Draw a few polygons
(making sure that the edges are straight and don’t cross) and try to break them up into triangles.
P.2 Existence and Complexity
If you play around with a few examples, you quickly discover that every triangulation of an
n
sided
has
n

2 triangles. You might even try to prove this observation by induction. The base case
n
=
3 is
trivial: there is only one triangulation of a triangle, and it obviously has only one triangle! To prove
the general case, let
P
be a polygon with
n
edges. Draw a diagonal between two vertices. This splits
P
into two smaller polygons. One of these polygons has
k
edges of
P
plus the diagonal, for some integer
k
between 2 and
n

2, for a total of
k
+
1 edges. So by the induction hypothesis, this polygon can be
broken into
k

1 triangles. The other polygon has
n

k
+
1 edges, and so by the induction hypothesis,
it can be broken into
n

k

1 tirangles. Putting the two pieces back together, we have a total of
(
k

1
) + (
n

k

1
) =
n

2 triangles.
Breaking a polygon into two smaller polygons with a diagonal.
1
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NonLecture P: Polygon Triangulation
This is a ﬁne induction proof, which any of you could have discovered on your own (right?), except
for one small problem. How do we know that every polygon
has
a diagonal? This seems patently
obvious, but it’s surprisingly hard to prove, and in fact many incorrect proofs were actually published as
late as 1975. The following proof is due to Meisters in 1975.
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 Spring '09
 A
 Line segment, Meisters

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