Computing the triangulation of a polygon is a fundamental algorithm in
computational geometry. In computer graphics, polygon triangulation algorithms
are widely used for tessellating curved geometries, as are described by splines
[Kumar and Manocha 1994]. Methods of triangulation include greedy algorithms [O'Rourke 1994], convex hull differences [Tor and Middleditch 1984] and horizontal decompositions [Seidel 1991].
This Gem describes an implementation based on Seidel's algorithm (op.
cit.) for triangulating simple polygons having no holes (The code has
since then been extended to handle holes). It is an incremental
randomized algorithm whose expected complexity is O(n log*n). In
practice, it is almost linear time for a simple polygon having n
vertices. The triangulation does not introduce any additional vertices and
decomposes the polygon into n-2 triangles. Furthermore, the algorithm
generates a query structure that can be used to determine the location of a
point in logarithmic time. Related gems include incremental Delaunay
triangulation of a set of points [Lischinski 1994] and polygonization of implicit surfaces [Bloomenthal 1994].
Overview of the Triangulation Algorithm
Generating monotone polygons from the trapezoid formation
The algorithm proceeds in three steps as shown in Figure 1.
Let S be a set of non-horizontal, non-intersecting
line segments of the polygon . The randomized algorithm is used to create the
trapezoidal decomposition of the X-Y plane arising due the segments of
set S. This is done by taking a random ordering s1 .. sN of the
segments in S and adding one segment at a time to incrementally construct
the trapezoids. This divides the polygon into trapezoids (which can degenerate
into a triangle if any of the horizontal segments of the trapezoid is of zero
length). The restriction that the segments be non-horizontal is necessary to
limit the number of neighbors of any trapezoid. However, no generality is lost
due to this assumption as it can be simulated using lexicographic ordering. That
is, if two points have the same y-coordinate then the one with larger
x-coordinate is considered higher. The number of trapezoids is linear
in the number of segments. Seidel proves that if each permutation of s1 .. sN
is equally likely then trapezoid formation takes O(n log*n) expected
A monotone polygon is a polygon whose boundary consists of two
y-monotone chains. These polygons are computed from the trapezoidal
decomposition by checking whether the two vertices of the original polygon lie
on the same side. This is a linear time operation.
A monotone polygon can be triangulated in linear time by using
a simple greedy algorithm which repeatedly cuts off the convex corners of the
polygon [Fournier and
Montuno 1984]. Hence, all the monotone polygons can be triangulated in
Data Structures for Implementation
All the data-structures used in the implementation are statically allocated.
The trapezoid formation requires a structure where the neighbors of each
trapezoid and its neighboring segments can be determined in constant time.
Therefore, for every trapezoid, the indices of its neighbors and the segments
are stored in its table-entry T.
The query-structure Q, used to determine the location of a point, is
implemented as described by Seidel. The same Q can be later used for fast
point-location queries. Both Q and T are updated as a new segment
is added into the existing trapezoid formation. This entails splitting in two
the trapezoid(s) in which the endpoints of the segment lie, then traversing
along the edge of the segment to merge in any neighboring trapezoids which both
share the same left and right edges and also share a horizontal edge. All the
monotone polygons are stored in a single linked list with pointers to the first
vertex in the list stored in a table.
shows the average running time of the algorithm for randomly generated data sets
of various sizes. All the measurements were taken on an HP Series 735 with
execution times averaged over one hundred repetitions.
Empirical testing has proven the method robust across wide class of input
data. The present implementation uses an epsilon tolerance when testing for
floating-point equality. This computation occurs when determining whether a
point lies to the left (right) of a segment or when detecting coincident points.
This tolerance could potentially be removed by substituting a well-crafted
point-in-polygon test [Haines
The triangulation code is invoked through the main interface routine,
int triangulate_polygon(n, vertices, triangles);
n-sided polygon given for input (the vertices are specified in canonical
anticlockwise order with no duplicate points). The output is an array of
n-2 triangles (with vertices also in anticlockwise order). Once the
triangulated, point-location queries can be invoked as
details appear in the C source code which accompanies this Gem.
Note/FTPing the code
implementation has been extended to handle polygons with holes. Click Click herezip to get the code. (Прим. сайта: оригинальная ссылка здесь).
R. Seidel. A simple and fast incremental randomized algorithm for
computing trapezoidal decompositions and for triangulating polygons.
Computational Geometry: Theory and Applications, 1(1):51-64, 1991.