2D Polygon Partitioning
Reference Manual

Susan Hert

A partition of a polygon is a set of polygons such that the interiors of the polygons do not intersect and the union of the polygons is equal to the interior of the original polygon. Functions are available for partitioning planar polygons into two types of subpolygons - y-monotone polygons and convex polygons.

The function that produces a y-monotone partitioning is based on the algorithm presented in [dBvKOS97] which requires O(n log n) time and O(n) space for a polygon with n vertices and guarantees nothing about the number of polygons produced with respect to the optimal number. Three functions are provided for producing convex partitions. Two of these functions produce approximately optimal partitions and one results in an optimal partition, where ``optimal'' is defined in terms of the number of partition polygons. The two functions that implement approximation algorithms are guaranteed to produce no more than four times the optimal number of convex pieces. The optimal partitioning function provides an implementation of Greene's dynamic programming algorithm [Gre83], which requires O(n4) time and O(n3) space to produce a convex partitioning. One of the approximation algorithms is also due to Greene [Gre83] and requires O(n log n) time and O(n) space to produce a convex partitioning given a y-monotone partitioning. The other approximation algorithm is a result of Hertel and Mehlhorn [HM83], which requires O(n) time and space to produce a convex partitioning from a triangulation of a polygon. Each of the partitioning functions uses a traits class to supply the primitive types and predicates used by the algorithms.

Assertions

The assertion flags for this package use PARTITION in their names (e.g., CGAL_PARTITION_NO_POSTCONDITIONS). The precondition checks for the planar polygon partitioning functions are: counterclockwise ordering of the input vertices and simplicity of the polygon these vertices represent. The postcondition checks are: simplicity, counterclockwise orientation, and convexity (or y-monotonicity) of the partition polygons and validity of the partition (i.e., the partition polygons are nonoverlapping and the union of these polygons is the same as the original polygon) .

22.4   Classified Reference Pages

Concepts

ConvexPartitionIsValidTraits_2
IsYMonotoneTraits_2
OptimalConvexPartitionTraits_2
PartitionTraits_2
PartitionIsValidTraits_2
YMonotonePartitionIsValidTraits_2
YMonotonePartitionTraits_2

Function Object Concepts

PolygonIsValid

Classes

Partition_is_valid_traits_2<Traits, PolygonIsValid>
Partition_traits_2<R>

Function Object Classes

Is_convex_2<Traits>
Is_vacuously_valid<Traits>
Is_y_monotone_2<Traits>

Functions

approx_convex_partition_2
convex_partition_is_valid_2
greene_approx_convex_partition_2
is_y_monotone_2
optimal_convex_partition_2
partition_is_valid_2
y_monotone_partition_2
y_monotone_partition_is_valid_2

22.5   Alphabetical List of Reference Pages