User Manual

Author

Ken Arroyo Ohori

Introduction

This package implements polygon repair methods. Starting from possibly invalid input in the form of a polygon, polygon with holes or multipolygon with holes, the method computes an arrangement of the input edges, labels each face according to what it represents (exterior, polygon interior or hole), and reconstructs the polygon(s) represented by the arrangement. The method returns valid output stored in a multipolygon with holes.

Different labeling heuristics are possible. This package offers the even-odd rule, the non-zero rule, the union rule, as well as the intersection rule. The even-odd rule results in areas that are alternately assigned as polygon interiors and exterior/holes each time that an input edge is passed. It does not distinguish between edges that are part of outer boundaries from those of inner boundaries.

The non-zero rule results in areas with a non-zero winding number.

The union and intersection rules are useful when given two or more similar valid polygons with holes. They compute either their union or their intersection, and they are hence conservative by bounding from the interior or the exterior.

Definitions

• A valid polygon (without holes) is a point set in \( \mathbb{R}^2\) that is bounded by a cycle of linear edges, which is known as its outer boundary. This outer boundary should be simple, meaning that the interiors of its edges are pairwise disjoint and all of its vertices have a degree of two. It is thus topologically equivalent to a disk and is represented internally as the sequence of points at the common end points of the edges around its outer boundary.
• A valid polygon with holes is a point set in \( \mathbb{R}^2\) that is bounded by one outer boundary and zero or more inner boundaries, where each inner boundary represents a hole in the polygon. Considered independently, each boundary should be simple. The different boundaries of a polygon are allowed to intersect tangentially at their common vertices (with no common edges), forming vertices with degrees of a multiple of two the tangential points. The interior of a polygon with holes should form a connected point set. Note that a valid polygon can also be represented as a valid polygon with holes (where the number of holes is zero).
• A valid multipolygon with holes is a point set in \( \mathbb{R}^2\) that is represented by a set of zero or more valid polygons with holes. The interiors of the polygons with holes should be pairwise disjoint, but they are allowed to intersect tangentially at their common vertices. Note that a valid polygon with holes can also be represented as a valid multipolygon with holes (with only one polygon).

Figure 16.1 Valid: (a) polygon, (b-c) polygons with holes, and (d-e) multipolygons with holes. (c) and (e) show cases where boundaries intersect tangentially at a single vertex.

Figure 16.2 Invalid: (a) self-intersecting polygon self-intersection, (b) self-touching polygon, (c-d) polygons with badly nested holes, (e) polygon with hole touching at edge, (f) polygon with hole that separates interior into two parts, (g) multipolygon with overlapping polygons, and (h) multipolygon with polygons that touch at an edge.

Stricter Conditions for Output

The conditions listed above are sufficient to define valid polygons, polygons with holes and multipolygons with holes for most applications. However, in order to ensure unique deterministic output from the repair algorithm, the valid multipolygons with holes returned by the package conform to more strict criteria:

• Adjacent collinear edges touching at vertices of degree two are merged
• The sequence of vertices representing a boundary starts from its lexicographically smallest vertex
• Outer boundaries are oriented counter-clockwise and inner boundaries are oriented clockwise
• The inner boundaries of a polygon with holes are stored in lexicographic order
• The polygons with holes of a multipolygon with holes are also stored in lexicographic order

Even-Odd Rule

Figure 16.3 Examples of polygons with holes (a-d) and multipolygons with holes (e-h) before (left) and after (right) being repaired with the even-odd rule.

Non-Zero Rule

Tbd.

Union and Intersection Rule

Given several valid polygons this rule computes their union or intersection.

Notes on the Output

If the input is already valid, the method will return a valid output representing the same area. However, the output might be different in order to conform to the stricter conditions to generate deterministic output (see Stricter Conditions for Output).

Also, it is worth noting that even the repair of a single polygon without holes but with self-intersections can result in a multipolygon with holes. This is why the repair function will always return a multipolygon with holes. The user can then check whether it consists of a single polygon with holes, and if a polygon with holes has zero holes and extract these if needed.

Implementation

Broadly, the algorithm consists of three steps:

1. Arrangement: the edges in the polygon, polygon with holes or multipolygon with holes are added as edges in the arrangement.
2. Labeling of the faces: the resulting faces are labeled with ids according to what they represent (exterior, polygon interior or hole).
3. Reconstruction of the multipolygon: each boundary is reconstructed, then these are assembled into individual polygons with holes and put into a single multipolygon with holes.

Arrangement

For the purposes of the repair operation, the input polygon, polygon with holes or multipolygon is merely used as a container of input line segments. These line segments are added to the arrangement as edges. Internally, this is done using a constrained triangulation where they are added as constraints.

With the even-odd rule, only the edges that are present an odd number of times in the input will be edges in the final arrangement. When these edges are only partially overlapping, only the parts that overlap an odd number of times will be edges in the final arrangement.

This procedure is done in two steps: 1. preprocessing to eliminate identical edges that are present an even number of times, and 2. adding edges incrementally while applying an even-odd counting mechanism, which erases existing (parts of) edges when new overlapping ones are added.

Labeling

First, the polygon exterior is labeled. For this, all of the faces that can be accessed from the exterior without passing through an edge are labeled as exterior faces.

Then, all other faces are labeled. For the even-odd rule, the label applied alternates between polygon interior and hole every time that an edge is passed.

Reconstruction of the Multipolygon

The algorithm reconstructs the multipolygon boundary by boundary, obtaining counter-clockwise cycles for outer boundaries and clockwise cycles for inner boundaries. Once all boundaries have been reconstructed, the boundaries are assembled into multipolygons using the face labels to know which polygon with holes inner/outer boundaries belong to, and using the orientation to distinguish between the outer and inner boundaries of each polygon with holes.

Examples

Repairing a (Multi)polygon with the Even-Odd Rule

It is possible to repair a polygon, polygon with holes or multipolygon with holes using the even-odd rule by calling the Polygon_repair::repair() function as shown in the following example. This function returns a repaired multipolygon with holes.

File Polygon_repair/repair_polygon_2.cpp

#include <iostream>
#include <fstream>
#include <string>
 
#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Polygon_repair/repair.h>
#include <CGAL/IO/WKT.h>
 
using Kernel = CGAL::Exact_predicates_inexact_constructions_kernel;
using Point_2 = Kernel::Point_2;
using Polygon_2 = CGAL::Polygon_2<Kernel>;
using Polygon_with_holes_2 = CGAL::Polygon_with_holes_2<Kernel>;
using Multipolygon_with_holes_2 = CGAL::Multipolygon_with_holes_2<Kernel>;
 
int main(int argc, char* argv[]) {
 
  const char* filename = (argc > 1) ? argv[1] : "data/bridge-edge.wkt";
 
  std::ifstream in(filename);
  Polygon_with_holes_2 pin;
  CGAL::IO::read_polygon_WKT(in, pin);
 
  Multipolygon_with_holes_2 mp = CGAL::Polygon_repair::repair(pin);
  if (mp.number_of_polygons_with_holes() > 1) {
    CGAL::IO::write_multi_polygon_WKT(std::cout, mp);
  } else {
    CGAL::IO::write_polygon_WKT(std::cout, mp.polygons_with_holes()[0]);
  }
 
  return 0;
}

Performance

The method can repair large invalid polygons of millions of vertices in a few seconds as long as the number of intersections between line segments is limited. This is a realistic assumption with many invalid data sets, which only have relatively minor issues involving a small number of their vertices/edges. However, it is worth noting that there can be a potentially quadratic number of intersection between edges in the worst case, leading to much worse performance since all of these intersections need to be calculated in the overlay.

Polygon	Vertices	Holes	Time
	101973	298	0.652 sec
	43925	125	0.190 sec

History

The polygon repair method as originally developed is described by Ledoux et al. [1] and implemented in the prepair software. This package is a reimplementation of the method with a new approach to label and reconstruct the multipolygons. It also incorporates improvements later added to prepair, such as the application of the even-odd counting heuristics to edges, which enables correct counting even on partially overlapping edges.

Ken Arroyo Ohori developed this package during the Google Summer of Code 2023 mentored by Sébastien Loriot and Andreas Fabri. The GSoC project was limited to the even-odd rule. Further rules were added with CGAL 6.1 by Andreas Fabri.