### Contents

- Boost.Polygon Main Page
- Design Overview
- Isotropy
- Coordinate Concept
- Interval Concept
- Point Concept
- Segment Concept
- Rectangle Concept
- Polygon 90 Concept
- Polygon 90 With Holes Concept
- Polygon 45 Concept
- Polygon 45 With Holes Concept
- Polygon Concept
- Polygon With Holes Concept
- Polygon 90 Set Concept
- Polygon 45 Set Concept
- Polygon Set Concept
- Connectivity Extraction 90
- Connectivity Extraction 45
- Connectivity Extraction
- Property Merge 90
- Property Merge 45
- Property Merge
- Voronoi Main Page
- Voronoi Benchmark
- Voronoi Builder
- Voronoi Diagram

### Other Resources

- GTL Boostcon 2009 Paper
- GTL Boostcon 2009 Presentation
- Performance Analysis
- Layout Versus Schematic Tutorial
- Minkowski Sum Tutorial
- Voronoi Basic Tutorial
- Voronoi Advanced Tutorial

### Polygon Sponsor

# THE BOOST POLYGON VORONOI EXTENSIONS

The Voronoi extensions of the Boost Polygon library provide functionality to construct a Voronoi diagram of a set of points and linear segments in 2D space with the following set of limitations:

- Coordinates of the input points and endpoints of the input segments should have integral type. The int32 data type is supported by the default implementation. Support for the other data types (e.g. int64) could be achieved through the configuration of the coordinate type traits (Voronoi Advanced tutorial).
- Input points and segments should not overlap except their endpoints. This means that input point should not lie inside the input segment and input segments should not intersect except their endpoints.

While the first restriction is permanent (it allows to give the exact warranties about the output precision and algorithm execution flow), the second one may be resolved using the Boost.Polygon segment utils. The strong sides of the library and the main benefits comparing to the other implementations are discussed in the following paragraphs.

## Fully Functional with Segments

There are just a few implementations of the Voronoi diagram construction algorithm that can handle input data sets that contain linear segments, even considering the commercial libraries. Support of the segments allows to discretize any input geometry (sampled floating-point coordinates can be scaled and snapped to the integer grid): circle, ellipse, parabola. This functionality allows to compute the medial axis transform of the arbitrary set of input geometries, with direct applications in the computer vision projects.

## Robustness and Efficiency

Robustness issues can be divided onto the two main categories: memory management issues and numeric stability issues. The implementation avoids the first type of the issues using pure STL data structures, thus there is no presence of the new operator in the code. The second category of the problems is resolved using the multiprecision geometric predicates. Even for the commercial libraries, usage of such predicates results in a vast performance slowdown. The library implementation overcomes this by avoiding the multiprecision computations in the 95% of the cases by using the efficient, floating-point based predicates. Such predicates don't produce the correct result always, however the library embeds the relative error arithmetic apparatus to identify such situations and switch to the higher precision predicates when appropriate. As the result, the implementation has a solid performance comparing to the other known libraries (more details in the benchmarks).

## Precision of the Output Structures

The implementation guaranties, that the relative error of the coordinates of the output geometries is at most 64 machine epsilons (6 bits of mantissa, for the IEEE-754 floating-point type), while on average it's slightly lower. This means, that the precision of the output geometries can be increased simply by using a floating-point type with the larger mantissa. The practical point of this statements is explained in the following table:

Output Coordinate Type | Output Coordinate Value | Max Absolute Error | Precise Value Range
double (53 bit mantissa) | 1 | 2-53 * 26 = 2-47 | 1 ± 2-47
double (53 bit mantissa) | 231 | 2-53 * 231 * 26 = 2-16 | 231 ± 2-16
long double (64 bit mantissa) | 1 | 2-64 * 26 = 2-58 | 1 ± 2-58
long double (64 bit mantissa) | 231 | 2-64 * 231 * 26 = 2-27 | 231 ± 2-27

Detailed description of the absolute and relative errors evaluation can be found in the article: "What Every Computer Scientist Should Know About Floating-Point Arithmetic".
During the finalization step the implementation unites Voronoi vertices whose both coordinates are located within the relative error range equal to 128 machine epsilons and removes any Voronoi edges between those. This is the only case, that might cause differences between the algorithm output topology and theoretically precise one, and practically means the following: for a Voronoi diagram of a set of solid bodies inside the Solar System (radius 242 metres) and the long double (64 bit mantissa) output coordinate type the maximum absolute error within the Solar System rectangle will be equal to 2-64 * 242 * 26 = 2-18 metres; as the result, vertices with both coordinates that are within 2-18 metres (8 micrometres or the size of a bacteria) will be considered equal and united.

## Simple Interface

The boost/polygon/ voronoi.hpp library header defines the following static functions to integrate the Voronoi extensions functionality with the Boost.Polygon interfaces:

template <typename Point, typename VB> size_t insert(const Point &point, VB *vb) | Inserts a point into a Voronoi builder data structure. Point type should model the point concept. Returns index of the inserted site.
template <typename PointIterator, typename VB> void insert(PointIterator first, PointIterator last, VB *vb) | Inserts an iterator range of points into a Voronoi builder data structure. Corresponding point type should model the point concept.
template <typename Segment, typename VB> size_t insert(const Segment &segment, VB *vb) | Inserts a segment into a Voronoi builder data structure. Segment type should model the segment concept. Returns index of the inserted site.
template <typename SegmentIterator, typename VB> void insert(SegmentIterator first, SegmentIterator last, VB *vb) | Inserts an iterator range of segments into a Voronoi builder data structure. Corresponding segment type should model the segment concept.
template <typename PointIterator, typename VD> void construct_voronoi(PointIterator first, PointIterator last, VD *vd) | Constructs a Voronoi diagram of a set of points. Corresponding point type should model the point concept. Complexity: O(N * log N), memory usage: O(N), where N is the total number of input points.
template <typename SegmentIterator, typename VD> void construct_voronoi(SegmentIterator first, SegmentIterator last, VD *vd) | Constructs a Voronoi diagram of a set of segments. Corresponding segment type should model the segment concept. Complexity: O(N * log N), memory usage: O(N), where N is the total number of input segments.

template <typename PointIterator,
typename SegmentIterator,
typename VD>
void construct_voronoi(PointIterator p_first,
PointIterator p_last,
SegmentIterator s_first,
SegmentIterator s_last,
VD *vd)

Constructs a Voronoi diagram of a set of points and segments.
Corresponding point type should model the point concept.
Corresponding segment type should model the segment concept.
Complexity: O(N* log N), memory usage: O(N), where N is the total number of input points and segments.

The following two lines of code construct a Voronoi diagram of a set of points (as long as the corresponding input geometry type satisfies the Boost.Polygon concept model):
voronoi_diagram<double> vd;
construct_voronoi(points.begin(), points.end(), &vd);
The library provides the clear interfaces to associate the user data with the output geometries and efficiently traverse the Voronoi graph. More details on those topics are covered in the basic Voronoi tutorial. Advanced usage of the library with the configuration of the coordinate types is explained in the advanced Voronoi tutorial. The library allows users to implement their own Voronoi diagram / Delaunay triangulation construction routines based on the Voronoi builder API.

## No Third Party Dependencies

The Voronoi extensions of the Boost Polygon library doesn't depend on any 3rd party code and contains single dependency on the Boost libraries: boost/cstdint.hpp. All the required multiprecision types and related functionality are encapsulated as part of the implementation. The library is fast to compile (3 public and 4 private heades), has strong cohesion between its components and is clearly modularized from the rest of the Boost Polygon library, with the optional integration through the voronoi.hpp header.

## Extensible for the User Provided Coordinate Types

The implementation is coordinate type agnostic. As long as the user provided types satisfy the set of the requirements of the Voronoi builder coordinate type traits, no additional changes are required neither to the algorithm, nor to the implementation of the predicates. For example, it's possible to construct the Voronoi diagram with the 256-bit integer input coordinate type and 512-bit output floating-point type without making any changes to the library.

## Future Development

Below one may find the list of the possible directions for the future development of the library.

- Delaunay triangulation data structure implementation.
- Medial axis data structure implementation.
- Serialization utilities for the Voronoi diagram data structure.
- Drop the restriction on the non-intersecting input geometries.
- Support for the different types of the distance metrics.

## Theoretical Research

The Voronoi extensions were developed as part of the Google Summer of Code 2010. The library was actively maintained for the last three years and involved deep mathematical research in the field of algorithms, data structures, relative error arithmetic and numerical robustness. Upon the community request, more details on the theoretical aspects of the implementation will be published. The authors would like to acknowledge the Steven Fortune's article "A Sweepline algorithm for Voronoi diagrams", that covers fundamental ideas of the current implementation.

Copyright: | Copyright © Andrii Sydorchuk 2010-2013.
License: | Distributed under the Boost Software License, Version 1.0. (See accompanying file `LICENSE_1_0.txt` or copy at http://www.boost.org/LICENSE_1_0.txt)