\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)

CGAL 4.3 - Optimal Distances

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Optimal Distances Reference

##

Kaspar Fischer, Bernd Gärtner, Thomas Herrmann, Michael Hoffmann, and Sven Schönherr
This package provides algorithms for computing the distance between the convex hulls of two point sets in d-dimensional space, without explicitely constructing the convex hulls. It further provides an algorithm to compute the width of a point set, and the furthest point for each vertex of a convex polygon.

Introduced in: CGAL 1.1
BibTeX: cgal:fghhs-od-13b
License: GPL

## Assertions

The optimization code uses infix `OPTIMISATION` in the assertions, e.g. defining the compiler flag `CGAL_OPTIMISATION_NO_PRECONDITIONS` switches precondition checking off, cf. Section Checks.

## Classified Reference Pages

## All furthest neighbors

- `CGAL::all_furthest_neighbors_2`
- `AllFurthestNeighborsTraits_2`

## Width

- `CGAL::Width_3<Traits>`
- `CGAL::Width_default_traits_3<K>`
- `WidthTraits_3`

## Polytope Distance

- `CGAL::Polytope_distance_d<Traits>`
- `CGAL::Polytope_distance_d_traits_2<K,ET,NT>`
- `CGAL::Polytope_distance_d_traits_3<K,ET,NT>`
- `CGAL::Polytope_distance_d_traits_d<K,ET,NT>`
- `PolytopeDistanceDTraits`

## Modules
| Concepts

## Classes
class | CGAL::Polytope_distance_d< Traits >
| An object of the class `Polytope_distance_d` represents the (squared) distance between two convex polytopes, given as the convex hulls of two finite point sets in \( d\)-dimensional Euclidean space \( \E^d\). More...
class | CGAL::Polytope_distance_d_traits_2< K, ET, NT >
| The class `Polytope_distance_d_traits_2` is a traits class for the \( d\)-dimensional optimisation algorithms using the two-dimensional CGAL kernel. More...
class | CGAL::Polytope_distance_d_traits_3< K, ET, NT >
| The class `Polytope_distance_d_traits_3` is a traits class for the \( d\)-dimensional optimisation algorithms using the three-dimensional CGAL kernel. More...
class | CGAL::Polytope_distance_d_traits_d< K, ET, NT >
| The class `Polytope_distance_d_traits_d` is a traits class for the \( d\)-dimensional optimisation algorithms using the \( d\)-dimensional CGAL kernel. More...
class | CGAL::Width_3< Traits >
| Given a set of points \( \mathcal{S}=\left\{p_1,\ldots , p_n\right\}\) in \( \mathbb{R}^3\). More...
class | CGAL::Width_default_traits_3< K >
| The class `Width_default_traits_3` is a traits class for `Width_3<Traits>` using the three-dimensional CGAL kernel. More...

## Functions
OutputIterator | CGAL::all_furthest_neighbors_2 (RandomAccessIterator points_begin, RandomAccessIterator points_end, OutputIterator o, Traits t=Default_traits)
| computes all furthest neighbors for the vertices of the convex polygon described by the range [`points_begin`, `points_end`), writes their indices (relative to `points_begin`) to `o` the furthest neighbor of `points_begin[i]` is `points_begin[i-th number written to o]` and returns the past-the-end iterator of this sequence. More...