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

CGAL 4.8 - Bounding Volumes

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MinSphereOfSpheresTraits Concept Reference
Concepts

## Definition

A model of concept `MinSphereOfSpheresTraits` must provide the following constants, types, predicates and operations.

Has Models:

  `CGAL::Min_sphere_of_spheres_d_traits_2<K,FT,UseSqrt,Algorithm>`

  `CGAL::Min_sphere_of_spheres_d_traits_3<K,FT,UseSqrt,Algorithm>`

  `CGAL::Min_sphere_of_spheres_d_traits_d<K,FT,Dim,UseSqrt,Algorithm>`

## Constants
typedef unspecified_type | D
| specifies the dimension of the spheres you want to compute the minsphere of.

## Types
typedef unspecified_type | Sphere
| is a typedef to to some class representing a sphere. More...
typedef unspecified_type | FT
| is a (exact or inexact) field number type. More...
typedef unspecified_type | Cartesian_const_iterator
| non-mutable model of the STL concept `ForwardIterator` with value type `FT`. More...
typedef unspecified_type | Use_square_roots
| must typedef to either `CGAL::Tag_true` or `CGAL::Tag_false`. More...
typedef unspecified_type | Algorithm
| selects the method to compute the minsphere with. More...

## Access Functions
FT | radius (const Sphere &s)
| returns the radius of sphere `s`. More...
Cartesian_const_iterator | center_cartesian_begin (const Sphere &s)
| returns an iterator referring to the first of the `D` Cartesian coordinates of the center of `s`.

## Member Typedef Documentation

typedef unspecified_type MinSphereOfSpheresTraits::Algorithm

selects the method to compute the minsphere with.

It must typedef to either `CGAL::Default_algorithm`, `CGAL::LP_algorithm` or `CGAL::Farthest_first_heuristic`. The recommended choice is the first, which is a synonym to the one of the other two methods which we consider "the best in practice." In case of `CGAL::LP_algorithm`, the minsphere will be computed using the LP-algorithm [11], which in our implementation has maximal expected running time \( O(2^d n)\) (in the number of operations on the number type `FT`). In case of `CGAL::Farthest_first_heuristic`, a simple heuristic will be used instead which seems to work fine in practice, but comes without a guarantee on the running time. For an inexact number type `FT` we strongly recommend `CGAL::Default_algorithm`, or, if you want, `CGAL::Farthest_first_heuristic`, since these handle most degeneracies in a satisfying manner. Notice that this compile-time flag is taken as a hint only. Should one of the methods not be available anymore in a future release, then the default algorithm will be chosen.

typedef unspecified_type MinSphereOfSpheresTraits::Cartesian_const_iterator

non-mutable model of the STL concept `ForwardIterator` with value type `FT`.

Used to access the center coordinates of a sphere.

typedef unspecified_type MinSphereOfSpheresTraits::FT

is a (exact or inexact) field number type.

Template Parameters

  FT | must either be `double` or `float`, or an exact field number type. (An exact number type is one which evaluates arithmetic expressions involving the four basic operations and comparisions with infinite precision, that is, like in \( \mathbb{R}\).)

typedef unspecified_type MinSphereOfSpheresTraits::Sphere

is a typedef to to some class representing a sphere.

(The package will compute the minsphere of spheres of type `Sphere`.) The type `Sphere` must provide a copy constructor.

typedef unspecified_type MinSphereOfSpheresTraits::Use_square_roots

must typedef to either `CGAL::Tag_true` or `CGAL::Tag_false`.

The algorithm uses (depending on the type `MinSphereOfSpheresTraits::Algorithm`) floating-point arithmetic internally for some intermediate computations. The type `Use_square_roots` affects how these calculations are done: When `Use_square_roots` is `Tag_true`, the algorithm computing the minsphere will perform square-root operations on `double`s and `float`s where appropriate. On the other hand, if `Use_square_roots` is `CGAL::Tag_false`, the algorithm will work without doing square-roots.

Note: On some platforms the algorithm is much faster when square-roots are disabled (due to lacking hardware support).

## Member Function Documentation

FT MinSphereOfSpheresTraits::radius | ( | const Sphere & | s) |  |

returns the radius of sphere `s`.

Postcondition
  The returned number is greater or equal to \( 0\).