CGAL 5.4 - Bounding Volumes
CGAL::Min_sphere_of_spheres_d< Traits > Class Template Reference

#include <CGAL/Min_sphere_of_spheres_d.h>

## Definition

An object of the class Min_sphere_of_spheres_d is a data structure that represents the unique sphere of smallest volume enclosing a finite set of spheres in $$d$$-dimensional Euclidean space $$\E^d$$.

For a set $$S$$ of spheres we denote by $$ms(S)$$ the smallest sphere that contains all spheres of $$S$$; we call $$ms(S)$$ the minsphere of $$S$$. $$ms(S)$$ can be degenerate, i.e., $$ms(S)=\emptyset$$, if $$S=\emptyset$$ and $$ms(S)=\{s\}$$, if $$S=\{s\}$$. Any sphere in $$S$$ may be degenerate, too, i.e., any sphere from $$S$$ may be a point. Also, $$S$$ may contain several copies of the same sphere.

An inclusion-minimal subset $$R$$ of $$S$$ with $$ms(R)=ms(S)$$ is called a support set for $$ms(S)$$; the spheres in $$R$$ are the support spheres. A support set has size at most $$d+1$$, and all its spheres lie on the boundary of $$ms(S)$$. (A sphere $$s'$$ is said to lie on the boundary of a sphere $$s$$, if $$s'$$ is contained in $$s$$ and if their boundaries intersect.) In general, the support set is not unique.

The algorithm computes the center and the radius of $$ms(S)$$, and finds a support set $$R$$ (which remains fixed until the next insert(), clear() or set() operation). We also provide a specialization of the algorithm for the case when the center coordinates and radii of the input spheres are floating-point numbers. This specialized algorithm uses floating-point arithmetic only, is very fast and especially tuned for stability and robustness. Still, it's output may be incorrect in some (rare) cases; termination is guaranteed.

When default constructed, an instance of type Min_sphere_of_spheres_d<Traits> represents the set $$S=\emptyset$$, together with its minsphere $$ms(S)=\emptyset$$. You can add spheres to the set $$S$$ by calling insert(). Querying the minsphere is done by calling the routines is_empty(), radius() and center_cartesian_begin(), among others.

In general, the radius and the Euclidean center coordinates of $$ms(S)$$ need not be rational. Consequently, the algorithm computing the exact minsphere will have to deal with algebraic numbers. Fortunately, both the radius and the coordinates of the minsphere are numbers of the form $$a_i+b_i\sqrt{t}$$, where $$a_i,b_i,t\in \Q$$ and where $$t\ge 0$$ is the same for all coordinates and the radius. Thus, the exact minsphere can be described by the number $$t$$, which is called the sphere's discriminant, and by $$d+1$$ pairs $$(a_i,b_i)\in\Q^2$$ (one for the radius and $$d$$ for the center coordinates).

Note
This class (almost) replaces CGAL::Min_sphere_d<Traits>, which solves the less general problem of finding the smallest enclosing ball of a set of points. Min_sphere_of_spheres_d is faster than CGAL::Min_sphere_d<Traits>, and in contrast to the latter provides a specialized implementation for floating-point arithmetic which ensures correct results in a large number of cases (including highly degenerate ones). The only advantage of CGAL::Min_sphere_d<Traits> over Min_sphere_of_spheres_d is that the former can deal with points in homogeneous coordinates, in which case the algorithm is division-free. Thus, CGAL::Min_sphere_d<Traits> might still be an option in case your input number type cannot (efficiently) divide.
Template Parameters
 Traits must be a model of the concept MinSphereOfSpheresTraits as its template argument.
CGAL::Min_sphere_d<Traits>
CGAL::Min_circle_2<Traits>

Implementation

We implement two algorithms, the LP-algorithm and a heuristic [11]. As described in the documentation of concept MinSphereOfSpheresTraits, each has its advantages and disadvantages: Our implementation of the LP-algorithm has maximal expected running time $$O(2^d n)$$, while the heuristic comes without any complexity guarantee. In particular, the LP-algorithm runs in linear time for fixed dimension $$d$$. (These running times hold for the arithmetic model, so they count the number of operations on the number type FT.)

On the other hand, the LP-algorithm is, for inexact number types FT, much worse at handling degeneracies and should therefore not be used in such a case. (For exact number types FT, both methods handle all kinds of degeneracies.)

Currently, we require Traits::FT to be either an exact number type or double or float; other inexact number types are not supported at this time. Also, the current implementation only handles spheres with Cartesian coordinates; homogenous representation is not supported yet.

Example

// Computes the minsphere of some random spheres.
#include <CGAL/Cartesian_d.h>
#include <CGAL/Random.h>
#include <CGAL/Exact_rational.h>
#include <CGAL/Min_sphere_of_spheres_d.h>
#include <vector>
const int N = 1000; // number of spheres
const int D = 3; // dimension of points
const int LOW = 0, HIGH = 10000; // range of coordinates and radii
typedef CGAL::Exact_rational FT;
//typedef double FT;
typedef CGAL::Cartesian_d<FT> K;
typedef K::Point_d Point;
typedef Traits::Sphere Sphere;
int main () {
std::vector<Sphere> S; // n spheres
FT coord[D]; // d coordinates
CGAL::Random r; // random number generator
for (int i=0; i<N; ++i) {
for (int j=0; j<D; ++j)
coord[j] = r.get_int(LOW,HIGH);
Point p(D,coord,coord+D); // random center...
}
Min_sphere ms(S.begin(),S.end()); // check in the spheres
CGAL_assertion(ms.is_valid());
}
Examples:
Min_circle_2/min_circle_2.cpp, Min_sphere_d/min_sphere_3.cpp, Min_sphere_of_spheres_d/benchmark.cpp, Min_sphere_of_spheres_d/min_sphere_of_spheres_d_2.cpp, Min_sphere_of_spheres_d/min_sphere_of_spheres_d_3.cpp, and Min_sphere_of_spheres_d/min_sphere_of_spheres_d_d.cpp.

## Types

typedef unspecified_type Sphere
is a typedef to Traits::Sphere.

typedef unspecified_type FT
is a typedef to Traits::FT.

typedef unspecified_type Result
is the type of the radius and of the center coordinates of the computed minsphere: When FT is an inexact number type (double, for instance), then Result is simply FT. More...

typedef unspecified_type Algorithm
is either CGAL::LP_algorithm or CGAL::Farthest_first_heuristic. More...

typedef unspecified_type Support_iterator
non-mutable model of the STL concept BidirectionalIterator with value type Sphere. More...

typedef unspecified_type Cartesian_const_iterator
non-mutable model of the STL concept BidirectionalIterator to access the center coordinates of the minsphere.

## Creation

Min_sphere_of_spheres_d (const Traits &traits=Traits())
creates a variable of type Min_sphere_of_spheres_d and initializes it to $$ms(\emptyset)$$. More...

template<typename InputIterator >
Min_sphere_of_spheres_d (InputIterator first, InputIterator last, const Traits &traits=Traits())
creates a variable minsphere of type Min_sphere_of_spheres_d and inserts (cf. More...

## Access Functions

Support_iterator support_begin () const
returns an iterator referring to the first support sphere of minsphere.

Support_iterator support_end () const
returns the corresponding past-the-end iterator.

const FTdiscriminant () const
returns the discriminant of minsphere. More...

returns the radius of minsphere. More...

Cartesian_const_iterator center_cartesian_begin () const
returns a const-iterator to the first of the Traits::D center coordinates of minsphere. More...

Cartesian_const_iterator center_cartesian_end () const
returns the corresponding past-the-end iterator, i.e. More...

## Predicates

bool is_empty () const
returns true, iff minsphere is empty, i.e. iff $$ms(S)=\emptyset$$.

## Modifiers

void clear ()
resets minsphere to $$ms(\emptyset)$$, with $$S:= \emptyset$$.

template<class InputIterator >
void set (InputIterator first, InputIterator last)
sets minsphere to the $$ms(S)$$, where $$S$$ is the set of spheres in the range [first,last). More...

void insert (const Sphere &s)
inserts the sphere s into the set $$S$$ of instance minsphere.

template<class InputIterator >
void insert (InputIterator first, InputIterator last)
inserts the spheres in the range [first,last) into the set $$S$$ of instance minsphere. More...

## Validity Check

An object minsphere is valid, iff

• minsphere contains all spheres of its defining set $$S$$,
• minsphere is the smallest sphere containing its support set $$R$$, and
• $$R$$ is minimal, i.e., no support sphere is redundant.
bool is_valid () const
returns true, iff minsphere is valid. More...

## Miscellaneous

const Traits & traits () const
returns a const reference to the traits class object.

## ◆ Algorithm

template<typename Traits>
 typedef unspecified_type CGAL::Min_sphere_of_spheres_d< Traits >::Algorithm

is either CGAL::LP_algorithm or CGAL::Farthest_first_heuristic.

As is described in the documentation of concept MinSphereOfSpheresTraits, the type Algorithm reflects the method which is used to compute the minsphere. (Normally, Algorithm coincides with Traits::Algorithm. However, if the method Traits::Algorithm should not be supported anymore in a future release, then Algorithm will have another type.)

## ◆ Result

template<typename Traits>
 typedef unspecified_type CGAL::Min_sphere_of_spheres_d< Traits >::Result

is the type of the radius and of the center coordinates of the computed minsphere: When FT is an inexact number type (double, for instance), then Result is simply FT.

However, when FT is an exact number type, then Result is a typedef to a derived class of std::pair<FT,FT>; an instance of this type represents the number $$a+b\sqrt{t}$$, where $$a$$ is the first and $$b$$ the second element of the pair and where the number $$t$$ is accessed using the member function discriminant() of class Min_sphere_of_spheres_d<Traits>.

## ◆ Support_iterator

template<typename Traits>
 typedef unspecified_type CGAL::Min_sphere_of_spheres_d< Traits >::Support_iterator

non-mutable model of the STL concept BidirectionalIterator with value type Sphere.

Used to access the support spheres defining the smallest enclosing sphere.

## ◆ Min_sphere_of_spheres_d() [1/2]

template<typename Traits>
 CGAL::Min_sphere_of_spheres_d< Traits >::Min_sphere_of_spheres_d ( const Traits & traits = Traits() )

creates a variable of type Min_sphere_of_spheres_d and initializes it to $$ms(\emptyset)$$.

If the traits parameter is not supplied, the class Traits must provide a default constructor.

## ◆ Min_sphere_of_spheres_d() [2/2]

template<typename Traits>
template<typename InputIterator >
 CGAL::Min_sphere_of_spheres_d< Traits >::Min_sphere_of_spheres_d ( InputIterator first, InputIterator last, const Traits & traits = Traits() )

creates a variable minsphere of type Min_sphere_of_spheres_d and inserts (cf.

insert()) the spheres from the range [first,last).

Template Parameters
 InputIterator is a model of InputIterator with Sphere as value type. If the traits parameter is not supplied, the class Traits must provide a default constructor.

## ◆ center_cartesian_begin()

template<typename Traits>
 Cartesian_const_iterator CGAL::Min_sphere_of_spheres_d< Traits >::center_cartesian_begin ( ) const

returns a const-iterator to the first of the Traits::D center coordinates of minsphere.

The iterator returns objects of type Result. If FT is an exact number type, then a center coordinate is represented by a pair $$(a,b)$$ describing the real number $$a+b\sqrt{t}$$, where $$t$$ is the minsphere's discriminant (cf. discriminant()).

Precondition
minsphere is not empty.

## ◆ center_cartesian_end()

template<typename Traits>
 Cartesian_const_iterator CGAL::Min_sphere_of_spheres_d< Traits >::center_cartesian_end ( ) const

returns the corresponding past-the-end iterator, i.e.

\ center_cartesian_begin()+Traits::D.

Precondition
minsphere is not empty.

## ◆ discriminant()

template<typename Traits>
 const FT& CGAL::Min_sphere_of_spheres_d< Traits >::discriminant ( ) const

returns the discriminant of minsphere.

This number is undefined when FT is an inexact number type. When FT is exact, the center coordinates and the radius of the minsphere are numbers of the form $$a+b\sqrt{t}$$, where $$t$$ is the discriminant of the minsphere as returned by this function.

Precondition
minsphere is not empty, and FT is an exact number type.

## ◆ insert()

template<typename Traits>
template<class InputIterator >
 void CGAL::Min_sphere_of_spheres_d< Traits >::insert ( InputIterator first, InputIterator last )

inserts the spheres in the range [first,last) into the set $$S$$ of instance minsphere.

Template Parameters
 InputIterator is a model of InputIterator with Sphere as value type.

## ◆ is_valid()

template<typename Traits>
 bool CGAL::Min_sphere_of_spheres_d< Traits >::is_valid ( ) const

returns true, iff minsphere is valid.

When FT is inexact, this routine always returns true.

template<typename Traits>
 Result CGAL::Min_sphere_of_spheres_d< Traits >::radius ( ) const

If FT is an exact number type then the radius of the minsphere is the real number $$a+b\sqrt{t}$$, where $$t$$ is the minsphere's discriminant, $$a$$ is the first and $$b$$ the second component of the pair returned by radius().

Precondition
minsphere is not empty.

## ◆ set()

template<typename Traits>
template<class InputIterator >
 void CGAL::Min_sphere_of_spheres_d< Traits >::set ( InputIterator first, InputIterator last )

sets minsphere to the $$ms(S)$$, where $$S$$ is the set of spheres in the range [first,last).

Template Parameters
 InputIterator is a model of InputIterator with Sphere as value type.