#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.

See Also: 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

File Min_sphere_of_spheres_d/min_sphere_of_spheres_d_d.cpp

// Computes the minsphere of some random spheres.
// This example illustrates how to use CGAL::Point_d and CGAL::
// Weighted_point with the Min_sphere_of_spheres_d package.
#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 CGAL::Min_sphere_of_spheres_d_traits_d<K,FT,D> Traits;
typedef CGAL::Min_sphere_of_spheres_d<Traits> Min_sphere;
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...
    S.push_back(Sphere(p,r.get_int(LOW,HIGH))); // ...and random radius
  }
  Min_sphere ms(S.begin(),S.end());       // check in the spheres
  CGAL_assertion(ms.is_valid());
}

Examples:: 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 FT &	discriminant () const
	returns the discriminant of `minsphere`. More...

Result	radius () const
	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. `center_cartesian_begin()+Traits::D`. 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.

Member Typedef Documentation

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.)

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>.

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.

Constructor & Destructor Documentation

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.

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.

Member Function Documentation

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.

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.

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.

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.

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

returns the radius of minsphere.

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.

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.

Definition

Types

Creation

Access Functions

Predicates

Modifiers

Validity Check

Miscellaneous

Member Typedef Documentation

Constructor & Destructor Documentation

Member Function Documentation