#include <CGAL/Min_circle_2.h>

Definition

An object of the class Min_circle_2 is the unique circle of smallest area enclosing a finite (multi)set of points in two-dimensional Euclidean space \( \E^2\).

For a point set \( P\) we denote by \( mc(P)\) the smallest circle that contains all points of \( P\). Note that \( mc(P)\) can be degenerate, i.e. \( mc(P)=\emptyset\) if \( P=\emptyset\) and \( mc(P)=\{p\}\) if \( P=\{p\}\).

An inclusion-minimal subset \( S\) of \( P\) with \( mc(S)=mc(P)\) is called a support set, the points in \( S\) are the support points. A support set has size at most three, and all its points lie on the boundary of \( mc(P)\). In general, neither the support set nor its size are necessarily unique.

The underlying algorithm can cope with all kinds of input, e.g. \( P\) may be empty or points may occur more than once. The algorithm computes a support set \( S\) which remains fixed until the next insert or clear operation.

Please note: This class is (almost) obsolete. The class CGAL::Min_sphere_of_spheres_d<Traits> solves a more general problem and is faster then Min_circle_2 even if used only for points in two dimensions as input. Most importantly, CGAL::Min_sphere_of_spheres_d<Traits> has a specialized implementation for floating-point arithmetic which ensures correct results in a large number of cases (including highly degenerate ones). In contrast, Min_circle_2 is not tuned for floating-point computations. The only advantage of Min_circle_2 over CGAL::Min_sphere_of_spheres_d<Traits> is that the former can deal with points in homogeneous coordinates, in which case the algorithm is division-free. Thus, Min_circle_2 might still be an option in case your input number type cannot (efficiently) divide.

Template Parameters

Traits must be a model for MinCircle2Traits.

We provide the model CGAL::Min_circle_2_traits_2 using the two-dimensional CGAL kernel.

See Also: CGAL::Min_ellipse_2<Traits>; CGAL::Min_sphere_d<Traits>; CGAL::Min_sphere_of_spheres_d<Traits>; CGAL::Min_circle_2_traits_2<K>; MinCircle2Traits

Implementation

We implement the incremental algorithm of Welzl, with move-to-front heuristic [16]. The whole implementation is described in [5].

If randomization is chosen, the creation time is almost always linear in the number of points. Access functions and predicates take constant time, inserting a point might take up to linear time, but substantially less than computing the new smallest enclosing circle from scratch. The clear operation and the check for validity each takes linear time.

Example

To illustrate the creation of Min_circle_2 and to show that randomization can be useful in certain cases, we give an example.

File Min_circle_2/min_circle_2.cpp

#include <CGAL/Exact_predicates_exact_constructions_kernel.h>
#include <CGAL/Min_circle_2.h>
#include <CGAL/Min_circle_2_traits_2.h>
#include <iostream>
// typedefs
typedef  CGAL::Exact_predicates_exact_constructions_kernel K;
typedef  CGAL::Min_circle_2_traits_2<K>  Traits;
typedef  CGAL::Min_circle_2<Traits>      Min_circle;
typedef  K::Point_2                      Point;
int
main( int, char**)
{
    const int n = 100;
    Point P[n];
    for ( int i = 0; i < n; ++i)
    P[ i] = Point( (i%2 == 0 ? i : -i), 0);
    // (0,0), (-1,0), (2,0), (-3,0), ...
    Min_circle  mc1( P, P+n, false);    // very slow
    Min_circle  mc2( P, P+n, true);     // fast
    CGAL::set_pretty_mode( std::cout);
    std::cout << mc2;
    return 0;
}

Examples:: Min_circle_2/min_circle_2.cpp.

Related Functions
(Note that these are not member functions.)
std::ostream &	operator<< (std::ostream &os, const Min_circle_2< Traits > &min_circle)
	writes `min_circle` to output stream `os`. More...

std::istream &	operator>> (std::istream &is, Min_circle_2< Traits > min_circle &)
	reads `min_circle` from input stream `is`. More...

Types
typedef unspecified_type	Point
	typedef to `Traits::Point`.

typedef unspecified_type	Circle
	typedef to `Traits::Circle`.

typedef unspecified_type	Point_iterator
	non-mutable model of the STL concept BidirectionalIterator with value type `Point`. More...

typedef unspecified_type	Support_point_iterator
	non-mutable model of the STL concept RandomAccessIterator with value type `Point`. More...

Creation
A `Min_circle_2` object can be created from an arbitrary point set \( P\) and by specialized construction methods expecting no, one, two or three points as arguments. The latter methods can be useful for reconstructing \( mc(P)\) from a given support set \( S\) of \( P\).
template<class InputIterator >
	Min_circle_2 (InputIterator first, InputIterator last, bool randomize, Random &random=CGAL::default_random, const Traits &traits=Traits())
	initializes `min_circle` to \( mc(P)\) with \( P\) being the set of points in the range [`first`,`last`). More...

	Min_circle_2 (const Traits &traits=Traits())
	initializes `min_circle` to \( mc(\emptyset)\), the empty set. More...

	Min_circle_2 (const Point &p, const Traits &traits=Traits())
	initializes `min_circle` to \( mc(\{p\})\), the set \( \{p\}\). More...

	Min_circle_2 (const Point &p1, const Point &p2, const Traits &traits=Traits())
	initializes `min_circle` to \( mc(\{p1,p2\})\), the circle with diameter equal to the segment connecting \( p1\) and \( p2\).

	Min_circle_2 (const Point &p1, const Point &p2, const Point &p3, const Traits &traits=Traits())
	initializes `min_circle` to \( mc(\{p1,p2,p3\})\).

Access Functions
int	number_of_points () const
	returns the number of points of `min_circle`, i.e. \( \|P\|\).

int	number_of_support_points () const
	returns the number of support points of `min_circle`, i.e. \( \|S\|\).

Point_iterator	points_begin () const
	returns an iterator referring to the first point of `min_circle`.

Point_iterator	points_end () const
	returns the corresponding past-the-end iterator.

Support_point_iterator	support_points_begin () const
	returns an iterator referring to the first support point of `min_circle`.

Support_point_iterator	support_points_end () const
	returns the corresponding past-the-end iterator.

const Point &	support_point (int i) const
	returns the `i`-th support point of `min_circle`. More...

const Circle &	circle () const
	returns the current circle of `min_circle`.

Predicates
By definition, an empty `Min_circle_2` has no boundary and no bounded side, i.e. its unbounded side equals the whole space \( \E^2\).
CGAL::Bounded_side	bounded_side (const Point &p) const
	returns `CGAL::ON_BOUNDED_SIDE`, `CGAL::ON_BOUNDARY`, or `CGAL::ON_UNBOUNDED_SIDE` iff `p` lies properly inside, on the boundary of, or properly outside of `min_circle`, resp.

bool	has_on_bounded_side (const Point &p) const
	returns `true`, iff `p` lies properly inside `min_circle`.

bool	has_on_boundary (const Point &p) const
	returns `true`, iff `p` lies on the boundary of `min_circle`.

bool	has_on_unbounded_side (const Point &p) const
	returns `true`, iff `p` lies properly outside of `min_circle`.

bool	is_empty () const
	returns `true`, iff `min_circle` is empty (this implies degeneracy).

bool	is_degenerate () const
	returns `true`, iff `min_circle` is degenerate, i.e. if `min_circle` is empty or equal to a single point, equivalently if the number of support points is less than 2.

Modifiers
New points can be added to an existing `min_circle`, allowing to build \( mc(P)\) incrementally, e.g. if \( P\) is not known in advance. Compared to the direct creation of \( mc(P)\), this is not much slower, because the construction method is incremental itself.
void	insert (const Point &p)
	inserts `p` into `min_circle` and recomputes the smallest enclosing circle.

template<class InputIterator >
void	insert (InputIterator first, InputIterator last)
	inserts the points in the range [`first`,`last`) into `min_circle` and recomputes the smallest enclosing circle by calling `insert(p)` for each point `p` in [`first`,`last`). More...

void	clear ()
	deletes all points in `min_circle` and sets `min_circle` to the empty set. More...

Validity Check
An object `min_circle` is valid, iff `min_circle` contains all points of its defining set \( P\), `min_circle` is the smallest circle spanned by its support set \( S\), and \( S\) is minimal, i.e. no support point is redundant.
bool	is_valid (bool verbose=false, int level=0) const
	returns `true`, iff `min_circle` 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_circle_2< Traits >::Point_iterator

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

Used to access the points of the smallest enclosing circle.

template<typename Traits>

typedef unspecified_type CGAL::Min_circle_2< Traits >::Support_point_iterator

non-mutable model of the STL concept RandomAccessIterator with value type Point.

Used to access the support points of the smallest enclosing circle.

Constructor & Destructor Documentation

template<typename Traits>

template<class InputIterator >

CGAL::Min_circle_2< Traits >::Min_circle_2	(	InputIterator	first,
		InputIterator	last,
		bool	randomize,
		Random &	random = `CGAL::default_random`,
		const Traits &	traits = `Traits()`
	)

initializes min_circle to \( mc(P)\) with \( P\) being the set of points in the range [first,last).

If randomize is true, a random permutation of \( P\) is computed in advance, using the random numbers generator random. Usually, this will not be necessary, however, the algorithm's efficiency depends on the order in which the points are processed, and a bad order might lead to extremely poor performance (see example below).

Requires:: The value type of first and last is Point.

template<typename Traits>

CGAL::Min_circle_2< Traits >::Min_circle_2 ( const Traits & traits = Traits())

initializes min_circle to \( mc(\emptyset)\), the empty set.

Postcondition: min_circle.is_empty() = true.

template<typename Traits>

CGAL::Min_circle_2< Traits >::Min_circle_2	(	const Point &	p,
		const Traits &	traits = `Traits()`
	)

initializes min_circle to \( mc(\{p\})\), the set \( \{p\}\).

Postcondition: min_circle.is_degenerate() = true.

Member Function Documentation

template<typename Traits>

void CGAL::Min_circle_2< Traits >::clear ( )

deletes all points in min_circle and sets min_circle to the empty set.

Postcondition: min_circle.is_empty() = true.

template<typename Traits>

template<class InputIterator >

void CGAL::Min_circle_2< Traits >::insert	(	InputIterator	first,
		InputIterator	last
	)

inserts the points in the range [first,last) into min_circle and recomputes the smallest enclosing circle by calling insert(p) for each point p in [first,last).

Requires:: The value type of first and last is Point.

template<typename Traits>

bool CGAL::Min_circle_2< Traits >::is_valid	(	bool	verbose = `false`,
		int	level = `0`
	)		const

returns true, iff min_circle is valid.

If verbose is true, some messages concerning the performed checks are written to standard error stream. The second parameter level is not used, we provide it only for consistency with interfaces of other classes.

template<typename Traits>

const Point& CGAL::Min_circle_2< Traits >::support_point ( int i) const

returns the i-th support point of min_circle.

Between two modifying operations (see below) any call to min_circle.support_point(i) with the same i returns the same point.

Precondition: \( 0 \leq i< \) min_circle.number_of_support_points().

Friends And Related Function Documentation

template<typename Traits>

std::ostream & operator<<	(	std::ostream &	os,
		const Min_circle_2< Traits > &	min_circle
	)

Requires:: The output operator is defined for Point (and for Circle, if pretty printing is used).

template<typename Traits>

std::istream & operator>>	(	std::istream &	is,
		Min_circle_2< Traits > min_circle &
	)

Requires:: The input operator is defined for Point.

Definition

Related Functions

Types

Creation

Access Functions

Predicates

Modifiers

Validity Check

Miscellaneous

Member Typedef Documentation

Constructor & Destructor Documentation

Member Function Documentation

Friends And Related Function Documentation