CGAL::Min_circle_2<Traits>

Definition

An object of the class Min_circle_2<Traits> is the unique circle of smallest area enclosing a finite (multi)set of points in two-dimensional Euclidean space $$ ₂. For a point set $$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)=Ø$$ if $$P=Ø$$ and $$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<Traits> 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<Traits> is not tuned for floating-point computations. The only advantage of Min_circle_2<Traits> 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<Traits> might still be an option in case your input number type cannot (efficiently) divide.

#include <CGAL/Min_circle_2.h>

Requirements

The template parameter Traits is a model for MinCircle2Traits.

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

Types

Creation

A Min_circle_2<Traits> 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.

Access Functions

int	min_circle.number_of_points ()	returns the number of points of min_circle, i.e. $$|P|.
int	min_circle.number_of_support_points ()
		returns the number of support points of min_circle, i.e. $$|S|.
Point_iterator	min_circle.points_begin ()	returns an iterator referring to the first point of min_circle.
Point_iterator	min_circle.points_end ()	returns the corresponding past-the-end iterator.
Support_point_iterator	min_circle.support_points_begin ()
		returns an iterator referring to the first support point of min_circle.
Support_point_iterator	min_circle.support_points_end ()	returns the corresponding past-the-end iterator.
Point	min_circle.support_point ( int i)	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:  $$0 i< min_circle.number_of_support_points()$$.
Circle	min_circle.circle ()	returns the current circle of min_circle.

Predicates

By definition, an empty Min_circle_2<Traits> has no boundary and no bounded side, i.e. its unbounded side equals the whole space $$ 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.

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.

Miscellaneous

I/O

#include <CGAL/IO/Window_stream.h>

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 [Wel91]. The whole implementation is described in [GS98a].

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<Traits> and to show that randomization can be useful in certain cases, we give an example.

File: examples/Min_circle_2/min_circle_2.cpp

#include <CGAL/Homogeneous.h>
#include <CGAL/Min_circle_2.h>
#include <CGAL/Min_circle_2_traits_2.h>
#include <CGAL/Gmpz.h>
#include <iostream>

// typedefs
typedef  CGAL::Gmpz                      NT;
typedef  CGAL::Homogeneous<NT>           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;
}