## CGAL::Min_ellipse_2<Traits>

### Definition

An object of the class Min_ellipse_2<Traits> is the unique ellipse of smallest area enclosing a finite (multi)set of points in two-dimensional euclidean space 2. For a point set P we denote by me(P) the smallest ellipse that contains all points of P. Note that me(P) can be degenerate, i.e. me(P)=Ø if P=Ø, me(P)={p} if P={p}, and me(P) = { (1-l)p + l q | 0 <= l <= 1 } if P={p,q}.

An inclusion-minimal subset S of P with me(S)=me(P) is called a support set, the points in S are the support points. A support set has size at most five, and all its points lie on the boundary of me(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.

#include <CGAL/Min_ellipse_2.h>

### Requirements

The template parameter Traits is a model for MinEllipse2Traits.

We provide the model CGAL::Min_ellipse_2_traits_2<K> using the two-dimensional CGAL kernel.

### Types

Min_ellipse_2<Traits>::Point
| Typedef to Traits::Point .
Min_ellipse_2<Traits>::Ellipse
| Typedef to Traits::Ellipse. If you are using the predefined traits class CGAL::Min_ellipse_2_traits_2<K>, you can access the coefficients of the ellipse, see the documentation of CGAL::Min_ellipse_2_traits_2<K> and the example below.
Min_ellipse_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 ellipse.
Min_ellipse_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 ellipse.

### Creation

A Min_ellipse_2<Traits> object can be created from an arbitrary point set P and by specialized construction methods expecting no, one, two, three, four or five points as arguments. The latter methods can be useful for reconstructing me(P) from a given support set S of P.

template < class InputIterator >

Min_ellipse_2<Traits> min_ellipse ( | InputIterator first, InputIterator last, bool randomize, Random& random = default_random, Traits traits = Traits());

initializes min_ellipse to me(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).
Requirement: The value type of first and last is Point.

Min_ellipse_2<Traits> min_ellipse ( Traits traits = Traits());
| creates a variable min_ellipse of type Min_ellipse_2<Traits>. It is initialized to me(Ø), the empty set. Postcondition: min_ellipse.is_empty() = true.
Min_ellipse_2<Traits> min_ellipse ( Point p, Traits traits = Traits());
| initializes min_ellipse to me({p}), the set {p}. Postcondition: min_ellipse.is_degenerate() = true.

Min_ellipse_2<Traits> min_ellipse ( | Point p, Point q, Traits traits = Traits());

| initializes min_ellipse to me({p,q}), the set { (1-l)p + l q | 0 <= l <= 1 }. Postcondition: min_ellipse.is_degenerate() = true.

Min_ellipse_2<Traits> min_ellipse ( | Point p1, Point p2, Point p3, Traits traits = Traits());

| initializes min_ellipse to me({p1,p2,p3}).

Min_ellipse_2<Traits> min_ellipse ( | Point p1, Point p2, Point p3, Point p4, Traits traits = Traits());

| initializes min_ellipse to me({p1,p2,p3,p4}).

Min_ellipse_2<Traits> min_ellipse ( | Point p1, Point p2, Point p3, Point p4, Point p5, Traits traits = Traits());

| initializes min_ellipse to me({p1,p2,p3,p4,p5}).

### Access Functions

int | min_ellipse.number_of_points ()
|  | returns the number of points of min_ellipse, i.e. |P|.
int | min_ellipse.number_of_support_points ()
|  | returns the number of support points of min_ellipse, i.e. |S|.
Point_iterator | min_ellipse.points_begin ()
|  | returns an iterator referring to the first point of min_ellipse.
Point_iterator | min_ellipse.points_end ()
|  | returns the corresponding past-the-end iterator.
Support_point_iterator
| min_ellipse.support_points_begin ()
|  | returns an iterator referring to the first support point of min_ellipse.
Support_point_iterator
| min_ellipse.support_points_end ()
|  | returns the corresponding past-the-end iterator.
Point | min_ellipse.support_point ( int i)
|  | returns the i-th support point of min_ellipse. Between two modifying operations (see below) any call to min_ellipse.support_point(i) with the same i returns the same point. Precondition: 0 i< min_ellipse.number_of_support_points().
Ellipse | min_ellipse.ellipse ()
|  | returns the current ellipse of min_ellipse.

### Predicates

By definition, an empty Min_ellipse_2<Traits> has no boundary and no bounded side, i.e. its unbounded side equals the whole space 2.

CGAL::Bounded_side | min_ellipse.bounded_side ( Point p)
|  | 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_ellipse, resp.
bool | min_ellipse.has_on_bounded_side ( Point p)
|  | returns true, iff p lies properly inside min_ellipse.
bool | min_ellipse.has_on_boundary ( Point p)
|  | returns true, iff p lies on the boundary of min_ellipse.
bool | min_ellipse.has_on_unbounded_side ( Point p)
|  | returns true, iff p lies properly outside of min_ellipse.
bool | min_ellipse.is_empty ()
|  | returns true, iff min_ellipse is empty (this implies degeneracy).
bool | min_ellipse.is_degenerate ()
|  | returns true, iff min_ellipse is degenerate, i.e. if min_ellipse is empty, equal to a single point or equal to a segment, equivalently if the number of support points is less than 3.

### Modifiers

New points can be added to an existing min_ellipse, allowing to build me(P) incrementally, e.g. if P is not known in advance. Compared to the direct creation of me(P), this is not much slower, because the construction method is incremental itself.

void | min_ellipse.insert ( Point p)
|  | inserts p into min_ellipse and recomputes the smallest enclosing ellipse.
template < class InputIterator >

void

min_ellipse.insert ( | InputIterator first, InputIterator last)

|  | inserts the points in the range [first,last) into min_ellipse and recomputes the smallest enclosing ellipse by calling insert(p) for each point p in [first,last). Requirement: The value type of first and last is Point.
void | min_ellipse.clear ()
|  | deletes all points in min_ellipse and sets min_ellipse to the empty set. Postcondition: min_ellipse.is_empty() = true.

### Validity Check

An object min_ellipse is valid, iff

- min_ellipse contains all points of its defining set P,
- min_ellipse is the smallest ellipse spanned by its support set S, and
- S is minimal, i.e. no support point is redundant.

Note: In this release only the first item is considered by the validity check.

bool

min_ellipse.is_valid ( | bool verbose = false, int level = 0)

|  | returns true, iff min_ellipse contains all points of its defining set P. 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.

### Miscellaneous

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

### I/O

std::ostream& | std::ostream& os << min_ellipse
|  | writes min_ellipse to output stream os. Requirement: The output operator is defined for Point (and for Ellipse, if pretty printing is used).

std::istream& | std::istream& is >> min_ellipse&
|  | reads min_ellipse from input stream is. Requirement: The input operator is defined for Point.

#include <CGAL/IO/Window_stream.h>

CGAL::Window_stream&
| CGAL::Window_stream& ws << min_ellipse
|  | writes min_ellipse to window stream ws. Requirement: The window stream output operator is defined for Point and Ellipse.

### See Also

CGAL::Min_circle_2<Traits>
CGAL::Min_ellipse_2_traits_2<K>
MinEllipse2Traits

### Implementation

We implement the incremental algorithm of Welzl, with move-to-front heuristic [Wel91], using the primitives as described in [GS97a, GS97b]. The whole implementation is described in [GS98b].

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 ellipse from scratch. The clear operation and the check for validity each takes linear time.

### Example

To illustrate the usage of Min_ellipse_2<Traits> and to show that randomization can be useful in certain cases, we give an example. The example also shows how the coefficents of the constructed ellipse can be accessed.

    // file: examples/Min_ellipse_2/example_Min_ellipse_2.C

    // includes
    #include <cassert>
    #include <CGAL/Cartesian.h>
    #include <CGAL/Point_2.h>
    #include <CGAL/Min_ellipse_2.h>
    #include <CGAL/Min_ellipse_2_traits_2.h>
    #include <CGAL/Gmpq.h>

    // typedefs
    typedef  CGAL::Gmpq                       NT;
    typedef  CGAL::Cartesian<NT>              K;
    typedef  CGAL::Point_2<K>                 Point;
    typedef  CGAL::Min_ellipse_2_traits_2<K>  Traits;
    typedef  CGAL::Min_ellipse_2<Traits>      Min_ellipse;

    // main
    int
    main( int, char**)
    {
        int     n = 200;
        Point*  P = new Point[ n];

        for ( int i = 0; i < n; ++i)
    	P[ i] = Point( i % 2 ? i : -i , 0);
        // (0,0), (-1,0), (2,0), (-3,0)

        std::cout << "Computing ellipse (without randomization)...";
        std::cout.flush();
        Min_ellipse  me1( P, P+n, false);    // very slow
        std::cout << "done." << std::endl;

        std::cout << "Computing ellipse (with randomization)...";
        std::cout.flush();
        Min_ellipse  me2( P, P+n, true);     // fast
        std::cout << "done." << std::endl;

        // because all input points are collinear, the ellipse is
        // degenerate and equals a line segment; the ellipse has
        // two support points
        assert(me2.is_degenerate());
        assert(me2.number_of_support_points()==2);

        // prettyprinting
        CGAL::set_pretty_mode( std::cout);
        std::cout << me2;

        // in general, the ellipse is not explicitly representable
        // over the input number type NT; when you use the default
        // traits class CGAL::Min_ellipse_2_traits_2<K>, you can
        // get double approximations for the coefficients of the
        // underlying conic curve. NOTE: this curve only exists
        // in the nondegenerate case!

        me2.insert(Point(0,1)); // resolves the degeneracy
        assert(!me2.is_degenerate());

        // get the coefficients
        double r,s,t,u,v,w;
        me2.ellipse().double_coefficients( r, s, t, u, v, w);
        std::cout << "ellipse has the equation " <<
          r << " x^2 + " <<
          s << " y^2 + " <<
          t << " xy + " <<
          u << " x + " <<
          v << " y + " <<
          w << " = 0." << std::endl;

        delete[] P;

        return( 0);
    }

    // ===== EOF ==================================================================