\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)

CGAL 4.8 - Bounding Volumes

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CGAL::Min_ellipse_2< Traits > Class Template Reference
Bounding Volumes Reference

`#include <CGAL/Min_ellipse_2.h>`

## Definition

An object of the class `Min_ellipse_2` is the unique ellipse 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 \( me(P)\) the smallest ellipse that contains all points of \( P\). Note that \( me(P)\) can be degenerate, i.e. \( me(P)=\emptyset\) if \( P=\emptyset\), \( me(P)=\{p\}\) if \( P=\{p\}\), and \( me(P) = \{ (1-\lambda)p + \lambda q \mid 0 \leq \lambda \leq 1 \}\) 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.

Template Parameters

  Traits | must be a model for `MinEllipse2Traits`.

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

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 [16], using the primitives as described in [4], [3]. The whole implementation is described in [6].

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` 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 Min_ellipse_2/min_ellipse_2.cpp

#include <CGAL/Cartesian.h>
#include <CGAL/Min_ellipse_2.h>
#include <CGAL/Min_ellipse_2_traits_2.h>
#include <CGAL/Exact_rational.h>
#include <cassert>
typedef CGAL::Exact_rational 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;
int
main( int, char**)
{
const int n = 200;
Point P[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;
return 0;
}

Examples:
  Min_ellipse_2/min_ellipse_2.cpp.

## Related Functions
(Note that these are not member functions.)
std::ostream & | operator<< (std::ostream &os, const Min_ellipse_2< Traits > &min_ellipse)
| writes `min_ellipse` to output stream `os`. More...
std::istream & | operator>> (std::istream &is, Min_ellipse_2< Traits > min_ellipse &)
| reads `min_ellipse` from input stream `is`. More...

## Types
typedef unspecified_type | Point
| Typedef to `Traits::Point`.
typedef unspecified_type | Ellipse
| Typedef to `Traits::Ellipse`. More...
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_ellipse_2` 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 (InputIterator first, InputIterator last, bool randomize, Random &random=default_random, const Traits &traits=Traits())
| initializes `min_ellipse` to \( me(P)\) with \( P\) being the set of points in the range [`first`,`last`). More...
| Min_ellipse_2 (const Traits &traits=Traits())
| creates a variable `min_ellipse` of type `Min_ellipse_2`. More...
| Min_ellipse_2 (const Point &p, const Traits &traits=Traits())
| initializes `min_ellipse` to \( me(\{p\})\), the set \( \{p\}\). More...
| Min_ellipse_2 (const Point &p, const Point &q, const Traits &traits=Traits())
| initializes `min_ellipse` to \( me(\{p,q\})\), the set \( \{ (1-\lambda) p + \lambda q \mid 0 \leq \lambda \leq 1 \}\) More...
| Min_ellipse_2 (const Point &p1, const Point &p2, const Point &p3, const Traits &traits=Traits())
| initializes `min_ellipse` to \( me(\{p1,p2,p3\})\).
| Min_ellipse_2 (const Point &p1, const Point &p2, const Point &p3, const Point &p4, const Traits &traits=Traits())
| initializes `min_ellipse` to \( me(\{p1,p2,p3,p4\})\).
| Min_ellipse_2 (const Point &p1, const Point &p2, const Point &p3, const Point &p4, const Point &p5, const Traits &traits=Traits())
| initializes `min_ellipse` to \( me(\{p1,p2,p3,p4,p5\})\).

## Access Functions
int | number_of_points () const
| returns the number of points of `min_ellipse`, i.e. \( |P|\).
int | number_of_support_points () const
| returns the number of support points of `min_ellipse`, i.e. \( |S|\).
Point_iterator | points_begin () const
| returns an iterator referring to the first point of `min_ellipse`.
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_ellipse`.
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_ellipse`. More...
const Ellipse & | ellipse () const
| returns the current ellipse of `min_ellipse`.

## Predicates
By definition, an empty `Min_ellipse_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_ellipse`, resp.
bool | has_on_bounded_side (const Point &p) const
| returns `true`, iff `p` lies properly inside `min_ellipse`.
bool | has_on_boundary (const Point &p) const
| returns `true`, iff `p` lies on the boundary of `min_ellipse`.
bool | has_on_unbounded_side (const Point &p) const
| returns `true`, iff `p` lies properly outside of `min_ellipse`.
bool | is_empty () const
| returns `true`, iff `min_ellipse` is empty (this implies degeneracy).
bool | is_degenerate () const
| 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 | insert (const Point &p)
| inserts `p` into `min_ellipse` and recomputes the smallest enclosing ellipse.
template<class InputIterator >
void | 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`). More...
void | clear ()
| deletes all points in `min_ellipse` and sets `min_ellipse` to the empty set. More...

## 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 | is_valid (bool verbose=false, int level=0) const
| returns `true`, iff `min_ellipse` contains all points of its defining set \( P\). 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_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.

template<typename Traits>

typedef unspecified_type CGAL::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.

template<typename Traits>

typedef unspecified_type CGAL::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.

## Constructor & Destructor Documentation

template<typename Traits>

CGAL::Min_ellipse_2< Traits >::Min_ellipse_2 | ( | const Traits & | traits = `Traits()`) |  |

creates a variable `min_ellipse` of type `Min_ellipse_2`.

It is initialized to \( me(\emptyset)\), the empty set.

Postcondition
  `min_ellipse.is_empty()` = `true`.

template<typename Traits>

CGAL::Min_ellipse_2< Traits >::Min_ellipse_2 | ( | const Point & | p,
|  | const Traits & | traits = `Traits()`
| ) |  |

initializes `min_ellipse` to \( me(\{p\})\), the set \( \{p\}\).

Postcondition
  `min_ellipse.is_degenerate()` = `true`.

template<typename Traits>

CGAL::Min_ellipse_2< Traits >::Min_ellipse_2 | ( | const Point & | p,
|  | const Point & | q,
|  | const Traits & | traits = `Traits()`
| ) |  |

initializes `min_ellipse` to \( me(\{p,q\})\), the set \( \{ (1-\lambda) p + \lambda q \mid 0 \leq \lambda \leq 1 \}\)

Postcondition
  `min_ellipse.is_degenerate()` = `true`.

## Member Function Documentation

template<typename Traits>

void CGAL::Min_ellipse_2< Traits >::clear | ( | ) |  |

deletes all points in `min_ellipse` and sets `min_ellipse` to the empty set.

Postcondition
  `min_ellipse.is_empty()` = `true`.

template<typename Traits>
template<class InputIterator >

void CGAL::Min_ellipse_2< Traits >::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`).

Template Parameters

  InputIterator | is a model of `InputIterator` with `Point` as value type.

template<typename Traits>

bool CGAL::Min_ellipse_2< Traits >::is_valid | ( | bool | verbose = `false`,
|  | int | level = `0`
| ) |  | const

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.

template<typename Traits>
template<class InputIterator >

CGAL::Min_ellipse_2< Traits >::Min_Ellipse_2 | ( | InputIterator | first,
|  | InputIterator | last,
|  | bool | randomize,
|  | Random & | random = `default_random`,
|  | const 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).

Template Parameters

  InputIterator | is a model of `InputIterator` with `Point` as value type.

template<typename Traits>

const Point& CGAL::Min_ellipse_2< Traits >::support_point | ( | int | i) |  | const

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
  \( 0 \leq i<\) `min_ellipse.number_of_support_points()`.

## Friends And Related Function Documentation

template<typename Traits>

std::ostream & operator<< | ( | std::ostream & | os,
|  | const Min_ellipse_2< Traits > & | min_ellipse
| ) |  |
| related

writes `min_ellipse` to output stream `os`.

An overload of `operator<<` must be defined for `Point` (and for `Ellipse`, if pretty printing is used).

template<typename Traits>

std::istream & operator>> | ( | std::istream & | is,
|  | Min_ellipse_2< Traits > min_ellipse & |
| ) |  |
| related

reads `min_ellipse` from input stream `is`.

An overload of `operator>>` must be defined for `Point`.