CGAL::Sphere_d< Kernel > Class Template Reference

#include <CGAL/Kernel_d/Sphere_d.h>

Definition

An instance $S$ of the data type Sphere_d is an oriented sphere in some $d$-dimensional space.

A sphere is defined by $d+1$ points (class Point_d<Kernel>). We use $A$ to denote the array of the defining points. A set $A$ of defining points is legal if either the points are affinely independent or if the points are all equal. Only a legal set of points defines a sphere in the geometric sense and hence many operations on spheres require the set of defining points to be legal. The orientation of $S$ is equal to the orientation of the defining points, i.e., orientation(A).

Implementation

Spheres are implemented by a vector of points as a handle type. All operations like creation, initialization, tests, input and output of a sphere $s$ take time $O(s.dimension())$. dimension(), point access take constant time. The center()-operation takes time $O(d^3)$ on its first call and constant time thereafter. The sidedness and orientation tests take time $O(d^3)$. The space requirement for spheres is $O(s.dimension())$ neglecting the storage room of the points.

Related Functions
(Note that these are not member functions.)
bool	weak_equality (const Sphere_d< Kernel > &S1, const Sphere_d< Kernel > &S2)
	Test for equality as unoriented spheres. More...

Types
typedef unspecified_type	LA
	the linear algebra layer.
typedef unspecified_type	point_iterator
	a read-only iterator for points defining the sphere.

Creation
	Sphere_d ()
	introduces a variable S of type Sphere_d<Kernel>.
template<class ForwardIterator >
	Sphere_d (int d, ForwardIterator first, ForwardIterator last)
	introduces a variable S of type Sphere_d<Kernel>. More...

Operations
int	dimension ()
	returns the dimension of the ambient space.
Point_d< Kernel >	point (int i)
	returns the $i$th defining point. More...
point_iterator	points_begin ()
	returns an iterator pointing to the first defining point.
point_iterator	points_end ()
	returns an iterator pointing beyond the last defining point.
bool	is_degenerate ()
	returns true iff the defining points are not full dimensional.
bool	is_legal ()
	returns true iff the set of defining points is legal. More...
Point_d< Kernel >	center ()
	returns the center of S. More...
FT	squared_radius ()
	returns the squared radius of the sphere. More...
Orientation	orientation ()
	returns the orientation of S.
Oriented_side	oriented_side (const Point_d< Kernel > &p)
	returns either the constant ON_ORIENTED_BOUNDARY, ON_POSITIVE_SIDE, or ON_NEGATIVE_SIDE, iff p lies on the boundary, properly on the positive side, or properly on the negative side of sphere, resp. More...
Bounded_side	bounded_side (const Point_d< Kernel > &p)
	returns ON_BOUNDED_SIDE, ON_BOUNDARY, or ON_UNBOUNDED_SIDE iff p lies properly inside, on the boundary, or properly outside of sphere, resp. More...
bool	has_on_positive_side (const Point_d< Kernel > &p)
	returns S.oriented_side(p)==ON_POSITIVE_SIDE. More...
bool	has_on_negative_side (const Point_d< Kernel > &p)
	returns S.oriented_side(p)==ON_NEGATIVE_SIDE. More...
bool	has_on_boundary (const Point_d< Kernel > &p)
	returns S.oriented_side(p)==ON_ORIENTED_BOUNDARY, which is the same as S.bounded_side(p)==ON_BOUNDARY. More...
bool	has_on_bounded_side (const Point_d< Kernel > &p)
	returns S.bounded_side(p)==ON_BOUNDED_SIDE. More...
bool	has_on_unbounded_side (const Point_d< Kernel > &p)
	returns S.bounded_side(p)==ON_UNBOUNDED_SIDE. More...
Sphere_d< Kernel >	opposite ()
	returns the sphere with the same center and squared radius as S but with opposite orientation.
Sphere_d< Kernel >	operator+ (const Vector_d< Kernel > &v)
	returns the sphere translated by v. More...

Constructor & Destructor Documentation

Sphere_d()

template<typename Kernel >

template<class ForwardIterator >

CGAL::Sphere_d< Kernel >::Sphere_d	(	int	d,
		ForwardIterator	first,
		ForwardIterator	last
	)

introduces a variable S of type Sphere_d<Kernel>.

S is initialized to the sphere through the points in A = tuple [first,last).

Precondition

$A$ consists of $d+1$ $d$-dimensional points.

Template Parameters

ForwardIterator

has Point_d<Kernel> as value type.

Member Function Documentation

bounded_side()

template<typename Kernel >

Bounded_side CGAL::Sphere_d< Kernel >::bounded_side

(

const Point_d< Kernel > &

p

)

returns ON_BOUNDED_SIDE, ON_BOUNDARY, or ON_UNBOUNDED_SIDE iff p lies properly inside, on the boundary, or properly outside of sphere, resp.

Precondition

S.dimension()==p.dimension().

center()

template<typename Kernel >

Point_d<Kernel> CGAL::Sphere_d< Kernel >::center

(

)

returns the center of S.

Precondition

S is legal.

has_on_boundary()

template<typename Kernel >

bool CGAL::Sphere_d< Kernel >::has_on_boundary

(

const Point_d< Kernel > &

p

)

returns S.oriented_side(p)==ON_ORIENTED_BOUNDARY, which is the same as S.bounded_side(p)==ON_BOUNDARY.

Precondition

S.dimension()==p.dimension().

has_on_bounded_side()

template<typename Kernel >

bool CGAL::Sphere_d< Kernel >::has_on_bounded_side

(

const Point_d< Kernel > &

p

)

returns S.bounded_side(p)==ON_BOUNDED_SIDE.

Precondition

S.dimension()==p.dimension().

has_on_negative_side()

template<typename Kernel >

bool CGAL::Sphere_d< Kernel >::has_on_negative_side

(

const Point_d< Kernel > &

p

)

returns S.oriented_side(p)==ON_NEGATIVE_SIDE.

Precondition

S.dimension()==p.dimension().

has_on_positive_side()

template<typename Kernel >

bool CGAL::Sphere_d< Kernel >::has_on_positive_side

(

const Point_d< Kernel > &

p

)

returns S.oriented_side(p)==ON_POSITIVE_SIDE.

Precondition

S.dimension()==p.dimension().

has_on_unbounded_side()

template<typename Kernel >

bool CGAL::Sphere_d< Kernel >::has_on_unbounded_side

(

const Point_d< Kernel > &

p

)

returns S.bounded_side(p)==ON_UNBOUNDED_SIDE.

Precondition

S.dimension()==p.dimension().

is_legal()

template<typename Kernel >

bool CGAL::Sphere_d< Kernel >::is_legal

(

)

returns true iff the set of defining points is legal.

A set of defining points is legal iff their orientation is non-zero or if they are all equal.

operator+()

template<typename Kernel >

Sphere_d<Kernel> CGAL::Sphere_d< Kernel >::operator+

(

const Vector_d< Kernel > &

v

)

returns the sphere translated by v.

Precondition

S.dimension()==v.dimension().

oriented_side()

template<typename Kernel >

Oriented_side CGAL::Sphere_d< Kernel >::oriented_side

(

const Point_d< Kernel > &

p

)

returns either the constant ON_ORIENTED_BOUNDARY, ON_POSITIVE_SIDE, or ON_NEGATIVE_SIDE, iff p lies on the boundary, properly on the positive side, or properly on the negative side of sphere, resp.

Precondition

S.dimension()==p.dimension().

point()

template<typename Kernel >

Point_d<Kernel> CGAL::Sphere_d< Kernel >::point

(

int

i

)

returns the $i$th defining point.

Precondition

$0 \le i \le dim$

squared_radius()

template<typename Kernel >

FT CGAL::Sphere_d< Kernel >::squared_radius

(

)

returns the squared radius of the sphere.

Precondition

S is legal.

Friends And Related Function Documentation

weak_equality()

template<typename Kernel >

bool weak_equality ( const Sphere_d< Kernel > &  S1, const Sphere_d< Kernel > &  S2  )

related

Test for equality as unoriented spheres.

Precondition

S1.dimension()==S2.dimension().