#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
	)

Precondition: S1.dimension()==S2.dimension().

Definition

Related Functions

Types

Creation

Operations

Constructor & Destructor Documentation

◆ Sphere_d()

Member Function Documentation

◆ bounded_side()

◆ center()

◆ has_on_boundary()

◆ has_on_bounded_side()

◆ has_on_negative_side()

◆ has_on_positive_side()

◆ has_on_unbounded_side()

◆ is_legal()

◆ operator+()

◆ oriented_side()

◆ point()

◆ squared_radius()

Friends And Related Function Documentation

◆ weak_equality()