CGAL::Sphere_3< Kernel > Class Template Reference

#include <CGAL/Sphere_3.h>

Definition

An object of type Sphere_3 is a sphere in the three-dimensional Euclidean space \( \E^3\).

The sphere is oriented, i.e. its boundary has clockwise or counterclockwise orientation. The boundary splits \( \E^3\) into a positive and a negative side, where the positive side is to the left of the boundary. The boundary also splits \( \E^3\) into a bounded and an unbounded side. Note that the sphere can be degenerated, i.e. the squared radius may be zero.

Is Model Of:

Kernel::Sphere_3

Hashable if Kernel is a cartesian kernel and if Kernel::FT is Hashable

Creation
	Sphere_3 (const Point_3< Kernel > &center, const Kernel::FT &squared_radius, const Orientation &orientation=COUNTERCLOCKWISE)
	introduces a variable c of type Sphere_3. More...
	Sphere_3 (const Point_3< Kernel > &p, const Point_3< Kernel > &q, const Point_3< Kernel > &r, const Point_3< Kernel > &s)
	introduces a variable c of type Sphere_3. More...
	Sphere_3 (const Point_3< Kernel > &p, const Point_3< Kernel > &q, const Point_3< Kernel > &r, const Orientation &o=COUNTERCLOCKWISE)
	introduces a variable c of type Sphere_3. More...
	Sphere_3 (const Point_3< Kernel > &p, const Point_3< Kernel > &q, const Orientation &o=COUNTERCLOCKWISE)
	introduces a variable c of type Sphere_3. More...
	Sphere_3 (const Point_3< Kernel > &center, const Orientation &orientation=COUNTERCLOCKWISE)
	introduces a variable c of type Sphere_3. More...
	Sphere_3 (const Circle_3< Kernel > &c)
	introduces a variable c of type Sphere_3. More...

Access Functions
const Point_3< Kernel > &	center () const
	returns the center of c.
Kernel::FT const &	squared_radius () const
	returns the squared radius of c.
Orientation const &	orientation () const
	returns the orientation of c.
bool	operator== (const Sphere_3< Kernel > &sphere2) const
	returns true, iff c and sphere2 are equal, i.e. if they have the same center, same squared radius and same orientation.
bool	operator!= (const Sphere_3< Kernel > &sphere2) const
	returns true, iff c and sphere2 are not equal.

Predicates
bool	is_degenerate () const
	returns true, iff c is degenerate, i.e. if c has squared radius zero.
Oriented_side	oriented_side (const Point_3< Kernel > &p) const
	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 c, resp.
Bounded_side	bounded_side (const Point_3< Kernel > &p) const
	returns ON_BOUNDED_SIDE, ON_BOUNDARY, or ON_UNBOUNDED_SIDE iff p lies properly inside, on the boundary, or properly outside of c, resp.
bool	has_on_positive_side (const Point_3< Kernel > &p) const
bool	has_on_negative_side (const Point_3< Kernel > &p) const
bool	has_on_boundary (const Point_3< Kernel > &p) const
bool	has_on_bounded_side (const Point_3< Kernel > &p) const
bool	has_on_unbounded_side (const Point_3< Kernel > &p) const
bool	has_on (const Point_3< Kernel > &p) const
bool	has_on (const Circle_3< Kernel > &p) const

Miscellaneous
Sphere_3< Kernel >	opposite () const
	returns the sphere with the same center and squared radius as c but with opposite orientation.
Sphere_3< Kernel >	orthogonal_transform (Aff_transformation_3< Kernel > const &at) const
	returns the sphere obtained by applying at on c. More...
Bbox_3	bbox () const
	returns a bounding box containing c.

Constructor & Destructor Documentation

Sphere_3() [1/6]

template<typename Kernel >

CGAL::Sphere_3< Kernel >::Sphere_3	(	const Point_3< Kernel > &	center,
		const Kernel::FT &	squared_radius,
		const Orientation &	orientation = COUNTERCLOCKWISE
	)

introduces a variable c of type Sphere_3.

It is initialized to the sphere with center center, squared radius squared_radius and orientation orientation.

Precondition

orientation \( \neq\) COPLANAR, and furthermore, squared_radius \( \geq\) 0.

Sphere_3() [2/6]

template<typename Kernel >

CGAL::Sphere_3< Kernel >::Sphere_3	(	const Point_3< Kernel > &	p,
		const Point_3< Kernel > &	q,
		const Point_3< Kernel > &	r,
		const Point_3< Kernel > &	s
	)

introduces a variable c of type Sphere_3.

It is initialized to the unique sphere which passes through the points p, q, r and s. The orientation of the sphere is the orientation of the point quadruple p, q, r, s.

Precondition

p, q, r, and s are not coplanar.

Sphere_3() [3/6]

template<typename Kernel >

CGAL::Sphere_3< Kernel >::Sphere_3	(	const Point_3< Kernel > &	p,
		const Point_3< Kernel > &	q,
		const Point_3< Kernel > &	r,
		const Orientation &	o = COUNTERCLOCKWISE
	)

introduces a variable c of type Sphere_3.

It is initialized to the smallest sphere which passes through the points p, q, and r. The orientation of the sphere is o.

Precondition

o is not COPLANAR.

Sphere_3() [4/6]

template<typename Kernel >

CGAL::Sphere_3< Kernel >::Sphere_3	(	const Point_3< Kernel > &	p,
		const Point_3< Kernel > &	q,
		const Orientation &	o = COUNTERCLOCKWISE
	)

introduces a variable c of type Sphere_3.

It is initialized to the smallest sphere which passes through the points p and q. The orientation of the sphere is o.

Precondition

o is not COPLANAR.

Sphere_3() [5/6]

template<typename Kernel >

CGAL::Sphere_3< Kernel >::Sphere_3	(	const Point_3< Kernel > &	center,
		const Orientation &	orientation = COUNTERCLOCKWISE
	)

introduces a variable c of type Sphere_3.

It is initialized to the sphere with center center, squared radius zero and orientation orientation.

Precondition

orientation \( \neq\) COPLANAR.

Postcondition

c.is_degenerate() = true.

Sphere_3() [6/6]

template<typename Kernel >

CGAL::Sphere_3< Kernel >::Sphere_3

(

const Circle_3< Kernel > &

c

)

introduces a variable c of type Sphere_3.

It is initialized to the diametral sphere of the circle.

Member Function Documentation

orthogonal_transform()

template<typename Kernel >

Sphere_3<Kernel> CGAL::Sphere_3< Kernel >::orthogonal_transform

(

Aff_transformation_3< Kernel > const &

at

)

const

returns the sphere obtained by applying at on c.

Precondition

at is an orthogonal transformation.