CGAL 5.3 - 2D and 3D Linear Geometry Kernel
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:

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< Kernelopposite () const
returns the sphere with the same center and squared radius as c but with opposite orientation.

Sphere_3< Kernelorthogonal_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.

## ◆ 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.

## ◆ orthogonal_transform()

template<typename Kernel >
 Sphere_3 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.