\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)
CGAL 4.7 - dD Geometry Kernel
 All Classes Namespaces Files Functions Variables Typedefs Enumerations Enumerator Groups Pages
List of all members | Related Functions
CGAL::Sphere_d< Kernel > Class Template Reference
Kernel Objects

#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

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.
Requires:
The value type of ForwardIterator is Point_d<Kernel>.

Member Function Documentation

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().
template<typename Kernel >
Point_d<Kernel> CGAL::Sphere_d< Kernel >::center ( )

returns the center of S.

Precondition
S is legal.
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().
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().
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().
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().
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().
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.

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().
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().
template<typename Kernel >
Point_d<Kernel> CGAL::Sphere_d< Kernel >::point ( int  i)

returns the \( i\)th defining point.

Precondition
\( 0 \le i \le dim \)
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

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().