CGAL::Nef_polyhedron_S2< Traits >::Sphere_segment Class Reference

#include <CGAL/Nef_polyhedron_S2.h>

Definition

An object s of type Sphere_segment is a segment in the surface of a unit sphere that is part of a great circle through the origin.

Sphere segments are represented by two sphere points \( p\) and \( q\) plus an oriented plane \( h\) that contains \( p\) and \( q\). The plane determines the sphere segment as follows. Let \( c\) be the circle in the intersection of \( h\) and \( S_2\). Then \( s\) is that part of \( c\) that is swept, when we rotate \( p\) into \( q\) in counterclockwise rotation around the normal vector of \( h\) as seen from the positive halfspace.

Creation
	Sphere_segment ()
	creates some sphere segment.
	Sphere_segment (const Sphere_point &p1, const Sphere_point &p2, bool shorter_arc=true)
	creates a spherical segment spanning the shorter arc from p1 to p2 if shorter_arc == true. More...
	Sphere_segment (const Sphere_point &p1, const Sphere_point &p2, const Sphere_circle &c)
	creates a spherical segment spanning the arc from p1 to p2 as part of the oriented circle c (p1 == p2 or p1 == p2.opposite() are possible.) More...
	Sphere_segment (const Sphere_circle &c1, const Sphere_circle &c2)
	creates the spherical segment as part of c1 that is part of the halfsphere left of the oriented circle c2. More...

Operations
const Sphere_point &	source ()
	the source point of s.
const Sphere_point &	target ()
	the target point of s.
const Sphere_circle &	sphere_circle ()
	the great circle supporting s.
Sphere_segment	opposite ()
	returns the sperical segment oriented from target() to source() with the same point set as s.
Sphere_segment	complement ()
	returns the sperical segment oriented from target() to source() with the point set completing s to a full circle.
bool	is_short ()
	a segment is short iff it is shorter than a half-circle.
bool	is_long ()
	a segment is long iff it is longer than a half-circle.
bool	is_degenerate ()
	return true iff s is degenerate, i.e. source and target are the same.
bool	is_halfcircle ()
	return true iff s is a perfect half-circle, i.e. source().antipode == target().
bool	has_on (const Sphere_point &p)
	return true iff s contains p.
bool	has_in_relative_interior (const Sphere_point &p)
	return true iff s contains p in its relative interior.

Constructor & Destructor Documentation

Sphere_segment() [1/3]

template<typename Traits >

CGAL::Nef_polyhedron_S2< Traits >::Sphere_segment::Sphere_segment	(	const Sphere_point &	p1,
		const Sphere_point &	p2,
		bool	shorter_arc = true
	)

creates a spherical segment spanning the shorter arc from p1 to p2 if shorter_arc == true.

Otherwise the longer arc is created.

Precondition

p1 != p2 and p1 != p2.opposite().

Sphere_segment() [2/3]

template<typename Traits >

CGAL::Nef_polyhedron_S2< Traits >::Sphere_segment::Sphere_segment	(	const Sphere_point &	p1,
		const Sphere_point &	p2,
		const Sphere_circle &	c
	)

creates a spherical segment spanning the arc from p1 to p2 as part of the oriented circle c (p1 == p2 or p1 == p2.opposite() are possible.)

Precondition

p1 and p2 are contained in c.

Sphere_segment() [3/3]

template<typename Traits >

CGAL::Nef_polyhedron_S2< Traits >::Sphere_segment::Sphere_segment	(	const Sphere_circle &	c1,
		const Sphere_circle &	c2
	)

creates the spherical segment as part of c1 that is part of the halfsphere left of the oriented circle c2.

Precondition

c1 != c2 as unoriented circles.