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

#include <CGAL/Nef_polyhedron_S2.h>

Definition

template<typename Traits>
class CGAL::Nef_polyhedron_S2< Traits >::Sphere_segment

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.
	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.)
	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.

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 spherical segment oriented from target() to source() with the same point set as s.
Sphere_segment	complement ()
	returns the spherical 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.