3D Envelopes Reference

Dan Halperin, Michal Meyerovitch, Ron Wein, and Baruch Zukerman

This package consists of functions that compute the lower (or upper) envelope of a set of arbitrary surfaces in 3D. The output is represented as an 2D envelope diagram, namely a planar subdivision such that the identity of the surfaces that induce the envelope over each diagram cell is unique.

Introduced in: CGAL 3.3
Depends on: 2D Arrangements
BibTeX: cgal:mwz-e3-24b
License: GPL

This package consists of functions that compute the lower (or upper) envelope of a set of arbitrary surfaces in 3D. The output is represented as an 2D envelope diagram - a planar subdivision such that the identity of the surfaces that induce the envelope over each diagram cell is unique.

Classified Reference Pages

Functions

• CGAL::lower_envelope_3
• CGAL::upper_envelope_3
• CGAL::lower_envelope_xy_monotone_3
• CGAL::upper_envelope_xy_monotone_3

Concepts

• EnvelopeTraits_3

Classes

• CGAL::Envelope_diagram_2<EnvTraits>
• CGAL::Env_triangle_traits_3<Kernel, ArrLinearTraits>
• CGAL::Env_sphere_traits_3<ConicTraits>
• CGAL::Env_plane_traits_3<Kernel, ArrLinearTraits>
• CGAL::Env_surface_data_traits_3<Traits,XyData,SData,Cnv>

Modules
	Concepts

Classes
class	CGAL::Env_plane_traits_3< Kernel, ArrLinearTraits >
	The traits class template Env_plane_traits_3 models the EnvelopeTraits_3 concept, and is used for the construction of lower and upper envelopes of planes and half planes in the space. More...
class	CGAL::Env_sphere_traits_3< ConicTraits >
	The traits class Env_sphere_traits_3 models the EnvelopeTraits_3 concept, and is used for the construction of lower and upper envelopes of spheres. More...
class	CGAL::Env_surface_data_traits_3< Traits, XyData, SData, Cnv >
	The class Env_surface_data_traits_3 is a model of the EnvelopeTraits_3 concept and serves as a decorator class that allows the extension of the surfaces defined by the base traits-class (the Traits parameter), which serves as a geometric traits-class (a model of the EnvelopeTraits_3 concept), with extraneous (non-geometric) data fields. More...
class	CGAL::Env_triangle_traits_3< Kernel, ArrSegmentTraits >
	The traits class template Env_triangle_traits_3 models the EnvelopeTraits_3 concept, and is used for the construction of lower and upper envelopes of triangles in the space. More...
class	CGAL::Envelope_diagram_2< EnvTraits >
	The class-template Envelope_diagram_2 represents the minimization diagram that corresponds to the lower envelope of a set of curves, or the maximization diagram that corresponds to their upper envelope. More...

Functions
template<class InputIterator , class Traits >
void	CGAL::lower_envelope_3 (InputIterator begin, InputIterator end, Envelope_diagram_2< Traits > &diag)
	Computes the lower envelope of a set of surfaces in \( \mathbb{R}^3\), as given by the range [begin, end).
template<class InputIterator , class Traits >
void	CGAL::lower_envelope_xy_monotone_3 (InputIterator begin, InputIterator end, Envelope_diagram_2< Traits > &diag)
	Computes the lower envelope of a set of \( xy\)-monotone surfaces in \( \mathbb{R}^3\), as given by the range [begin, end).
template<class InputIterator , class Traits >
void	CGAL::upper_envelope_3 (InputIterator begin, InputIterator end, Envelope_diagram_2< Traits > &diag)
	Computes the upper envelope of a set of surfaces in \( \mathbb{R}^3\), as given by the range [begin, end).
template<class InputIterator , class Traits >
void	CGAL::upper_envelope_xy_monotone_3 (InputIterator begin, InputIterator end, Envelope_diagram_2< Traits > &diag)
	Computes the upper envelope of a set of \( xy\)-monotone surfaces in \( \mathbb{R}^3\), as given by the range [begin, end).

Function Documentation

lower_envelope_3()

template<class InputIterator , class Traits >

void CGAL::lower_envelope_3	(	InputIterator	begin,
		InputIterator	end,
		Envelope_diagram_2< Traits > &	diag
	)

#include <CGAL/envelope_3.h>

Computes the lower envelope of a set of surfaces in \( \mathbb{R}^3\), as given by the range [begin, end).

The lower envelope is represented using the output minimization diagram diag.

Template Parameters

Traits

must be a model of EnvelopeTraits_3. InputIterator must be an input iterator with value type Traits::Surface_3.

Examples

Envelope_3/envelope_planes.cpp, Envelope_3/envelope_spheres.cpp, and Envelope_3/envelope_triangles.cpp.

lower_envelope_xy_monotone_3()

template<class InputIterator , class Traits >

void CGAL::lower_envelope_xy_monotone_3	(	InputIterator	begin,
		InputIterator	end,
		Envelope_diagram_2< Traits > &	diag
	)

#include <CGAL/envelope_3.h>

Computes the lower envelope of a set of \( xy\)-monotone surfaces in \( \mathbb{R}^3\), as given by the range [begin, end).

The lower envelope is represented using the output minimization diagram diag.

Template Parameters

Traits

must be a model of EnvelopeTraits_3. InputIterator must be an input iterator with value type Traits::Xy_monotone_surface_3.

upper_envelope_3()

template<class InputIterator , class Traits >

void CGAL::upper_envelope_3	(	InputIterator	begin,
		InputIterator	end,
		Envelope_diagram_2< Traits > &	diag
	)

#include <CGAL/envelope_3.h>

Computes the upper envelope of a set of surfaces in \( \mathbb{R}^3\), as given by the range [begin, end).

The upper envelope is represented using the output maximization diagram diag.

Template Parameters

Traits

must be a model of EnvelopeTraits_3. InputIterator must be an input iterator with value type Traits::Surface_3.

Examples

Envelope_3/envelope_planes.cpp, and Envelope_3/envelope_triangles.cpp.

upper_envelope_xy_monotone_3()

template<class InputIterator , class Traits >

void CGAL::upper_envelope_xy_monotone_3	(	InputIterator	begin,
		InputIterator	end,
		Envelope_diagram_2< Traits > &	diag
	)

#include <CGAL/envelope_3.h>

Computes the upper envelope of a set of \( xy\)-monotone surfaces in \( \mathbb{R}^3\), as given by the range [begin, end).

The lower envelope is represented using the output maximization diagram diag.

Template Parameters

Traits

must be a model of EnvelopeTraits_3. InputIterator must be an input iterator with value type Traits::Xy_monotone_surface_3.