\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)

CGAL 4.8 - 3D Convex Hulls

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Convex Hull Functions
3D Convex Hulls Reference

##

The function `convex_hull_3()` computes the convex hull of a given set of three-dimensional points.

Two versions of this function are available. The first can be used when it is known that the result will be a polyhedron and the second when a degenerate hull may also be possible.

## Functions
template<class PlaneIterator , class Polyhedron >
void | CGAL::halfspace_intersection_3 (PlaneIterator begin, PlaneIterator end, Polyhedron &P, boost::optional< Polyhedron::Vertex::Point_3 > origin=boost::none)
| computes robustly the intersection of the halfspaces defined by the planes contained in the range [`begin`, `end`) without constructing the dual points. More...
template<class PlaneIterator , class Polyhedron , class Traits >
void | CGAL::halfspace_intersection_with_constructions_3 (PlaneIterator pbegin, PlaneIterator pend, Polyhedron &P, boost::optional< Polyhedron::Vertex::Point_3 > origin=boost::none, const Traits &ch_traits=Default_traits)
| computes the intersection of the halfspaces defined by the planes contained in the range [`begin`, `end`). More...
template<class InputIterator , class Polyhedron_3 , class Traits >
void | CGAL::convex_hull_3 (InputIterator first, InputIterator last, Polyhedron_3 &P, const Traits &ch_traits=Default_traits)
| computes the convex hull of the set of points in the range [`first`, `last`). More...
template<class InputIterator , class Traits >
void | CGAL::convex_hull_3 (InputIterator first, InputIterator last, Object &ch_object, const Traits &ch_traits=Default_traits)
| computes the convex hull of the set of points in the range [`first`, `last`). More...
template<class Triangulation , class Polyhedron >
void | CGAL::convex_hull_3_to_polyhedron_3 (const Triangulation &T, Polyhedron &P)
| fills a polyhedron with the convex hull of a set of 3D points contained in a 3D triangulation of CGAL. More...
template<class InputIterator , class Polyhedron >
void | CGAL::convex_hull_incremental_3 (InputIterator first, InputIterator beyond, Polyhedron &P, bool test_correctness=false)

## Function Documentation

template<class InputIterator , class Polyhedron_3 , class Traits >

void CGAL::convex_hull_3 | ( | InputIterator | first,
|  | InputIterator | last,
|  | Polyhedron_3 & | P,
|  | const Traits & | ch_traits = `Default_traits`
| ) |  |

computes the convex hull of the set of points in the range [`first`, `last`).

The polyhedron `P` is cleared, then the convex hull is stored in `P`. Note that the convex hull will be triangulated, that is `P` will contain only triangular facets.

Attention
  This function does not compute the plane equations of the faces of `P`.

Precondition
  There are at least four points in the range [`first`, `last`) not all of which are collinear.

Template Parameters

  InputIterator | must be an input iterator with a value type equivalent to `Traits::Point_3`.
  Polyhedron_3 | must be a model of `ConvexHullPolyhedron_3`.
  Traits | must be a model of the concept `ConvexHullTraits_3`. For the purposes of checking the postcondition that the convex hull is valid, `Traits` should also be a model of the concept `IsStronglyConvexTraits_3`.

If the kernel `R` of the points determined by the value type of `InputIterator` is a kernel with exact predicates but inexact constructions (in practice we check `R::Has_filtered_predicates_tag` is `Tag_true` and `R::FT` is a floating point type), then the default traits class of `convex_hull_3()` is `Convex_hull_traits_3<R>`, and `R` otherwise.

See Also
  `convex_hull_incremental_3()`

Implementation

The algorithm implemented by these functions is the quickhull algorithm of Barnard et al. [1].

`#include <CGAL/convex_hull_3.h>`

Examples:
  Convex_hull_3/quickhull_3.cpp.

template<class InputIterator , class Traits >

void CGAL::convex_hull_3 | ( | InputIterator | first,
|  | InputIterator | last,
|  | Object & | ch_object,
|  | const Traits & | ch_traits = `Default_traits`
| ) |  |

computes the convex hull of the set of points in the range [`first`, `last`).

The result, which may be a point, a segment, a triangle, or a polyhedron, is stored in `ch_object`. In the case the result is a polyhedron, the convex hull will be triangulated, that is the polyhedron will contain only triangular facets.

Attention
  This function does not compute the plane equations of the faces of `P` in case the result is a polyhedron.

Template Parameters

  InputIterator | must be an input iterator with a value type equivalent to `Traits::Point_3`.
  Traits | must be model of the concept `ConvexHullTraits_3`. For the purposes of checking the postcondition that the convex hull is valid, `Traits` should also be a model of the concept `IsStronglyConvexTraits_3`. Furthermore, `Traits` must define a type `Polyhedron_3` that is a model of `ConvexHullPolyhedron_3`.

If the kernel `R` of the points determined by the value type of `InputIterator` is a kernel with exact predicates but inexact constructions (in practice we check `R::Has_filtered_predicates_tag` is `Tag_true` and `R::FT` is a floating point type), then the default traits class of `convex_hull_3()` is `Convex_hull_traits_3<R>`, and `R` otherwise.

`#include <CGAL/convex_hull_3.h>`

template<class Triangulation , class Polyhedron >

void CGAL::convex_hull_3_to_polyhedron_3 | ( | const Triangulation & | T,
|  | Polyhedron & | P
| ) |  |

fills a polyhedron with the convex hull of a set of 3D points contained in a 3D triangulation of CGAL.

The polyhedron `P` is cleared and the convex hull of the set of 3D points is stored in `P`.

Attention
  This function does not compute the plane equations of the faces of `P`.

Precondition
  `T.dimension()`==3.

Template Parameters

  Triangulation | is a CGAL 3D triangulation.
  Polyhedron | is an instantiation of `CGAL::Polyhedron_3<Traits>`.

See Also
  `convex_hull_3`

`#include <CGAL/convex_hull_3_to_polyhedron_3.h>`

Examples:
  Convex_hull_3/dynamic_hull_3.cpp.

template<class InputIterator , class Polyhedron >

void CGAL::convex_hull_incremental_3 | ( | InputIterator | first,
|  | InputIterator | beyond,
|  | Polyhedron & | P,
|  | bool | test_correctness = `false`
| ) |  |

Deprecated:
  This function relies on a package deprecated since CGAL 4.6 and thus is also deprecated.

computes the convex hull polyhedron of the three-dimensional points in the range [`first`,`beyond`) and assigns it to `P`. If `test_correctness` is set to `true`, the tests described in [3] are used to determine the correctness of the resulting polyhedron.

Attention
  This function is provided for completeness and educational purposes. When an efficient incremental implementation is needed, using `Delaunay_triangulation_3` together with `convex_hull_3_to_polyhedron_3()` is highly recommended.

Template Parameters

  InputIterator | must be an input iterator where the value type must be `Polyhedron::Traits::Point`.
  Polyhedron | must provide a type `Polyhedron::Traits` that defines the following types `Polyhedron::Traits::R`, which is a model of the representation class `R` required by `Convex_hull_d_traits_3<R>` `Polyhedron::Traits::Point`

See Also
  `convex_hull_3()`
  `Convex_hull_d`

Implementation

This function uses the `d`-dimensional convex hull incremental construction algorithm [2] with `d` fixed to 3. The algorithm requires \( O(n^2)\) time in the worst case and \( O(n \log n)\) expected time.

`#include <CGAL/convex_hull_incremental_3.h>`

template<class PlaneIterator , class Polyhedron >

void CGAL::halfspace_intersection_3 | ( | PlaneIterator | begin,
|  | PlaneIterator | end,
|  | Polyhedron & | P,
|  | boost::optional< Polyhedron::Vertex::Point_3 > | origin = `boost::none`
| ) |  |

computes robustly the intersection of the halfspaces defined by the planes contained in the range [`begin`, `end`) without constructing the dual points.

The result is stored in the polyhedron `P`. If `origin` is given then it must be a point strictly inside the polyhedron. If an interior point is not given then it is computed using a linear program and thus is slower.

This version does not construct the dual points explicitely but uses a special traits class for the function `CGAL::convex_hull_3()` to handle predicates on dual points without constructing them.

Attention
  Halfspaces are considered as lower halfspaces that is to say if the plane's equation is \( a\, x +b\, y +c\, z + d = 0 \) then the corresponding halfspace is defined by \( a\, x +b\, y +c\, z + d \le 0 \) .
  The value type of `PlaneIterator` and the point type of `origin` must come from the same CGAL Kernel.

Precondition
  if provided, `origin` is inside the intersection of halfspaces defined by the range `[begin, end)`.
  The computed intersection must be a bounded convex polyhedron.

Template Parameters

  PlaneIterator | must be an input iterator where the value type must be `Polyhedron::Traits::Plane_3`
  Polyhedron | must be a model of `ConvexHullPolyhedron_3`.

See Also
  `halfspace_intersection_with_constructions_3()`

`#include <CGAL/Convex_hull_3/dual/halfspace_intersection_3.h>`

Examples:
  Convex_hull_3/halfspace_intersection_3.cpp.

template<class PlaneIterator , class Polyhedron , class Traits >

void CGAL::halfspace_intersection_with_constructions_3 | ( | PlaneIterator | pbegin,
|  | PlaneIterator | pend,
|  | Polyhedron & | P,
|  | boost::optional< Polyhedron::Vertex::Point_3 > | origin = `boost::none`,
|  | const Traits & | ch_traits = `Default_traits`
| ) |  |

computes the intersection of the halfspaces defined by the planes contained in the range [`begin`, `end`).

The result is stored in the polyhedron `P`. If `origin` is given then it must be a point strictly inside the polyhedron. If an interior point is not given then it is computed using a linear program and thus is slower. This version constructs explicitly the dual points using the convex hull algorithm parametrized with the given traits class.

Attention
  Halfspaces are considered as lower halfspaces that is to say if the plane's equation is \( a\, x +b\, y +c\, z + d = 0 \) then the corresponding halfspace is defined by \( a\, x +b\, y +c\, z + d \le 0 \) .
  The value type of `PlaneIterator` and the point type of `origin` must come from the same CGAL Kernel.

Precondition
  if provided, `origin` is inside the intersection of halfspaces defined by the range `[begin, end)`.
  The computed intersection must be a bounded convex polyhedron.

Template Parameters

  PlaneIterator | must be an input iterator where the value type must be `Polyhedron::Traits::Plane_3`
  Polyhedron | must be a model of `ConvexHullPolyhedron_3`.
  Traits | must be a model of the concept `ConvexHullTraits_3`.

See Also
  `halfspace_intersection_3()`

`#include <CGAL/Convex_hull_3/dual/halfspace_intersection_with_constructions_3.h>`