CGAL 6.2 - 3D Convex Hulls

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##

Susan Hert and Stefan Schirra
This package provides functions for computing convex hulls in three dimensions, as well as functions for checking if sets of points are strongly convex or not, and checking if two convex hulls intersect or not. One can compute the convex hull of a set of points in three dimensions in three ways: using a static algorithm, using a triangulation to get a fully dynamic computation, or using the specific data structure `Convex_hull_hierarchy_3` for optimal intersection testing.
Introduced in: CGAL 1.1
Depends on: The dynamic algorithms depend on 3D Triangulations.
BibTeX: cgal:hs-ch3-26a
License: GPL
Windows Demo: CGAL Lab

A subset \(S \subseteq \mathbb{R}^3 \) is convex if for any two points `p` and `q` in the set the line segment with endpoints `p` and `q` is contained in \(S\). The convex hull of a set \( S \) is the smallest convex set containing \( S \). The convex hull of a set of points `P` is a convex polytope with vertices in `P`. A point in `P` is an extreme point (with respect to `P`) if it is a vertex of the convex hull of `P`.

CGAL provides functions for computing convex hulls in two, three and arbitrary dimensions as well as functions for testing if a given set of points in is strongly convex or not and checking if two convex hulls intersect. This chapter describes the functions available for three dimensions.

## Assertions

For the convex hull algorithms, the postcondition check tests only convexity (if not disabled), but not containment of the input points in the polygon or polyhedron defined by the output points. The latter is considered an expensive checking and can be enabled by defining `CGAL_CHECK_EXPENSIVE`.

## Classified Reference Pages

## Concepts

- `ConvexHullTraits_3`
- `IsStronglyConvexTraits_3`

## Traits Classes

- `CGAL::Convex_hull_traits_3<R>`
- `CGAL::Extreme_points_traits_adapter_3`
- `CGAL::make_extreme_points_traits_adapter()`

## Convex Hull Functions

- `CGAL::convex_hull_3()`
- `CGAL::extreme_points_3()`

## Intersection and Directional Queries

- `CGAL::extreme_point_3()`
- `CGAL::extreme_vertex_3()`
- `CGAL::Convex_hull_3::do_intersect()`
- `CGAL::Convex_hull_hierarchy_3`

## Convexity Checking Function

- `CGAL::is_strongly_convex_3`

## Halfspace Intersection Functions

- `CGAL::halfspace_intersection_3()`
- `CGAL::halfspace_intersection_with_constructions_3()`
- `CGAL::halfspace_intersection_interior_point_3()`

## Modules
| Concepts
| Traits Classes
| Intersection Test Functions
| Directional Queries Functions
| Convex Hull Functions
| The function `CGAL::convex_hull_3()` computes the convex hull of a given set of three-dimensional points.
| Convexity Checking