CGAL 5.6.3 - 3D Convex Hulls

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3D Convex Hulls Reference

##

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. One can compute the convex hull of a set of points in three dimensions in two ways: using a static algorithm or using a triangulation to get a fully dynamic computation.
Introduced in: CGAL 1.1
Depends on: The dynamic algorithms depend on 3D Triangulations.
BibTeX: cgal:hs-ch3-25b
License: GPL
Windows Demo: Polyhedron demo

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. 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`

## Convex Hull Functions

- `CGAL::convex_hull_3`
- `CGAL::extreme_points_3`
- `CGAL::make_extreme_points_traits_adapter`

## 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
| Convex Hull Functions
| The function `convex_hull_3()` computes the convex hull of a given set of three-dimensional points.
| Convexity Checking