CGAL 4.7 - 3D Convex Hulls: Convexity Checking

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

CGAL 4.7 - 3D Convex Hulls

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Convexity Checking
3D Convex Hulls Reference

##

## Functions

template<class Polyhedron , class Traits >
bool | CGAL::is_strongly_convex_3 (Polyhedron &P, const Traits &traits=Default_traits)
| determines if the vertices of a given polyhedron represents a strongly convex set of points or not. More...

## Function Documentation

template<class Polyhedron , class Traits >

bool CGAL::is_strongly_convex_3 | ( | Polyhedron & | P,
|  | const Traits & | traits = `Default_traits`
| ) |  |

determines if the vertices of a given polyhedron represents a strongly convex set of points or not.

A set of points is said to be strongly convex if it consists of only extreme points (i.e., vertices of the convex hull).

The default traits class is the kernel in which the type `Polyhedron::Point_3` is defined.

Requirements

1. `Polyhedron::Point_3` is equivalent to `Traits::Point_3`.
2. `Traits` is a model of the concept `IsStronlyConvexTraits_3`
3. `Polyhedron` must define the following types:
   - `Polyhedron::Facet_iterator`
   - `Polyhedron::Vertex_iterator`
   and the following member functions:
   - `facets_begin()`
   - `facets_end()`
   - `vertices_begin()`
   - `vertices_end()`
   The vertex type of `Polyhedron` must be a model of `ConvexHullPolyhedronVertex_3`; the facet type must be `ConvexHullPolyhedronFacet_3`.

Implementation

This function implements the tests described in [3] to determine convexity and requires \( O(e + f)\) time for a polyhedron with \( e\) edges and \( f\) faces.

determines if the set of vertices of the polyhedron `P` represent a strongly convex set of points or not.

Precondition
  The equations of the facet planes of the polyhedron must have already been computed.

`#include <CGAL/convexity_check_3.h>`