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

CGAL 4.8 - 3D Convex Hulls

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ConvexHullTraits_3 Concept Reference
Concepts

## Definition

Requirements of the traits class of the function `CGAL::convex_hull_3()`.

Has Models:

  `CGAL::Convex_hull_traits_3`

  All models of `Kernel`

## Types
typedef unspecified_type | Point_3
| The point type on which the convex hull algorithm operates.
typedef unspecified_type | Plane_3
| a 3D plane
typedef unspecified_type | Segment_3
| a 3D segment
typedef unspecified_type | Triangle_3
| a 3D triangle
typedef unspecified_type | Construct_plane_3
| Function object type that provides `Plane_3 operator()(Point_3 p, Point_3 q, Point_3 r)`, which constructs and returns a plane passing through `p`, `q`, and `r` and oriented in a positive sense when seen from the positive side of the plane.
typedef unspecified_type | Construct_segment_3
| Function object type that provides `Segment_3 operator()(Point_3 p, Point_3 q)`, which constructs and returns the segment with source `p` and target `q`.
typedef unspecified_type | Construct_triangle_3
| Function object type that provides `Triangle_3 operator()(Point_3 p, Point_3 q, Point_3 r)`, which constructs and returns the triangle with vertices `p`, `q`, and `r`.
typedef unspecified_type | Equal_3
| Predicate object type that provides `bool operator()(Point_3 p, Point_3 q)`, which determines if points `p` and `q` are equal.
typedef unspecified_type | Collinear_3
| Predicate object type that provides `bool operator()(Point_3 p, Point_3 q, Point_3 r)`, which determines if points `p`, `q` and `r` are collinear.
typedef unspecified_type | Coplanar_3
| Predicate object type that provides `bool operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s)`, which determines if points `p`, `q`, `r`, and `s` are coplanar.
typedef unspecified_type | Has_on_positive_side_3
| Predicate object type that provides `bool operator()(Plane_3 h, Point_3 q)`, which determines if the point `q` is on the positive side of the halfspace `h`.
typedef unspecified_type | Less_distance_to_point_3
| Predicate object type that provides a constructor taking a single `Point_3` object and `bool operator()(Point_3 q, Point_3 r)`, which returns true iff the distance from `q` to `p` is smaller than the distance from `r` to `p`, where `p` is the point passed to the object at construction.
typedef unspecified_type | Less_signed_distance_to_plane_3
| Predicate object type that provides `bool operator()(Plane_3 p, Point_3 q, Point_3 r)`, which returns true iff the signed distance from `q` to `p` is smaller than the signed distance from `r` to `p`
typedef unspecified_type | Traits_xy_3
| A traits class providing the requirements of the template parameter `Traits` of the 2D convex hull function `CGAL::ch_bykat()` such that `Traits::Point_2` is `Point_3`, and the 2D points considered in the algorithm are the projections of the 3D points in the `xy`-plane. More...
typedef unspecified_type | Traits_yz_3
| Same as above but in the `yz`-plane.
typedef unspecified_type | Traits_xz_3
| Same as above but in the `xz`-plane.

## Creation
Only a copy constructor is required.
| ConvexHullTraits_3 (ConvexHullTraits_3 &ch)

## Operations
For each of the above function and predicate object types, `Func_obj_type`, a function must exist with the name `func_obj_type_object` that creates an instance of the function or predicate object type. For example:
Construct_plane_3 | construct_plane_3_object ()

## Member Typedef Documentation

typedef unspecified_type ConvexHullTraits_3::Traits_xy_3

A traits class providing the requirements of the template parameter `Traits` of the 2D convex hull function `CGAL::ch_bykat()` such that `Traits::Point_2` is `Point_3`, and the 2D points considered in the algorithm are the projections of the 3D points in the `xy`-plane.

If this type is not available, the function `CGAL::convex_hull_3()` will automatically use `CGAL::Projection_traits_xy< CGAL::Kernel_traits<Point_3>::Kernel >.`