CGAL 4.12 - Optimal Distances
WidthTraits_3 Concept Reference

## Definition

This concept defines the requirements for traits classes of Width_3<Traits>.

Operations

Whatever the coordinates of the points are, it is required for the width-algorithm to have access to the homogeneous representation of points.

Has Models:
CGAL::Width_default_traits_3
CGAL::Width_3<Traits>

## Related Functions

(Note that these are not member functions.)

Point_3 make_point (const RT &hx, const RT &hy, const RT &hz, const RT &hw) const
returns a point of type Point_3 with homogeneous coordinates $$hx$$, $$hy$$, $$hz$$ and $$hw$$.

Plane_3 make_plane (const RT &a, const RT &b, const RT &c, const RT &d) const
returns a plane of type Plane_3 with coefficients $$a$$, $$b$$, $$c$$ and $$d$$.

Vector_3 make_vector (const RT &a, const RT &b, const RT &c) const
returns a vector of type Vector_3 with the four homogeneous coefficients $$a$$, $$b$$, $$c$$ and 1.

## Types

Notes: If you want to compute the width of a polyhedron then you have to make sure that the point type in the traits class and the point type in the polyhedron class are the same! The same holds for Traits::Plane_3 and Polyhedron::Plane_3.

typedef unspecified_type Point_3
The point type. More...

typedef unspecified_type Plane_3
The plane type. More...

typedef unspecified_type Vector_3
The vector type. More...

typedef unspecified_type ChullTraits
The traits class for using the convex hull algorithm. More...

typedef unspecified_type RT
Ring type numbers. More...

## Creation

Only a default constructor is required.

WidthTraits_3 ()

RT get_hx (const Point_3 &p) const
returns the homogeneous $$x$$-coordinate of point $$p$$.

RT get_hy (const Point_3 &p) const
returns the homogeneous $$y$$-coordinate of point $$p$$.

RT get_hz (const Point_3 &p) const
returns the homogeneous $$z$$-coordinate of point $$p$$.

RT get_hw (const Point_3 &p) const
returns the homogenizing coordinate of point $$p$$.

void get_point_coordinates (const Point_3 &p, RT &px, RT &py, RT &pz, RT &ph) const
returns all homogeneous coordinates of point $$p$$ at once.

RT get_a (const Plane_3 &f) const
returns the first coefficient of plane $$f$$.

RT get_b (const Plane_3 &f) const
returns the second coefficient of plane $$f$$.

RT get_c (const Plane_3 &f) const
returns the third coefficient of plane $$f$$.

RT get_d (const Plane_3 &f) const
returns the fourth coefficient of plane $$f$$.

void get_plane_coefficients (const Plane_3 &f, RT &a, RT &b, RT &c, RT &d) const
returns all four plane coefficients of $$f$$ at once.

## ◆ ChullTraits

The traits class for using the convex hull algorithm.

It must be a model of the concept ConvexHullTraits_3. This class is used only if the width is computed from a set of points.

## ◆ Plane_3

The plane type.

Access to the coefficients of the plane is made via the get_.() functions. Constructing a plane is done with the make_plane() operation.

## ◆ Point_3

The point type.

The (in)equality tests must be available. Access to the point coordinates is done via the get_.() functions. Constructing a point is done with the make_point( ) operation.

## ◆ RT

 typedef unspecified_type WidthTraits_3::RT

Ring type numbers.

Internally all numbers are treated as ring type numbers, i.e., neither $$/$$-operator nor $$\sqrt{.}$$ nor other inexact operations are used. But because the algorithm does not use any divisions, but multiplication instead, the numbers can get really big. Therefore it is recommended to use a ring type number, that provides values of arbitrary length. Furthermore it is assumed that the underlying number type of Point_3, Plane_3 and Vector_3 equals RT.

## ◆ Vector_3

The vector type.

There is no need to access the coefficients of a vector; only constructing is required and is done with the make_vector operation.