## CGAL::Polyhedron_traits_with_normals_3<Kernel>

### Definition

The class Polyhedron_traits_with_normals_3<Kernel> is a model of the PolyhedronTraits_3 concept. It defines the geometric types and primitive operations used in the polyhedral surface data structure CGAL::Polyhedron_3<PolyhedronTraits_3>. Polyhedron_traits_with_normals_3<Kernel> uses the normal vector from Kernel for the plane equation in facets. It keeps a local copy of the kernel which makes it suitable for kernels with local state.

#include <CGAL/Polyhedron_traits_with_normals_3.h>

### Is Model for the Concept

PolyhedronTraits_3

### Types

Polyhedron_traits_with_normals_3<Kernel>::Kernel
| the Kernel model.

typedef Kernel::Point_3
| Point_3;
typedef Kernel::Vector_3
| Plane_3;
typedef Kernel::Construct_opposite_vector_3
| Construct_opposite_plane_3;

### Creation

Polyhedron_traits_with_normals_3<Kernel> traits;
| default constructor, uses Kernel() as local reference to the kernel.
Polyhedron_traits_with_normals_3<Kernel> traits ( Kernel kernel);
| stores kernel as local reference.

### Operations

Construct_opposite_plane_3
| traits.construct_opposite_plane_3_object ()
|  | forwarded to kernel.

### See Also

CGAL::Polyhedron_traits_3<Kernel>

### Example

We use this traits class to instantiate a polyhedral surface with a normal vector and no plane equation for each facet. We compute the normal vector assuming exact arithmetic (integers in this example) and convex planar facets.

    // file: examples/Polyhedron/polyhedron_prog_normals.C

    #include <CGAL/Homogeneous.h>
    #include <CGAL/Polyhedron_traits_with_normals_3.h>
    #include <CGAL/Polyhedron_3.h>
    #include <iostream>
    #include <algorithm>

    struct Normal_vector {
        template <class Facet>
        typename Facet::Plane_3 operator()( Facet& f) {
            typename Facet::Halfedge_handle h = f.halfedge();
            // Facet::Plane_3 is the normal vector type. We assume the
            // CGAL Kernel here and use its global functions.
            return CGAL::cross_product(
              h->next()->vertex()->point() - h->vertex()->point(),
              h->next()->next()->vertex()->point() - h->next()->vertex()->point());
        }
    };

    typedef CGAL::Homogeneous<int>                         Kernel;
    typedef Kernel::Point_3                                Point_3;
    typedef Kernel::Vector_3                               Vector_3;
    typedef CGAL::Polyhedron_traits_with_normals_3<Kernel> Traits;
    typedef CGAL::Polyhedron_3<Traits>                     Polyhedron;

    int main() {
        Point_3 p( 1, 0, 0);
        Point_3 q( 0, 1, 0);
        Point_3 r( 0, 0, 1);
        Point_3 s( 0, 0, 0);
        Polyhedron P;
        P.make_tetrahedron( p, q, r, s);
        std::transform( P.facets_begin(), P.facets_end(), P.planes_begin(),
                        Normal_vector());
        CGAL::set_pretty_mode( std::cout);
        std::copy( P.planes_begin(), P.planes_end(),
                   std::ostream_iterator<Vector_3>( std::cout, "\n"));
        return 0;
    }