Given a set of sample points issued from a surface and a query point p, the functions surface_neighbors_3() compute the neighbors of p on the surface within the sample points.

If the sampling is sufficiently dense, the neighbors are provably close to the point p on the surface (cf. the manual pages and [2],[5]). They are defined to be the neighbors of p in the regular triangulation dual to the power diagram which is equivalent to the intersection of the Voronoi cell of the query point p with the tangent plane to the surface at p.

The functions surface_neighbors_certified_3() also return, in addition, a Boolean value that certifies whether or not, the Voronoi cell of p can be affected by points that lie outside the input range, i.e. outside the ball centered on p passing through the furthest sample point from p in the range [first, beyond). If the sample points are collected by a k-nearest neighbor or a range search query, this permits to verify that a large enough neighborhood has been considered.

Requirements

Dt is equivalent to the class Delaunay_triangulation_3.
OutputIterator::value_type is equivalent to Dt::Point_3, i.e. a point type.
ITraits is equivalent to the class Voronoi_intersection_2_traits_3<K>.

See Also: CGAL::Voronoi_intersection_2_traits_3<K>; 3D Surface Neighbor Coordinates Functions

Implementation

These functions compute the regular triangulation of the sample points and the point p using a traits class equivalent to Voronoi_intersection_2_traits_3<K>. They determine the neighbors of p in this triangulation. The functions which certify the result need to compute, in addition, the Voronoi vertices of the cell of p in this diagram.

Functions
template<class OutputIterator , class InputIterator , class Kernel >
OutputIterator	CGAL::surface_neighbors_3 (InputIterator first, InputIterator beyond, const typename Kernel::Point_3 &p, const typename Kernel::Vector_3 &normal, OutputIterator out, const Kernel &K)
	The sample points \( \mathcal{P}\) are provided in the range `[first, beyond)`. More...

template<class OutputIterator , class InputIterator , class ITraits >
OutputIterator	CGAL::surface_neighbors_3 (InputIterator first, InputIterator beyond, const typename ITraits::Point_2 &p, OutputIterator out, const ITraits &traits)
	The same as above only that the traits class must be instantiated by the user. More...

template<class OutputIterator , class InputIterator , class Kernel >
std::pair< OutputIterator, bool >	CGAL::surface_neighbors_certified_3 (InputIterator first, InputIterator beyond, const typename Kernel::Point_3 &p, const typename Kernel::Vector_3 &normal, OutputIterator out, const Kernel &K)
	Similar to the first function. More...

template<class OutputIterator , class InputIterator , class Kernel >
std::pair< OutputIterator, bool >	CGAL::surface_neighbors_certified_3 (InputIterator first, InputIterator beyond, const typename Kernel::Point_3 &p, const typename Kernel::Vector_3 &normal, const typename Kernel::FT &max_distance, OutputIterator out, const Kernel &kernel)
	The same as above except that this function takes the maximal distance from `p` to the points in the range `[first, beyond)` as additional parameter.

template<class OutputIterator , class InputIterator , class ITraits >
std::pair< OutputIterator, bool >	CGAL::surface_neighbors_certified_3 (InputIterator first, InputIterator beyond, const typename ITraits::Point_2 &p, OutputIterator out, const ITraits &traits)
	The same as above only that the traits class must be instantiated by the user. More...

template<class OutputIterator , class InputIterator , class ITraits >
std::pair< OutputIterator, bool >	CGAL::surface_neighbors_certified_3 (InputIterator first, InputIterator beyond, const typename ITraits::Point_2 &p, const typename ITraits::FT &max_distance, OutputIterator out, const ITraits &traits)
	The same as above with the parameter `max_distance`.

template<class Dt , class OutputIterator >
OutputIterator	CGAL::surface_neighbors_3 (const Dt &dt, const typename Dt::Geom_traits::Point_3 &p, const typename Dt::Geom_traits::Vector_3 &normal, OutputIterator out, typename Dt::Cell_handle start=typename Dt::Cell_handle())
	computes the surface neighbor coordinates with respect to the points that are vertices of the Delaunay triangulation `dt`. More...

template<class Dt , class OutputIterator , class ITraits >
OutputIterator	CGAL::surface_neighbors_3 (const Dt &dt, const typename ITraits::Point_2 &p, OutputIterator out, const ITraits &traits, typename Dt::Cell_handle start=typename Dt::Cell_handle())
	The same as above only that the parameter `traits` instantiates the geometric traits class. More...

Function Documentation

template<class OutputIterator , class InputIterator , class Kernel >

OutputIterator CGAL::surface_neighbors_3	(	InputIterator	first,
		InputIterator	beyond,
		const typename Kernel::Point_3 &	p,
		const typename Kernel::Vector_3 &	normal,
		OutputIterator	out,
		const Kernel &	K
	)

The sample points \( \mathcal{P}\) are provided in the range [first, beyond).

InputIterator::value_type is the point type Kernel::Point_3. The tangent plane is defined by the point p and the vector normal. The parameter K determines the kernel type that will instantiate the template parameter of Voronoi_intersection_2_traits_3<K>.

The surface neighbors of p are computed which are the neighbors of p in the regular triangulation that is dual to the intersection of the 3D Voronoi diagram of \( \mathcal{P}\) with the tangent plane. The point sequence that is computed by the function is placed starting at out. The function returns an iterator that is placed past-the-end of the resulting point sequence.