3D Surface Neighbor Coordinates Functions

The functions surface_neighbor_coordinates_3() compute natural neighbor coordinates for surface points associated to a finite set of sample points issued from the surface.

The coordinates are computed from the intersection of the Voronoi cell of the query point p with the tangent plane to the surface at p. If the sampling is sufficiently dense, the coordinate system meets the properties described in the manual pages and in [2],[5]. The query point p needs to lie inside the convex hull of the projection of the sample points onto the tangent plane at p.

The functions surface_neighbor_coordinates_certified_3() return, in addition, a second Boolean value (the fourth value of the quadruple) 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 check whether the neighborhood which has been considered is large enough.

Requirements

1. Dt is equivalent to the class Delaunay_triangulation_3.
2. The value type of OutputIterator is equivalent to std::pair<Dt::Point_3, Dt::Geom_traits::FT>, i.e. a pair associating a point and its natural neighbor coordinate.
3. ITraits is equivalent to the class Voronoi_intersection_2_traits_3<K>.

See also

CGAL::linear_interpolation()

CGAL::sibson_c1_interpolation()

CGAL::farin_c1_interpolation()

CGAL::Voronoi_intersection_2_traits_3<K>

3D Surface Neighbors Functions

Implementation

This functions construct the regular triangulation of the input points instantiated with Voronoi_intersection_2_traits_3<Kernel> or ITraits if provided. They return the result of the function call CGAL::regular_neighbor_coordinates_2() with the regular triangulation and p as arguments.

Functions
template<class OutputIterator, class InputIterator, class Kernel>
CGAL::Triple< OutputIterator, typename Kernel::FT, bool >	CGAL::surface_neighbor_coordinates_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).
template<class OutputIterator, class InputIterator, class ITraits>
CGAL::Triple< OutputIterator, typename ITraits::FT, bool >	CGAL::surface_neighbor_coordinates_3 (InputIterator first, InputIterator beyond, const typename ITraits::Point_2 &p, OutputIterator out, const ITraits &traits)
	Same as above only that the traits class must be instantiated by the user.
template<class OutputIterator, class InputIterator, class Kernel>
CGAL::Quadruple< OutputIterator, typename Kernel::FT, bool, bool >	CGAL::surface_neighbor_coordinates_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.
template<class OutputIterator, class InputIterator, class Kernel>
CGAL::Quadruple< OutputIterator, typename Kernel::FT, bool, bool >	CGAL::surface_neighbor_coordinates_certified_3 (InputIterator first, InputIterator beyond, const typename Kernel::Point_3 &p, const typename Kernel::FT &max_distance, OutputIterator out, const Kernel &kernel)
	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>
CGAL::Quadruple< OutputIterator, typename ITraits::FT, bool, bool >	CGAL::surface_neighbor_coordinates_certified_3 (InputIterator first, InputIterator beyond, const typename ITraits::Point_2 &p, OutputIterator out, const ITraits &traits)
	Same as above only that the traits class must be instantiated by the user and without the parameter max_distance.
template<class OutputIterator, class InputIterator, class ITraits>
CGAL::Quadruple< OutputIterator, typename ITraits::FT, bool, bool >	CGAL::surface_neighbor_coordinates_certified_3 (InputIterator first, InputIterator beyond, const typename ITraits::Point_2 &p, const typename ITraits::FT &max_distance, OutputIterator out, const ITraits &traits)
	Same as above with the parameter max_distance.
template<class Dt, class OutputIterator>
CGAL::Triple< OutputIterator, typename Dt::Geom_traits::FT, bool >	CGAL::surface_neighbor_coordinates_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.
template<class Dt, class OutputIterator, class ITraits>
CGAL::Triple< OutputIterator, typename Dt::Geom_traits::FT, bool >	CGAL::surface_neighbor_coordinates_3 (const Dt &dt, const typename Dt::Geom_traits::Point_3 &p, OutputIterator out, const ITraits &traits, typename Dt::Cell_handle start=typename Dt::Cell_handle())
	Same as above only that the parameter traits instantiates the geometric traits class.

Function Documentation

surface_neighbor_coordinates_3() [1/4]

template<class Dt, class OutputIterator>

CGAL::Triple< OutputIterator, typename Dt::Geom_traits::FT, bool > CGAL::surface_neighbor_coordinates_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() )

#include <CGAL/surface_neighbor_coordinates_3.h>

Computes the surface neighbor coordinates with respect to the points that are vertices of the Delaunay triangulation dt.

The type Dt must be equivalent to Delaunay_triangulation_3<Gt, Tds>. The optional parameter start is used as a starting place for the search of the conflict zone. It may be the result of the call dt.locate(p). This function instantiates the template parameter ITraits to be Voronoi_intersection_2_traits_3<Dt::Geom_traits>.

This function allows to filter some potential neighbors of the query point p from \( \mathcal{P}\) via its three-dimensional Delaunay triangulation. All surface neighbors of p are necessarily neighbors in the Delaunay triangulation of \( \mathcal{P} \cup \{p\}\).

surface_neighbor_coordinates_3() [2/4]

template<class Dt, class OutputIterator, class ITraits>

CGAL::Triple< OutputIterator, typename Dt::Geom_traits::FT, bool > CGAL::surface_neighbor_coordinates_3	(	const Dt &	dt,
		const typename Dt::Geom_traits::Point_3 &	p,
		OutputIterator	out,
		const ITraits &	traits,
		typename Dt::Cell_handle	start = typename Dt::Cell_handle() )

#include <CGAL/surface_neighbor_coordinates_3.h>

Same as above only that the parameter traits instantiates the geometric traits class.

Its type ITraits must be equivalent to Voronoi_intersection_2_traits_3<K>.

surface_neighbor_coordinates_3() [3/4]

template<class OutputIterator, class InputIterator, class ITraits>

CGAL::Triple< OutputIterator, typename ITraits::FT, bool > CGAL::surface_neighbor_coordinates_3	(	InputIterator	first,
		InputIterator	beyond,
		const typename ITraits::Point_2 &	p,
		OutputIterator	out,
		const ITraits &	traits )

#include <CGAL/surface_neighbor_coordinates_3.h>

Same as above only that the traits class must be instantiated by the user.

ITraits must be equivalent to Voronoi_intersection_2_traits_3<K>.

surface_neighbor_coordinates_3() [4/4]

template<class OutputIterator, class InputIterator, class Kernel>

CGAL::Triple< OutputIterator, typename Kernel::FT, bool > CGAL::surface_neighbor_coordinates_3	(	InputIterator	first,
		InputIterator	beyond,
		const typename Kernel::Point_3 &	p,
		const typename Kernel::Vector_3 &	normal,
		OutputIterator	out,
		const Kernel &	K )

#include <CGAL/surface_neighbor_coordinates_3.h>

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

The value type of InputIterator 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 natural neighbor coordinates for p are computed in the power diagram that results from the intersection of the 3D Voronoi diagram of \( \mathcal{P}\) with the tangent plane. The sequence of point/coordinate pairs that is computed by the function is placed starting at out. The function returns a triple with an iterator that is placed past-the-end of the resulting sequence of point/coordinate pairs, the normalization factor of the coordinates and a Boolean value which is set to true iff the coordinate computation was successful, i.e. if p lies inside the convex hull of the projection of the points \( \mathcal{P}\) onto the tangent plane.

Examples

Interpolation/surface_neighbor_coordinates_3.cpp.

surface_neighbor_coordinates_certified_3() [1/2]

template<class OutputIterator, class InputIterator, class ITraits>

CGAL::Quadruple< OutputIterator, typename ITraits::FT, bool, bool > CGAL::surface_neighbor_coordinates_certified_3	(	InputIterator	first,
		InputIterator	beyond,
		const typename ITraits::Point_2 &	p,
		OutputIterator	out,
		const ITraits &	traits )

#include <CGAL/surface_neighbor_coordinates_3.h>

Same as above only that the traits class must be instantiated by the user and without the parameter max_distance.

ITraits must be equivalent to Voronoi_intersection_2_traits_3<K>.

surface_neighbor_coordinates_certified_3() [2/2]

template<class OutputIterator, class InputIterator, class Kernel>

CGAL::Quadruple< OutputIterator, typename Kernel::FT, bool, bool > CGAL::surface_neighbor_coordinates_certified_3	(	InputIterator	first,
		InputIterator	beyond,
		const typename Kernel::Point_3 &	p,
		const typename Kernel::Vector_3 &	normal,
		OutputIterator	out,
		const Kernel &	K )

#include <CGAL/surface_neighbor_coordinates_3.h>

Similar to the first function.

The additional fourth return value is true if the furthest point in the range [first, beyond) is further away from p than twice the distance from p to the furthest vertex of the intersection of the Voronoi cell of p with the tangent plane defined by (p,normal). It is false otherwise.