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CGAL 5.5 - 2D and Surface Function Interpolation
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CGAL 5.5 - 2D and Surface Function Interpolation
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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
Dt is equivalent to the class Delaunay_triangulation_3. 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. ITraits is equivalent to the class Voronoi_intersection_2_traits_3<K>. CGAL::linear_interpolation() CGAL::sibson_c1_interpolation() CGAL::farin_c1_interpolation() CGAL::Voronoi_intersection_2_traits_3<K> 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 PkgInterpolationRegularNeighborCoordinates2 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). More... | |
| 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. More... | |
| 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. More... | |
| 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. More... | |
| 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. More... | |
| 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. More... | |
| 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.
| 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>.
| 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 \( \mathcal{P} \cup \{p\}\).
| 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() |
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| ) |
#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>.
| 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.
| 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>.
1.8.13