cgal/Interpolation/doc/Interpolation/Interpolation.txt

namespace CGAL {
/*!

\mainpage User Manual
\anchor Chapter_2D_and_Surface_Function_Interpolation
\anchor chapinterpolation
\cgalAutoToc

\authors Julia Flötotto

This package implements various neighbor coordinate computation functions
as well as different methods for scattered data interpolation.
Computation of natural and regular neighbor coordinates in 2D Euclidean space
is described in Section \ref seccoordinates.
Coordinate and neighbor computations on surfaces are discussed in Section \ref secsurface.
Finally, we describe the different interpolation methods offered by this package
in Section \ref secinterpolation.

Scattered data interpolation solves the following problem: given
measures of a function on a set of discrete data points, how to interpolate
this function at an arbitrary query point.
More formally, let \f$ \mathcal{P}=\{\mathbf{p_1},\ldots ,\mathbf{p_n}\}\f$ be a set of
\f$ n\f$ points in \f$ \mathbb{R}^2\f$ or \f$ \mathbb{R}^3\f$ and \f$ \Phi\f$ be a scalar
function defined on the convex hull of \f$ \mathcal{P}\f$. We assume that
the function values are known at the points of \f$ \mathcal{P}\f$, i.e.\ to
each \f$ \mathbf{p_i} \in \mathcal{P}\f$, we associate \f$ z_i =
\Phi(\mathbf{p_i})\f$. Knowledge of the gradient of \f$ \Phi\f$ at \f$ \mathbf{p_i}\f$
is also sometimes required. It is then denoted \f$ \mathbf{g_i}= \nabla
\Phi(\mathbf{p_i})\f$. The interpolation is carried out at an arbitrary query point
\f$ \mathbf{x}\f$ on the convex hull of \f$ \mathcal{P}\f$.

\section seccoordinates Natural Neighbor Coordinates

Most interpolation methods offered by this package rely on 2D natural and regular
neighbor coordinates, which we describe in this section.

\subsection InterpolationIntroduction Introduction

Natural neighbor interpolation was introduced by Sibson
\cgalCite{s-bdnni-81} to interpolate multivariate scattered data. Given
a set of data points \f$ \mathcal{P}\f$, the natural neighbor coordinates
associated to \f$ \mathcal{P}\f$ are defined from the Voronoi diagram of
\f$ \mathcal{P}\f$. When simulating the insertion of a query point
\f$ \mathbf{x}\f$ into the Voronoi diagram of \f$ \mathcal{P}\f$, the potential
Voronoi cell of \f$ \mathbf{x}\f$ "steals" some parts from neighboring existing
cells.

\cgalFigureBegin{fignn_coords, nn_coords.svg}
2D example: \f$ \mathbf{x}\f$ has five natural neighbors \f$ \mathbf{p_1},\ldots, \mathbf{p_5}\f$.
The natural neighbor coordinate \f$ \lambda_3(\mathbf{x})\f$ is the ratio of the area of the purple
polygon, \f$ \pi_3(\mathbf{x})\f$, over the area of the total highlighted zone.
\cgalFigureEnd

Let \f$ \pi(\mathbf{x})\f$ denote the volume of the potential Voronoi cell
of \f$ \mathbf{x}\f$ and \f$ \pi_i(\mathbf{x})\f$ denote the volume of the
sub-cell that would be stolen from the cell of \f$ \mathbf{p_i}\f$ by the
cell of \f$ \mathbf{x}\f$. The natural neighbor coordinate of \f$ \mathbf{x}\f$
with respect to the data point \f$ \mathbf{p_i}\in \mathcal{P}\f$ is defined by
\f[
\lambda_i(\mathbf{x}) =
\frac{\pi_i(\mathbf{x})}{\pi(\mathbf{x})}. \f]
A two-dimensional example
is depicted in \cgalFigureRef{fignn_coords}.

Various papers (\cgalCite{b-scaps-97}, \cgalCite{f-sodt-90}, \cgalCite{hs-vbihc-00},
\cgalCite{cgal:p-plcbd-93}, \cgalCite{s-vidt-80}) show that
the natural neighbor coordinates exhibit the following properties:
- \f$ \mathbf{x} = \sum_{i=1}^n \lambda_i(\mathbf{x}) \mathbf{p_i}\f$
(barycentric coordinate property).
- For any \f$ i,j \leq n, \lambda_i(\mathbf{p_j}) =
\delta_{ij}\f$, where \f$ \delta_{ij}\f$ is the Kronecker symbol.
- \f$ \sum_{i=1}^n \lambda_i(\mathbf{x}) = 1\f$ (partition of unity
property).

When the query point x is located on the envelope of the convex hull of \f$ \mathcal{P}\f$,
the potential Voronoi cell of x becomes infinite and:
- \f$ \pi(\mathbf{x}) = \infty\f$
- \f$ \lambda_i(\mathbf{x}) = 0 \f$, for all data point \f$ \mathbf{p_i}\f$ of \f$ \mathcal{P}\f$
except for the two endpoints -- say \f$ \mathbf{p}\f$ and \f$ \mathbf{q}\f$ -- of the edge where \f$ x\f$ lies.

The natural neighbor coordinate of \f$ \mathbf{x}\f$ with respect to these endpoints \f$ \mathbf{p}\f$
and \f$ \mathbf{q}\f$ will be:
- \f$ \lambda_p(\mathbf{x}) = \frac{\|\mathbf{x} - \mathbf{q}\| }{ \|\mathbf{q} - \mathbf{p}\|} \f$
- \f$ \lambda_q(\mathbf{x}) = \frac{\|\mathbf{x} - \mathbf{p}\| }{ \|\mathbf{q} - \mathbf{p}\|} \f$

Furthermore, Piper \cgalCite{cgal:p-plcbd-93} shows that the coordinate
functions are continuous in the convex hull of \f$ \mathcal{P}\f$ and
continuously differentiable except on the data points \f$ \mathcal{P}\f$.<BR>

\subsection InterpolationIntroductionRegular Regular Neighbor Coordinates

The previous definition naturally extends to weighted Voronoi diagrams. These
diagrams, also known as <em>power diagrams</em>, are obtained by considering weighted points
(the weight being a scalar) and considering a weighted distance, the <em>power distance</em>,
defined between two weighted points \f$ (p, \omega_p) \f$ and \f$ (q, \omega_q) \f$ by
\f$ \Pi( (p, \omega_p), (q, \omega_q) ) = pq^2 - \omega_p - \omega_q \f$.
See \ref Subsection_2D_Triangulations_Regular_Description "this section"
of the package \ref PkgTriangulation2 for an in-depth description of power diagrams.

\cgalFigureBegin{figrn_coords, rn_coords.svg}
Illustration of regular neighbor coordinates. The point set is the same
as in \cgalFigureRef{fignn_coords} but weights have been added. These weights
are shown using circles, with the radius of each circle being equal to the
square root of the weight of the point.
\cgalFigureEnd

\warning Contrary to Voronoi diagrams, a weighted point \f$ p_i\f$ does not necessarily
possess a non-empty cell in the power diagram of \f$ \mathcal{P}\f$ (with
\f$ p_i\in\mathcal{P}\f$). When this is the case, that point is then said to be <em>hidden</em>
and all of its regular neighbor coordinates are null.

\subsection InterpolationImplementation Implementation

The interpolation package of \cgal provides functions to compute
natural and regular neighbor coordinates for \f$ 2D\f$ points.
Refer to the reference pages `natural_neighbor_coordinates_2()` and
`regular_neighbor_coordinates_2()`.
In addition, the package provides functions to compute natural
neighbor coordinates on well sampled point set surfaces. See
Section \ref secsurface and the reference page
`surface_neighbor_coordinates_3()` for further information.

\subsubsection InterpolationCoordinateComputationApproach Computation of the Coordinates

Given a Delaunay triangulation or a regular triangulation, our implementation
computes natural and regular neighbor coordinates in two steps.
Firstly, the vertices in conflict with the query point (that is, the vertices
from which the query point will "steal") are determined. Then, the areas
\f$ \pi_i(\mathbf{x})\f$ are computed by triangulating the Voronoi
sub-cells. The output is threefold:
- points (or vertices) with a non-null coordinate along with these coordinates \f$ \pi_i(\mathbf{x})\f$,
- the normalization factor \f$ \pi(\mathbf{x})\f$,
- a Boolean indicator on whether the coordinate computation was successful
(which is equivalent to a Boolean indicating whether the query point lies in the
convex hull or not).

Note that if the query point has already been located in the triangulation (for example using
functions such as \link Triangulation_2::locate() locate()\endlink) and/or the boundary
edges of the conflict zone are already determined, alternative functions
allow to avoid the re-computation (see \ref PkgInterpolationNaturalNeighborCoordinates2
and \ref PkgInterpolationRegularNeighborCoordinates2).

\subsection InterpolationExampleforCoordinates Examples

This section presents some examples of utilization of the natural and regular
neighbor coordinates computation methods. Sections \ref InterpolationExampleforNaturalNeighborCoordinates
and \ref InterpolationExampleforRegularNeighborCoordinates show basic examples
of the computation of natural and regular neighbor coordinates.
Section \ref InterpolationExampleforRegularNeighborCoordinates shows a more
advanced use case of the API, where a functor is passed to change the output.

\subsubsection InterpolationExampleforNaturalNeighborCoordinates Example for Natural Neighbor Coordinates

The signature of all coordinate computation functions is about the
same.
\cgalExample{Interpolation/nn_coordinates_2.cpp}

\subsubsection InterpolationExampleforRegularNeighborCoordinates Example for Regular Neighbor Coordinates

For regular neighbor coordinates, it is sufficient to replace the name
of the function and the type of triangulation passed as parameter, as shown
in the example below:

\cgalExample{Interpolation/rn_coordinates_2.cpp}

\subsubsection InterpolationExampleforOutputFunctor Formatting the Output Coordinates

Internally, natural or regular neighbor coordinates are associated to the vertices of
the triangulation via objects of type `std::pair<Vertex_handle, Coord_type>`,
where `Coord_type` is the number type of the coordinates.
By default and for backward compatibility reasons, this output is converted
into objects of type `std::pair<Point, Coord_type>`, where `Point` is a bare
(weightless) point.

It is however possible to collect the output as objects of any desired
type, for example the simple `Point` type, by passing a functor as extra parameter
of the coordinates computation function.
The argument type of this functor must be `std::pair<Vertex_handle, Coord_type>`
and the result type is chosen by the user, but must be consistent with the output iterator.
Usage of this parameter is demonstrated in the example below, where the output is
kept as objects of type `std::pair<Vertex_handle, Coord_type>`, which allows
us to then store the coordinate in the vertex, using the class
`CGAL::Triangulation_vertex_base_with_info_2`.

\cgalExample{Interpolation/nn_coordinates_with_info_2.cpp}

\section secsurface Surface Natural Neighbor Coordinates and Surface Neighbors

This section introduces functions to compute natural neighbor
coordinates and surface neighbors associated to a set of sample points
issued from a surface \f$ \mathcal{S}\f$ and given a query point
\f$ \mathbf{x}\f$ on \f$ \mathcal{S}\f$. We assume that \f$ \mathcal{S}\f$ is a
closed and compact surface of \f$ \mathbb{R}^3\f$, and let \f$ \mathcal{P}=
\{\mathbf{p_1}, \ldots,\mathbf{p_n}\}\f$ be an \f$ \epsilon\f$-sample of
\f$ \mathcal{S}\f$ (refer to Amenta and Bern \cgalCite{ab-srvf-99}). The
concepts are based on the definition of Boissonnat and Fl&ouml;totto
\cgalCite{bf-lcss-02}, \cgalCite{cgal:f-csapc-03}. Both references
contain a thorough description of the requirements and the
mathematical properties.

\subsection InterpolationIntroduction_1 Introduction

Two observations lead to the definition of surface neighbors and
surface neighbor coordinates: First, it is clear that the tangent
plane \f$ \mathcal{T}_x\f$ of the surface \f$ \mathcal{S}\f$ at the point
\f$ \mathbf{x} \in \mathcal{S}\f$ approximates \f$ \mathcal{S}\f$ in the
neighborhood of \f$ \mathbf{x}\f$. It has been shown in \cgalCite{bf-lcss-02}
that, if the surface \f$ \mathcal{S}\f$ is well sampled with respect to the
curvature and the local thickness of \f$ \mathcal{S}\f$, i.e.\ it is an \f$ \epsilon\f$-sample,
then the intersection of the tangent plane \f$ \mathcal{T}_x\f$ with the Voronoi cell of
\f$ \mathbf{x}\f$ in the Voronoi diagram of \f$ \mathcal{P} \cup
\{\mathbf{x}\}\f$ has a small diameter. Consequently, inside this
Voronoi cell, the tangent plane \f$ \mathcal{T}_x\f$ is a reasonable
approximation of \f$ \mathcal{S}\f$. Furthermore, the second observation
allows to compute this intersection diagram easily: one can show using
Pythagoras' Theorem that the intersection of a three-dimensional
Voronoi diagram with a plane \f$ \mathcal{H}\f$ is a two-dimensional power
diagram. The points defining the power diagram are the projections of
the points in \f$ \mathcal{P}\f$ onto \f$ \mathcal{H}\f$, each point weighted
with its negative square distance to \f$ \mathcal{H}\f$. Algorithms for the
computation of power diagrams via the dual regular triangulation are
well known and for example provided by \cgal in the class
`Regular_triangulation_2<Gt, Tds>`.

\subsection InterpolationImplementation_1 Implementation

\subsubsection InterpolationVoronoiIntersectionDiagrams Voronoi Intersection Diagrams

In \cgal, the regular triangulation dual to the intersection of a \f$ 3D\f$
Voronoi diagram with a plane \f$ \mathcal{H}\f$ can be computed by
instantiating the `Regular_triangulation_2<Gt, Tds>` class with
the traits class `Voronoi_intersection_2_traits_3<K>`. This traits
class contains a point and a vector as class member which define the
plane \f$ \mathcal{H}\f$. All predicates and constructions used by
`Regular_triangulation_2<Gt, Tds>` are replaced by the
corresponding operators on three-dimensional points. For example, the
power test predicate (which takes three weighted \f$ 2D\f$ points
\f$ p'\f$, \f$ q'\f$, \f$ r'\f$ of the regular triangulation and tests the power
distance of a fourth point \f$ t'\f$ with respect to the power circle orthogonal
to \f$ p\f$, \f$ q\f$, \f$ r\f$) is replaced by a
`Side_of_plane_centered_sphere_2_3` predicate that tests the
position of a \f$ 3D\f$ point \f$ t\f$ with respect to the sphere centered on
the plane \f$ \mathcal{H}\f$ passing through the \f$ 3D\f$ points \f$ p\f$, \f$ q\f$, \f$ r\f$.
This approach allows to avoid the explicit construction of the
projected points and the weights, operations which are very prone to rounding
errors.

\subsubsection InterpolationNaturalNeighborCoordinateson Natural Neighbor Coordinates on Surfaces

The computation of natural neighbor coordinates on surfaces is based
upon the computation of regular neighbor coordinates with respect to
the regular triangulation that is dual to \f$ {\rm Vor}(\mathcal{P}) \cap
\mathcal{T}_x\f$, the intersection of \f$ \mathcal{T}_x\f$ and the Voronoi
diagram of \f$ \mathcal{P}\f$, via the function
`regular_neighbor_coordinates_2()`.

Of course, we might introduce all data points \f$ \mathcal{P}\f$ into this
regular triangulation. However, this is not necessary because we are
only interested in the cell of \f$ \mathbf{x}\f$. It is sufficient to
guarantee that all surface neighbors of the query point \f$ \mathbf{x}\f$
are among the input points that are passed as argument to the
function. The sample points \f$ \mathcal{P}\f$ can be filtered for example
by distance, e.g. using range search or \f$ k\f$-nearest neighbor queries,
or with the help of the \f$ 3D\f$ Delaunay triangulation since the surface
neighbors are necessarily a subset of the natural neighbors of the
query point in this triangulation. \cgal provides a function that
encapsulates the filtering based on the \f$ 3D\f$ Delaunay triangulation.
For input points filtered by distance, functions are provided that
indicate whether or not points that lie outside the input range (i.e.\
points that are further from \f$ \mathbf{x}\f$ than the furthest input
point) can still influence the result. This allows to iteratively
enlarge the set of input points until the range is sufficient to
certify the result.

\subsubsection InterpolationSurfaceNeighbors Surface Neighbors

The surface neighbors of the query point are its neighbors in the
regular triangulation that is dual to \f$ {\rm Vor}(\mathcal{P}) \cap
\mathcal{T}_x\f$, the intersection of \f$ \mathcal{T}_x\f$ and the Voronoi
diagram of \f$ \mathcal{P}\f$. As for surface neighbor coordinates, this
regular triangulation is computed and the same kind of filtering of
the data points as well as the certification described above is
provided.

\subsection InterpolationExampleforSurfaceNeighborCoordinates Example for Surface Neighbor Coordinates

The example below describes the usage of the function `CGAL::surface_neighbor_coordinates_3()`.

\cgalExample{Interpolation/surface_neighbor_coordinates_3.cpp}

\section secinterpolation Interpolation Methods

We describe in this section the different interpolation methods offered by this package.
These can be regrouped into two large classes: values and gradient interpolation
methods.

\subsection InterpolationIntroduction_2 Interpolation of Function Values

The interpolation functions presented below are used to interpolate function values.

\subsubsection InterpolationLinearPrecisionInterpolation Linear Precision Interpolation

Sibson \cgalCite{s-bdnni-81} defines a very simple interpolant that
re-produces linear functions exactly. The interpolation of
\f$ \Phi(\mathbf{x})\f$ is given as the linear combination of the neighbors' function
values weighted by the coordinates:
\f[
Z^0(\mathbf{x}) = \ccSum{i}{}{ \lambda_i(\mathbf{x}) z_i}.
\f]
Indeed, if \f$ z_i=a + \mathbf{b}^t \mathbf{p_i}\f$ for all natural
neighbors of \f$ \mathbf{x}\f$, we have
\f[ Z^0(\mathbf{x}) = \ccSum{i}{}{ \lambda_i(\mathbf{x}) (a + \mathbf{b}^t\mathbf{p_i})} = a+\mathbf{b}^t \mathbf{x} \f]
by the barycentric coordinate property.

This interpolation is implemented through the function `        CGAL::linear_interpolation()`.
The first example in Subsection \ref subsecinterpol_examples shows how the function is
called.

\subsubsection InterpolationSibson Sibson's C^1 Continuous Interpolant

In \cgalCite{s-bdnni-81}, Sibson describes a second interpolation method
that relies also on the function gradient \f$ \mathbf{g_i}\f$ for all \f$ \mathbf{p_i} \in \mathcal{P}\f$. It is \f$ C^1\f$ continuous with gradient \f$ \mathbf{g_i}\f$ at
\f$ \mathbf{p_i}\f$. Spherical quadrics of the form \f$ \Phi(\mathbf{x}) =a +
\mathbf{b}^t \mathbf{x} +\gamma\ \mathbf{x}^t\mathbf{x}\f$ are reproduced
exactly. The
proof relies on the barycentric coordinate property of the natural
neighbor coordinates and assumes that the gradient of \f$ \Phi\f$ at the
data points is known or approximated from the function values as
described in \cgalCite{s-bdnni-81} (see Section \ref sgradient_fitting).

Sibson's \f$ Z^1\f$ interpolant is a combination of the linear interpolant
\f$ Z^0\f$ and an interpolant \f$ \xi\f$ which is the weighted sum of the first
degree functions
\f[ \xi_i(\mathbf{x}) = z_i
+\mathbf{g_i}^t(\mathbf{x}-\mathbf{p_i}),\qquad \xi(\mathbf{x})= \frac{\ccSum{i}{}{ \frac{\lambda_i(\mathbf{x})}
{\|\mathbf{x}-\mathbf{p_i}\|}\xi_i(\mathbf{x}) } }{\ccSum{i}{}{
\frac{\lambda_i(\mathbf{x})}{\|\mathbf{x}-\mathbf{p_i}\|}}}. \f]
Sibson observed that the combination of \f$ Z^0\f$ and \f$ \xi\f$ reconstructs exactly
a spherical quadric if they are mixed as follows:
\f[
Z^1(\mathbf{x}) = \frac{\alpha(\mathbf{x}) Z^0(\mathbf{x}) +
\beta(\mathbf{x}) \xi(\mathbf{x})}{\alpha(\mathbf{x}) +
\beta(\mathbf{x})} \mbox{ where } \alpha(\mathbf{x}) =
\frac{\ccSum{i}{}{ \lambda_i(\mathbf{x}) \frac{\|\mathbf{x} -
\mathbf{p_i}\|^2}{f(\|\mathbf{x} - \mathbf{p_i}\|)}}}{\ccSum{i}{}{
\frac{\lambda_i(\mathbf{x})} {f(\|\mathbf{x} - \mathbf{p_i}\|)}}}
\mbox{ and } \beta(\mathbf{x})= \ccSum{i}{}{ \lambda_i(\mathbf{x})
\|\mathbf{x} - \mathbf{p_i}\|^2}, \f]
where in Sibson's original work,
\f$ f(\|\mathbf{x} - \mathbf{p_i}\|) = \|\mathbf{x} - \mathbf{p_i}\|\f$.

This interpolation method can be used by calling `CGAL::sibson_c1_interpolation()`.

\cgal contains a second implementation using \f$ f(\|\mathbf{x} -
\mathbf{p_i}\|) = \|\mathbf{x} - \mathbf{p_i}\|^2\f$, `CGAL::sibson_c1_interpolation_square()`,
which is less demanding on the number type because it avoids the square-root
computation needed to compute the distance \f$ \|\mathbf{x} -
\mathbf{p_i}\|\f$. The theoretical guarantees are the same (see
\cgalCite{cgal:f-csapc-03}). Simply, the smaller the slope of \f$ f\f$
around \f$ f(0)\f$, the faster the interpolant approaches \f$ \xi_i\f$ as
\f$ \mathbf{x} \f$ goes to \f$ \mathbf{p_i}\f$.

\subsubsection InterpolationFarin Farin's C^1 Continuous Interpolant

Farin \cgalCite{f-sodt-90} extended Sibson's work and realizes a \f$ C^1\f$
continuous interpolant by embedding natural neighbor coordinates in
the Bernstein-B&eacute;zier representation of a cubic simplex. If the
gradient of \f$ \Phi\f$ at the data points is known, this interpolant
reproduces quadratic functions exactly. The function gradient can be
approximated from the function values by Sibson's method
\cgalCite{s-bdnni-81} (see Section \ref sgradient_fitting) which is exact only
for spherical quadrics.

The Farin \f$ C^1\f$-continuous interpolant is implemented in the function
`CGAL::farin_c1_interpolation()`.

\subsubsection InterpolationQuadraticPrecisionInterpolants Quadratic Precision Interpolants

Knowing the gradient \f$ \mathbf{g_i}\f$ for all \f$ \mathbf{p_i} \in
\mathcal{P}\f$, we formulate a very simple interpolant that reproduces
exactly quadratic functions. This interpolant is not \f$ C^1\f$ continuous
in general. It is defined as follows:
\f[
I^1(\mathbf{x}) = \ccSum{i}{}{ \lambda_i(\mathbf{x})
(z_i + \frac{1}{2} \mathbf{g_i}^t (\mathbf{x} - \mathbf{p_i}))}
\f]

This interpolation can be used with the method `CGAL::quadratic_interpolation()`.

\subsection sgradient_fitting Gradient Fitting

Sibson describes a method to approximate the gradient of the function
\f$ f\f$ from the function values on the data sites. For the data point
\f$ \mathbf{p_i}\f$, we determine
\f[ \mathbf{g_i}
= \min_{\mathbf{g}}
\ccSum{j}{}{
\frac{\lambda_j(\mathbf{p_i})}{\|\mathbf{p_i} - \mathbf{p_j}\|^2}
\left( z_j - (z_i + \mathbf{g}^t (\mathbf{p_j} -\mathbf{p_i})) \right)},
\f]
where \f$ \lambda_j(\mathbf{p_i})\f$ is the natural neighbor coordinate
of \f$ \mathbf{p_i}\f$ with respect to \f$ \mathbf{p_i}\f$ associated to
\f$ \mathcal{P} \setminus \{\mathbf{p_i}\}\f$. For spherical quadrics, the result is exact.

\cgal provides functions to approximate the gradients of all data
points that are inside the convex hull. There is one function for each
type of neighbor coordinates:
- `natural_neighbor_coordinates_2()`: `CGAL::sibson_gradient_fitting_nn_2()`
- `regular_neighbor_coordinates_2()`: `CGAL::sibson_gradient_fitting_rn_2()`.

\subsection subsecinterpol_examples Example for Linear Interpolation

The example below shows a typical example of the usage of `CGAL::linear_interpolation()`.
Note the use of `CGAL::Data_access` to transform the `std::map` in a functor
compatible with the function.

\cgalExample{Interpolation/linear_interpolation_2.cpp}

The next example shows the interpolation of non-scalar values, namely `Vector_3`.
It may be any type `T` which provides a  multiplication operator with a scalar, and
the addition operator for `T`.

\cgalExample{Interpolation/linear_interpolation_of_vector_3.cpp}

\subsection InterpolationExampleSibson Example for Sibson's C^1 Interpolation Scheme with Gradient Estimation

\cgalExample{Interpolation/sibson_interpolation_2.cpp}

The example \link Interpolation/interpolation_2.cpp interpolation_2.cpp \endlink
compares numerically the errors of the different interpolation functions
with respect to a known function.

\subsection InterpolationExampleVertices Example for Storing Values and Gradients in Vertices

In the previous examples, we have stored the values and gradients in a `std::map`
and wrapped them in a `CGAL::Data_access` object. We can avoid this "external" storage by
using the class `Triangulation_vertex_base_with_info_2` provided by the triangulation package:
values and gradients can be stored directly in the vertices of the triangulation.
Functors and output iterators requested by the gradient fitting and interpolation functions
are then simply wrappers providing read/write accesses to the `info` member of the vertices.
This approach is shown in the example below.

\cgalExample{Interpolation/sibson_interpolation_vertex_with_info_2.cpp}

\section Interpolation Design and Implementation History

The original version was written by Julia Fl&ouml;totto, while working towards her PhD thesis at Inria.
The possibility to use values and gradients in a `Triangulation_vertex_base_with_info_2` was introduced
by Andreas Fabri and Mael Rouxel-Labb&eacute; working at GeometryFactory.

*/
} /* namespace CGAL */