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Andreas Fabri 2013-01-11 11:04:19 +01:00
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Interpolation/doc/Interpolation/Interpolation.txt
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@ -114,13 +114,13 @@ returned. If the query point is already located and/or the boundary
edges of the conflict zone are already determined, alternative edges of the conflict zone are already determined, alternative
functions allow to avoid the re-computation. functions allow to avoid the re-computation.
\subsection InterpolationExampleforNaturalNeighborCoordinates Example for Natural Neighbor Coordinates \subsubsection InterpolationExampleforNaturalNeighborCoordinates Example for Natural Neighbor Coordinates
The signature of all coordinate computation functions is about the The signature of all coordinate computation functions is about the
same. same.
\cgalExample{Interpolation/nn_coordinates_2.cpp} \cgalExample{Interpolation/nn_coordinates_2.cpp}
\subsection InterpolationExampleforRegularNeighborCoordinates Example for Regular Neighbor Coordinates \subsubsection InterpolationExampleforRegularNeighborCoordinates Example for Regular Neighbor Coordinates
For regular neighbor coordinates, it is sufficient to replace the name For regular neighbor coordinates, it is sufficient to replace the name
of the function and the type of triangulation passed as parameter. A of the function and the type of triangulation passed as parameter. A
@ -169,7 +169,7 @@ well known and for example provided by \cgal in the class
\subsection InterpolationImplementation_1 Implementation \subsection InterpolationImplementation_1 Implementation
\subsection InterpolationVoronoiIntersectionDiagrams Voronoi Intersection Diagrams \subsubsection InterpolationVoronoiIntersectionDiagrams Voronoi Intersection Diagrams
In \cgal, the regular triangulation dual to the intersection of a \f$ 3D\f$ 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 Voronoi diagram with a plane \f$ \mathcal{H}\f$ can be computed by
@ -190,7 +190,7 @@ This approach allows to avoid the explicit constructions of the
projected points and the weights which are very prone to rounding projected points and the weights which are very prone to rounding
errors. errors.
\subsection InterpolationNaturalNeighborCoordinateson Natural Neighbor Coordinates on Surfaces \subsubsection InterpolationNaturalNeighborCoordinateson Natural Neighbor Coordinates on Surfaces
The computation of natural neighbor coordinates on surfaces is based The computation of natural neighbor coordinates on surfaces is based
upon the computation of regular neighbor coordinates with respect to upon the computation of regular neighbor coordinates with respect to
@ -217,7 +217,7 @@ point) can still influence the result. This allows to iteratively
enlarge the set of input points until the range is sufficient to enlarge the set of input points until the range is sufficient to
certify the result. certify the result.
\subsection InterpolationSurfaceNeighbors Surface Neighbors \subsubsection InterpolationSurfaceNeighbors Surface Neighbors
The surface neighbors of the query point are its neighbors in the The surface neighbors of the query point are its neighbors in the
regular triangulation that is dual to \f$ {\rm Vor}(\mathcal{P}) \cap regular triangulation that is dual to \f$ {\rm Vor}(\mathcal{P}) \cap
@ -235,7 +235,7 @@ provided.
\subsection InterpolationIntroduction_2 Introduction \subsection InterpolationIntroduction_2 Introduction
\subsection InterpolationLinearPrecisionInterpolation Linear Precision Interpolation \subsubsection InterpolationLinearPrecisionInterpolation Linear Precision Interpolation
Sibson \cite s-bdnni-81 defines a very simple interpolant that Sibson \cite s-bdnni-81 defines a very simple interpolant that
re-produces linear functions exactly. The interpolation of re-produces linear functions exactly. The interpolation of
@ -251,7 +251,7 @@ by the barycentric coordinate property. The first example in
Subsection \ref subsecinterpol_examples shows how the function is Subsection \ref subsecinterpol_examples shows how the function is
called. called.
\subsection InterpolationSibson Sibson's C^1 Continuous Interpolant \subsubsection InterpolationSibson Sibson's C^1 Continuous Interpolant
In \cite s-bdnni-81, Sibson describes a second interpolation method In \cite 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 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
@ -293,7 +293,7 @@ computation needed to compute the distance \f$ \|\mathbf{x} -
around \f$ f(0)\f$, the faster the interpolant approaches \f$ \xi_i\f$ as around \f$ f(0)\f$, the faster the interpolant approaches \f$ \xi_i\f$ as
\f$ \mathbf{x} \rightarrow \mathbf{p_i}\f$. \f$ \mathbf{x} \rightarrow \mathbf{p_i}\f$.
\subsection InterpolationFarin Farin's C^1 Continuous Interpolant \subsubsection InterpolationFarin Farin's C^1 Continuous Interpolant
Farin \cite f-sodt-90 extended Sibson's work and realizes a \f$ C^1\f$ Farin \cite f-sodt-90 extended Sibson's work and realizes a \f$ C^1\f$
continuous interpolant by embedding natural neighbor coordinates in continuous interpolant by embedding natural neighbor coordinates in
@ -304,7 +304,7 @@ approximated from the function values by Sibson's method
\cite s-bdnni-81 (see Section \ref sgradient_fitting) which is exact only \cite s-bdnni-81 (see Section \ref sgradient_fitting) which is exact only
for spherical quadrics. for spherical quadrics.
\subsection InterpolationQuadraticPrecisionInterpolants Quadratic Precision Interpolants \subsubsection InterpolationQuadraticPrecisionInterpolants Quadratic Precision Interpolants
Knowing the gradient \f$ \mathbf{g_i}\f$ for all \f$ \mathbf{p_i} \in Knowing the gradient \f$ \mathbf{g_i}\f$ for all \f$ \mathbf{p_i} \in
\mathcal{P}\f$, we formulate a very simple interpolant that reproduces \mathcal{P}\f$, we formulate a very simple interpolant that reproduces