diff --git a/Interpolation/doc/Interpolation/Interpolation.txt b/Interpolation/doc/Interpolation/Interpolation.txt index c0ec6c8b153..1a73f25cbf5 100644 --- a/Interpolation/doc/Interpolation/Interpolation.txt +++ b/Interpolation/doc/Interpolation/Interpolation.txt @@ -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 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 same. \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 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 InterpolationVoronoiIntersectionDiagrams Voronoi Intersection Diagrams +\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 @@ -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 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 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 certify the result. -\subsection InterpolationSurfaceNeighbors Surface Neighbors +\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 @@ -235,7 +235,7 @@ provided. \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 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 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 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 \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$ 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 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 \mathcal{P}\f$, we formulate a very simple interpolant that reproduces