- corrections

Packages/Interpolation/doc_tex/Interpolation/interpolation.tex

@@ -18,10 +18,10 @@ called.
 
 \subsubsection{Sibson's $C^1$ continuous interpolant}
 In \cite{s-bdnni-81}, Sibson describes a second interpolation method
-that is $C^1$ continuous with gradient $\mathbf{g_i}$ at
-$\mathbf{p_i}$. It
-re-produces spherical quadrics of the form $\Phi(\mathbf{x}) =a +
-\mathbf{b}^t \mathbf{x} +\gamma\ \mathbf{x}^t\mathbf{x}$ exactly. The
+that relies also on the function gradient $\mathbf{g_i}$ for all $\mathbf{p_i} \in \mathcal{P}$. It is $C^1$ continuous with gradient $\mathbf{g_i}$ at
+$\mathbf{p_i}$. Spherical quadrics of the form $\Phi(\mathbf{x}) =a +
+\mathbf{b}^t \mathbf{x} +\gamma\ \mathbf{x}^t\mathbf{x}$ are reproduced 
+exactly. The
 proof relies on the barycentric coordinate property of the natural
 neighbor coordinates and assumes that the gradient of $\Phi$ at the
 data points is known or approximated from the function values as
@@ -100,7 +100,7 @@ defined. 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 coordinate computation.
+type of natural neighbor coordinate (i.e.\ \ccc{natural_neighbor_coordinates_2}, \ccc{regular_neighbor_coordinates_2}).
 %\begin{ccAdvanced}
 %\end{ccAdvanced}
 

Packages/Interpolation/doc_tex/Interpolation/introduction.tex

@@ -1,7 +1,7 @@
 This chapter describes \cgal's interpolation package which implements
 natural neighbor coordinate functions as well as different
 methods for scattered data interpolation most of which are based on
-natural neighbor coordinates. The functions for the natural neighbor 
+natural neighbor coordinates. The functions for computing natural neighbor 
 coordinates in Euclidian space are described in 
 Section~\ref{sec:coordinates}, 
 the functions concerning the coordinate and neighbor 
@@ -19,5 +19,4 @@ each $\mathbf{p_i} \in \mathcal{P}$, we associate $z_i =
 \Phi(\mathbf{p_i})$. Sometimes, the gradient of $\Phi$ is also known
 at $\mathbf{p_i}$. It is denoted $\mathbf{g_i}= \nabla
 \Phi(\mathbf{p_i})$. The interpolation is carried out for a point
-$\mathbf{x}$ in the convex hull of $\mathcal{P}$.
-
+$\mathbf{x}$ in the convex hull of $\mathcal{P}$.

Packages/Interpolation/doc_tex/Interpolation/surface_coords.tex

@@ -20,7 +20,7 @@ plane $\mathcal{T}_x$ of the surface $\mathcal{S}$ at the point
 $\mathbf{x} \in \mathcal{S}$ approximates $\mathcal{S}$ in the
 neighborhood of $\mathbf{x}$. It has been shown in \cite{bf-lcss-02}
 that, if the surface $\mathcal{S}$ is well sampled with respect to the
-curvature and the local thickness of $\mathcal{S}$, the intersection
+curvature and the local thickness of $\mathcal{S}$, i.e.\ it is an $\epsilon$-sample, the intersection
 of the tangent plane $\mathcal{T}_x$ with the Voronoi cell of
 $\mathbf{x}$ in the Voronoi diagram of $\mathcal{P} \cup
 \{\mathbf{x}\}$ has a small diameter.  Consequently, inside this
@@ -30,7 +30,7 @@ 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 $\mathcal{H}$ is a two-dimensional power
 diagram. The points defining the power diagram are the projections of
-the points in $\mathcal{P}$ onto $\mathcal{H}$ each point weighted
+the points in $\mathcal{P}$ onto $\mathcal{H}$, each point weighted
 with its negative square distance to $\mathcal{H}$. Algorithms for the
 computation of power diagrams via the dual regular triangulation are
 well known and for example provided by \cgal\ in the class
@@ -54,8 +54,7 @@ Voronoi diagram with a plane $\mathcal{H}$ can be computed by
 instantiating the \ccc{Regular_triangulation_2<Gt, Tds>} class with
 the traits class \ccc{Voronoi_intersection_2_traits_3<K>}. This traits
 class contains a point and a vector as class member which define the
-plane $\mathcal{H}$. All predicates and constructions used by the
-triangulation algorithm are replaced by the corresponding operators on
+plane $\mathcal{H}$. All predicates and constructions used by \ccc{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 points $p$, $q$, $r$ of the regular triangulation
 and tests the power distance of a fourth point $t$ respect to the
@@ -75,9 +74,11 @@ upon the computation of regular neighbor coordinates via the function
 triangulation that is dual to the intersection of $\mathcal{T}_x$ and
 the Voronoi diagram of $\mathcal{P}$.
 
-Of course, we might introduce all data points into this regular
-triangulation. However, this is not necessary if we guarantee that all
-surface neighbors of the query point are within the input points. The
+Of course, we might introduce all data points $\mathcal{P}$ into 
+this regular triangulation. However, this is not necessary because we are only interested in the cell of $\mathbf{x}$. It is sufficient to  
+guarantee that all
+surface neighbors of the query point $\mathbf{x}$ are within the 
+input points. The
 sample points $\mathcal{P}$ can be filtered for example by distance,
 e.g.\ using range search or $k$-nearest neighbor queries, or with the
 help of the $3D$ Delaunay triangulation since the surface neighbors
@@ -86,7 +87,7 @@ in this triangulation. \cgal\ provides a function that encapsulates
 the filtering based on the $3D$ 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 the query point than the furthest input point)
+that are further from $\mathbf{x}$ 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.

Packages/Interpolation/doc_tex/basic/Interpolation/interpolation.tex

@@ -18,10 +18,10 @@ called.
 
 \subsubsection{Sibson's $C^1$ continuous interpolant}
 In \cite{s-bdnni-81}, Sibson describes a second interpolation method
-that is $C^1$ continuous with gradient $\mathbf{g_i}$ at
-$\mathbf{p_i}$. It
-re-produces spherical quadrics of the form $\Phi(\mathbf{x}) =a +
-\mathbf{b}^t \mathbf{x} +\gamma\ \mathbf{x}^t\mathbf{x}$ exactly. The
+that relies also on the function gradient $\mathbf{g_i}$ for all $\mathbf{p_i} \in \mathcal{P}$. It is $C^1$ continuous with gradient $\mathbf{g_i}$ at
+$\mathbf{p_i}$. Spherical quadrics of the form $\Phi(\mathbf{x}) =a +
+\mathbf{b}^t \mathbf{x} +\gamma\ \mathbf{x}^t\mathbf{x}$ are reproduced 
+exactly. The
 proof relies on the barycentric coordinate property of the natural
 neighbor coordinates and assumes that the gradient of $\Phi$ at the
 data points is known or approximated from the function values as
@@ -100,7 +100,7 @@ defined. 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 coordinate computation.
+type of natural neighbor coordinate (i.e.\ \ccc{natural_neighbor_coordinates_2}, \ccc{regular_neighbor_coordinates_2}).
 %\begin{ccAdvanced}
 %\end{ccAdvanced}
 

Packages/Interpolation/doc_tex/basic/Interpolation/introduction.tex

@@ -1,7 +1,7 @@
 This chapter describes \cgal's interpolation package which implements
 natural neighbor coordinate functions as well as different
 methods for scattered data interpolation most of which are based on
-natural neighbor coordinates. The functions for the natural neighbor 
+natural neighbor coordinates. The functions for computing natural neighbor 
 coordinates in Euclidian space are described in 
 Section~\ref{sec:coordinates}, 
 the functions concerning the coordinate and neighbor 
@@ -19,5 +19,4 @@ each $\mathbf{p_i} \in \mathcal{P}$, we associate $z_i =
 \Phi(\mathbf{p_i})$. Sometimes, the gradient of $\Phi$ is also known
 at $\mathbf{p_i}$. It is denoted $\mathbf{g_i}= \nabla
 \Phi(\mathbf{p_i})$. The interpolation is carried out for a point
-$\mathbf{x}$ in the convex hull of $\mathcal{P}$.
-
+$\mathbf{x}$ in the convex hull of $\mathcal{P}$.

Packages/Interpolation/doc_tex/basic/Interpolation/surface_coords.tex

@@ -20,7 +20,7 @@ plane $\mathcal{T}_x$ of the surface $\mathcal{S}$ at the point
 $\mathbf{x} \in \mathcal{S}$ approximates $\mathcal{S}$ in the
 neighborhood of $\mathbf{x}$. It has been shown in \cite{bf-lcss-02}
 that, if the surface $\mathcal{S}$ is well sampled with respect to the
-curvature and the local thickness of $\mathcal{S}$, the intersection
+curvature and the local thickness of $\mathcal{S}$, i.e.\ it is an $\epsilon$-sample, the intersection
 of the tangent plane $\mathcal{T}_x$ with the Voronoi cell of
 $\mathbf{x}$ in the Voronoi diagram of $\mathcal{P} \cup
 \{\mathbf{x}\}$ has a small diameter.  Consequently, inside this
@@ -30,7 +30,7 @@ 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 $\mathcal{H}$ is a two-dimensional power
 diagram. The points defining the power diagram are the projections of
-the points in $\mathcal{P}$ onto $\mathcal{H}$ each point weighted
+the points in $\mathcal{P}$ onto $\mathcal{H}$, each point weighted
 with its negative square distance to $\mathcal{H}$. Algorithms for the
 computation of power diagrams via the dual regular triangulation are
 well known and for example provided by \cgal\ in the class
@@ -54,8 +54,7 @@ Voronoi diagram with a plane $\mathcal{H}$ can be computed by
 instantiating the \ccc{Regular_triangulation_2<Gt, Tds>} class with
 the traits class \ccc{Voronoi_intersection_2_traits_3<K>}. This traits
 class contains a point and a vector as class member which define the
-plane $\mathcal{H}$. All predicates and constructions used by the
-triangulation algorithm are replaced by the corresponding operators on
+plane $\mathcal{H}$. All predicates and constructions used by \ccc{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 points $p$, $q$, $r$ of the regular triangulation
 and tests the power distance of a fourth point $t$ respect to the
@@ -75,9 +74,11 @@ upon the computation of regular neighbor coordinates via the function
 triangulation that is dual to the intersection of $\mathcal{T}_x$ and
 the Voronoi diagram of $\mathcal{P}$.
 
-Of course, we might introduce all data points into this regular
-triangulation. However, this is not necessary if we guarantee that all
-surface neighbors of the query point are within the input points. The
+Of course, we might introduce all data points $\mathcal{P}$ into 
+this regular triangulation. However, this is not necessary because we are only interested in the cell of $\mathbf{x}$. It is sufficient to  
+guarantee that all
+surface neighbors of the query point $\mathbf{x}$ are within the 
+input points. The
 sample points $\mathcal{P}$ can be filtered for example by distance,
 e.g.\ using range search or $k$-nearest neighbor queries, or with the
 help of the $3D$ Delaunay triangulation since the surface neighbors
@@ -86,7 +87,7 @@ in this triangulation. \cgal\ provides a function that encapsulates
 the filtering based on the $3D$ 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 the query point than the furthest input point)
+that are further from $\mathbf{x}$ 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.