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Fixed trailing whitespace in Interpolation/doc

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Mael Rouxel-Labbé 2018-01-15 15:18:34 +01:00
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54
Interpolation/doc/Interpolation/Interpolation.txt
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@ -1,7 +1,7 @@
namespace CGAL {
/*!
\mainpage User Manual
\mainpage User Manual
\anchor Chapter_2D_and_Surface_Function_Interpolation
\anchor chapinterpolation
\cgalAutoToc
@ -11,12 +11,12 @@ namespace CGAL {
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 computing natural neighbor
coordinates in Euclidean space are described in
Section \ref seccoordinates,
the functions concerning the coordinate and neighbor
computation on surfaces are discussed in Section \ref secsurface.
In Section \ref secinterpolation, we describe the different interpolation
natural neighbor coordinates. The functions for computing natural neighbor
coordinates in Euclidean space are described in
Section \ref seccoordinates,
the functions concerning the coordinate and neighbor
computation on surfaces are discussed in Section \ref secsurface.
In Section \ref secinterpolation, we describe the different interpolation
functions.
Scattered data interpolation solves the following problem: given
@ -32,7 +32,7 @@ at \f$ \mathbf{p_i}\f$. It is denoted \f$ \mathbf{g_i}= \nabla
\Phi(\mathbf{p_i})\f$. The interpolation is carried out for an arbitrary query point
\f$ \mathbf{x}\f$ on the convex hull of \f$ \mathcal{P}\f$.
\section seccoordinates Natural Neighbor Coordinates
\section seccoordinates Natural Neighbor Coordinates
\subsection InterpolationIntroduction Introduction
@ -46,7 +46,7 @@ Voronoi cell of \f$ \mathbf{x}\f$ "steals" some parts from the existing
cells.
\cgalFigureBegin{fignn_coords,nn_coords.png}
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 pink polygon, \f$ \pi_3(\mathbf{x})\f$, over the area of the total highlighted zone.
2D example: \f$ \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 pink 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
@ -54,7 +54,7 @@ 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[
\f[
\lambda_i(\mathbf{x}) =
\frac{\pi_i(\mathbf{x})}{\pi(\mathbf{x})}. \f]
A two-dimensional example
@ -94,7 +94,7 @@ The interpolation package of \cgal provides functions to compute
natural neighbor coordinates for \f$ 2D\f$ and \f$ 3D\f$ points with respect
to Voronoi diagrams as well as with respect to power diagrams (only
\f$ 2D\f$), i.e.\ for weighted points. Refer to the reference pages
`natural_neighbor_coordinates_2()` and
`natural_neighbor_coordinates_2()` and
`regular_neighbor_coordinates_2()`.
In addition, the package provides functions to compute natural
@ -127,7 +127,7 @@ special traits class is needed.
For surface neighbor coordinates, the surface normal at the query
point must be provided, see Section \ref secsurface.
\section secsurface Surface Natural Neighbor Coordinates and Surface Neighbors
\section secsurface Surface Natural Neighbor Coordinates and Surface Neighbors
This section introduces the functions to compute natural neighbor
coordinates and surface neighbors associated to a set of sample points
@ -209,7 +209,7 @@ neighbors are necessarily a subset of the natural neighbors of the
query point in this triangulation. \cgal provides a function that
encapsulates the filtering based on the \f$ 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.\
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
@ -229,7 +229,7 @@ provided.
\cgalExample{Interpolation/surface_neighbor_coordinates_3.cpp}
\section secinterpolation Interpolation Methods
\section secinterpolation Interpolation Methods
\subsection InterpolationIntroduction_2 Introduction
@ -239,7 +239,7 @@ 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[
\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
@ -254,7 +254,7 @@ called.
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
\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
@ -270,7 +270,7 @@ degree functions
\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[
\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}) =
@ -308,18 +308,18 @@ 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[
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]
\subsection sgradient_fitting Gradient Fitting
\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$, we determine
\f[ \mathbf{g_i}
= \min_{\mathbf{g}}
\f[ \mathbf{g_i}
= \min_{\mathbf{g}}
\ccSum{j}{}{
\frac{\lambda_j(\mathbf{p_i})}{\|\mathbf{p_i} - \mathbf{p_j}\|^2}
\left( z_j - (z_i + \mathbf{g}^t (\mathbf{p_j} -\mathbf{p_i})) \right)},
@ -332,7 +332,7 @@ of \f$ \mathbf{p_i}\f$ with respect to \f$ \mathbf{p_i}\f$ associated to
points that are inside the convex hull. There is one function for each
type of natural neighbor coordinate (i.e.\ `natural_neighbor_coordinates_2()`, `regular_neighbor_coordinates_2()`).
\subsection subsecinterpol_examples Example for Linear Interpolation
\subsection subsecinterpol_examples Example for Linear Interpolation
\cgalExample{Interpolation/linear_interpolation_2.cpp}
@ -352,8 +352,8 @@ so that the gradient fitting and interpolation function can access them.
\cgalExample{Interpolation/sibson_interpolation_vertex_with_info_2.cpp}
An additional example in the distribution compares numerically the errors of the different
interpolation functions with respect to a known function.
An additional example in the distribution compares numerically the errors of the different
interpolation functions with respect to a known function.
It is distributed in the examples directory.
\section Interpolation Design and Implementation History
@ -362,6 +362,6 @@ The original version was written by Julia Flötotto, while working towards h
The possibility to use values and gradients in a `Triangulation_vertex_base_with_info_2` was introduced
by Andreas Fabri and Mael Rouxel-Labb&eacut working at GeometryFactory.
*/
*/
} /* namespace CGAL */

6
Interpolation/doc/Interpolation/PackageDescription.txt
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@ -51,8 +51,8 @@ diagram of the data points. Interpolation methods based on natural
neighbor coordinates are particularly interesting because they adapt
easily to non-uniform and highly anisotropic data. This package
contains Sibson's \f$ C^1\f$ continuous interpolation method which
interpolates exactly spherical quadrics (of the form
\f$ \Phi(\mathbf{x})=a + \mathbf{b}^t \mathbf{x} +\gamma\ \mathbf{x}^t\mathbf{x}\f$)
interpolates exactly spherical quadrics (of the form
\f$ \Phi(\mathbf{x})=a + \mathbf{b}^t \mathbf{x} +\gamma\ \mathbf{x}^t\mathbf{x}\f$)
and Farin's \f$ C^1\f$ continuous interpolation method based on
Bernstein-Bézier techniques and interpolating exactly quadratic
functions - assuming that the function gradient is known. In
@ -63,7 +63,7 @@ This method is exact for spherical quadrics.
This \cgal package implements Sibson's and Farin's interpolation
functions as well as Sibson's function gradient fitting method.
Furthermore, it provides functions to compute the natural neighbor
coordinates with respect to a two-dimensional Voronoi diagram (i. e.,
coordinates with respect to a two-dimensional Voronoi diagram (i. e.,
from the Delaunay triangulation of the data points) and to a
two-dimensional power diagram for weighted points (i. e., from their
regular triangulation). Natural neighbor coordinates on closed and