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Jet_fitting_3/doc_tex/Jet_fitting_3/Jet_fitting_3_user.tex
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@ -2,8 +2,9 @@
%WARNING: amsmath not available
This chapter describes the \cgal's package for the estimation of local
differential quantities on sampled surfaces.
This chapter describes the \cgal's package geared towards the
estimation of local differential quantities on sampled surfaces, known
either as a mesh or a point cloud.
%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
@ -15,38 +16,68 @@ differential quantities on sampled surfaces.
Consider a sampled smooth surface, and assume we are given a
collection of points $P$ about a given sample $p$. We aim at
estimating the differential properties up to order $d'$ of the surface
at point $p$ from the point set $P^+ = P\cup \{ p\}$ --we note $N=\mid
P^+\mid$. More precisely, first order properties correspond to the
normal or the tangent plane; second order properties provide the
principal curvatures and directions, third order properties provide
the directional derivatives of the principal curvatures along the
curvature lines, etc. Most of the time, estimating first and second
order differential quantities is sufficient. However, some
applications involving shape analysis require estimating third and
fourth order differential quantities.
estimating the differential properties up to any fixed order of the
surface at point $p$ from the point set $P^+ = P\cup \{ p\}$ ---we
denote $N=\mid P^+\mid$. More precisely, first order properties
correspond to the normal or the tangent plane; second order properties
provide the principal curvatures and directions, third order
properties provide the directional derivatives of the principal
curvatures along the curvature lines, etc. Most of the time,
estimating first and second order differential quantities is
sufficient. However, some applications involving shape analysis
require estimating third and fourth order differential quantities.
%%
Many different estimators have been proposed in the vast literature of
applied geometry \cite{cgal:p-smrqt-01}. They all need to define a
neighborhood around the point at which the estimation is computed and
use at different level the geometry of the mesh. On one hand, methods
relying on {\em discrete differential geometry} only use the
information provided by the mesh
\cite{cgal:pp-cdmsc-93,cgal:mdsb-ddgot-02,cgal:csm-rdtnc-03}. On the
other hand, fitting methods rely on smooth differential geometry and
approximation, and hence are able to process point clouds directly.
applied geometry \cite{cgal:p-smrqt-01}, and all of them need to
define a neighborhood around the point at which the estimation is
computed. These methods may be classified into two groups. The first
one features methods geared towards meshes
\cite{cgal:pp-cdmsc-93,cgal:mdsb-ddgot-02,cgal:csm-rdtnc-03}.
Such methods exploit the geometry and the topology of the mesh, and
are sometimes referred to as methods from {\em discrete differential
geometry}.
%%
The second one comprises methods relying on smooth differential
geometry calculations, carried out on smooth objects {\em fitted} from
the sample points provided. Such naturally accommodate meshes, but also
point clouds.
By now, there are two methods to extract local differential properties
coming with theoretical analysis of their convergence rates. On one
hand, the normal cycle theory is used in
\cite{cgal:csm-rdtnc-03} to provide an estimate of the integral of the
Weingarten map of surface discretized by a mesh. On the other hand,
%On the other hand, fitting methods rely on smooth differential
%geometry and approximation, and hence are able to process point clouds
%directly.
%and use at different level the geometry of the mesh. On one hand,
%methods relying on {\em discrete differential geometry} only use the
%information provided by the mesh
%By now, there are two methods to extract local differential properties
%coming with theoretical analysis of their convergence rates.
Estimating differential quantities naturally subsumes (i)\ a smooth
surface exists (ii)\ one wishes to recover its differential
properties. This two observations call for the convergence analysis of
the methods developed, and by now, two estimation methods come with
approximation guarantees measuring the convergence rate of the
discrepancy between the estimates and the true (unknown) values.
On one hand, the normal cycle theory is used in
\cite{cgal:csm-rdtnc-03} to provide an estimate of the integral of the
$3D$ embedding \footnote{Recall that the Weingarten map is a bilinear
form of the tangent space, so that integrating it requires embedding
it in $3D$ since the tangent plane rotates.} of the Weingarten map of
surface discretized by a mesh. On the other hand,
\cite{cgal:cp-edqpf-05} uses polynomial fitting to extract the
coefficients of the local representation of the surface as a height
function. This method has three advantages~: first, differential
properties of any order can be retrieved; second, the coefficients
estimated feature the best error bounds known so far; third, it
process point samples (and does not need a mesh).
function. The former method provides spatial averages, and is
particularly well suited to meshes. Yet, it is limited to second order
differential properties.
%%
The later one carries three advantages~: first, differential
properties of any order can be retrieved; second, the convergence
rates provided are the best ones known so far ---they are actually
optimal; third, point clouds can be directly handled ---no topological
information is required.
\subsection{Jets, Monge form and polynomial fitting}
@ -56,10 +87,10 @@ process point samples (and does not need a mesh).
%
To present the method, we shall need the following notions. Consider a
smooth surface. About one of its points, consider a coordinate system
whose $z$-axis is not aligned with the normal of the surface. In such
a frame, the surface can locally be written as the graph of a
bivariate function. Denoting h $\hot$ standing for {\em
higher order terms}, one has~:
whose $z$-axis does not belong to the tangent space. In such a frame,
the surface can locally be written as the graph of a bivariate
function. Denoting h $\hot$ standing for {\em higher order terms}, one
has~:
%
%Assume the surface is given, locally and in a suitable , as a height
%function ---with $\hot$ standing for {\em higher order terms}~:
@ -69,17 +100,19 @@ J_{B,d}(x,y)=\sum_{k=0}^{k=d}(\sum_{i=0}^{i=k}
\frac{B_{k-i,i}x^{k-i}y^{i}}{i!(k-i)!}).
\end{equation}
The degree $d$ polynomial $J_{B,d}$ is the Taylor expansion of the
function $z$, we call it its $d$-jet. Notice that a $d$-jet contains
function $z$, an is called its $d$-jet. Notice that a $d$-jet contains
$N_d=(d+1)(d+2)/2$ coefficients.
At any point of the surface which is not an umbilic, principal
directions $d_1, d_2$ are well defined, and these (non oriented)
directions together with the normal vector $n$ define two direct
orthonormal frames. If $v_1$ is a unit vector of direction $d_1$ then
there exists a unique unit vector $v_2$ so that $(v_1,v_2,n)$ is
direct; and the other possible frame is $(-v_1,-v_2,n)$. In one of
these Monge coordinate systems, the surface is said to be given in the
Monge form and its jet has the following canonical form~:
Recall that an umbilical point of a surface ---or umbilic for short,
is a point where both principal curvatures are identical. At any
point of the surface which is not an umbilic, principal directions
$d_1, d_2$ are well defined, and these (non oriented) directions
together with the normal vector $n$ define two direct orthonormal
frames. If $v_1$ is a unit vector of direction $d_1$ then there exists
a unique unit vector $v_2$ so that $(v_1,v_2,n)$ is direct; and the
other possible frame is $(-v_1,-v_2,n)$. In one of these Monge
coordinate systems, the surface is said to be given in the Monge form
and its jet has the following canonical form~:
\begin{eqnarray}
\label{eq:monge}
@ -94,20 +127,23 @@ respective curvature lines, while $b_1,b_2$ are the directional
derivatives of $k_1,k_2$ along the other curvature lines.
The Monge coordinate system can be computed from any $d$-jet ($d\geq
2$), and so are the Monge coefficients. These data, characterizing the
geometry of the surface in a canonical way, will naturally be the
output of the algorithm.
2$), and so are the Monge coefficients. These informations
characterize the local geometry of the surface in a canonical way, and
are the quantities returned by our algorithm.
\paragraph{Interpolating or approximating the $d$-jet.}
%
The idea is to fit the $d$-jet, in a well chosen coordinate system,
using bivariate polynomial interpolation or approximation on the point
set $P^+$.
In a well chosen coordinate system, the idea is to fit the $d$-jet of
the surface using bivariate polynomial interpolation or approximation
on the point set $P^+$.
%
More precisely, the fitting consists of finding the coefficients
$A_{i,j}$ of the degree $d$ polynomial $J_{A,d}=
\sum_{k=0}^{k=d}(\sum_{i=0}^{i=k}
\frac{A_{k-i,i}x^{k-i}y^{i}}{i!(k-i)!})$.
$A_{i,j}$ of the degree $d$ polynomial
\begin{equation}
\label{eq-answer}
J_{A,d}= \sum_{k=0}^{k=d}(\sum_{i=0}^{i=k}
\frac{A_{k-i,i}x^{k-i}y^{i}}{i!(k-i)!}).
\end{equation}
Denote $p_i=(x_i,y_i,z_i), \ i=1,\ldots , N$ the coordinates of the
@ -118,21 +154,24 @@ i=1,\ldots,N$, and for approximation one has to minimize $\sum_{i=1}^N
(A(x_i,y_i)-z_i)^2$. The linear algebra formulation of the problem is
given by
%
\begin{eqnarray*}
\begin{eqnarray}
\label{eq:fit-linalg}
A = & (A_{0,0}, A_{1,0},A_{0,1}, \ldots , A_{0,d})^T \\
Z= &(z_1, z_2,\ldots , z_N)^T \\
M= &(1,x_i,\ y_i,\ \frac{x_i^2}{2},\ldots ,
\ \frac{x_iy_i^{d-1}}{(d-1)!},\ \frac{y_i^d}{d!})_{i=1,...,N}\\
\end{eqnarray*}
\end{eqnarray}
%
The equations for interpolation become $MA=Z$ and for approximation
$\min ||MA-Z||_2$.
The equations for interpolation become $MA=Z$. For approximation, one
seeks the matrix $A$ such that $A = \arg \min_A ||MA-Z||_2$.
The following theorem precisely states the order of convergence of the
polynomial fitting method. Given a parameter $h$ measuring the
sampling step, the following theorem, provising the best asymptotic
estimates known to date, is proved in \cite{cgal:cp-edqpf-05}~:
To assess the convergence guarantees, assume the sampling step is
given by a parameter $h$, which means all samples are at distance
$O(h)$ of the point where the calculation is carried out.
%%
The following theorem, proved in \cite{cgal:cp-edqpf-05}, provides the
asymptotic error estimates ---which are the best known to date:
%%
\begin{theorem}
A polynomial fitting of degree $d$ estimates any $k^{th}$-order
differential quantity to accuracy $O(h^{d-k+1})$~:
@ -160,11 +199,11 @@ Based on the above concepts, the algorithm consists of 4 steps.
%
\begin{enumerate}
\item
We perform a PCA on $P^+$. This analysis outputs three orthonormal
eigenvectors and the associated eigenevalues. If the
surface is well sampled, we expect the PCA to provide one small and
two large eigenvalues, the eigenvector associated to the small one
approximating the normal vector.
We perform a Principal Component Analysis (PCA) on $P^+$. This
analysis outputs three orthonormal eigenvectors and the associated
eigenvalues. If the surface is well sampled, we expect the PCA to
provide one small and two large eigenvalues, the eigenvector
associated to the small one approximating the normal vector.
\item
We perform a change of coordinates to move the samples into the
coordinate system defined by the PCA eigenvectors. We then resort to
@ -179,17 +218,22 @@ Finally, we compute the Monge coefficients.
\end{enumerate}
For the fitting, we do not assume the z-axis of the fitting to
be the normal of the surface. Hence we keep the first order
coefficients of the polynomial $J_{A,d}$. It is prooved that such a choice
improves the estimations \cite{cgal:cp-edqpf-05}.
For the fitting, we do not assume the $z$-axis of the fitting to be
the normal of the surface.
As proved in \cite{cgal:cp-edqpf-05}, two stages methods estimating
the tangent plane first incur of higher order quantities the error
made on first order quantities.
%%
Thus, we keep the first order coefficients of the polynomial
$J_{A,d}$, which provide the estimate for the tangent plane.
Note that we do not aim at identifying exactly special points such as
umbilics where both principal curvatures are equal. This would only be
possible if the original surface and the fitted surface coincide. This
implying that the original surface is a graph of a bivariate
polynomial of a lower degree than the fitted polynomial and that the
coordinate system used for the fitting has the same height direction.
Finally, notice that we do not aim at identifying exactly special
points such as umbilics where both principal curvatures are
equal. This would only be possible if the original surface and the
fitted surface coincide. This implying that the original surface is a
graph of a bivariate polynomial of a lower degree than the fitted
polynomial and that the coordinate system used for the fitting has the
same height direction.
%For the sake of clarity and wlog, we assume the study point is at the
%origin.
@ -198,14 +242,22 @@ coordinate system used for the fitting has the same height direction.
\label{sec:deg-cases}
%%%%%%%%%%%%%%%%%%%%%
As usual, the fitting procedure may run into (almost) degenerate
cases:
%%
\begin{itemize}
\item
The PCA used to determine a rough normal vector does not yield an
eigenvalue significantly smaller then the two remaining ones.
\item
When solving the linear system, the condition number is too large.
\end{itemize}
Due to poor sampling, the PCA used to determine a rough normal vector
may not yield an eigenvalue significantly smaller than the two
remaining ones.
\item
As observed in \cite{cgal:cp-edqpf-05}, the interpolating problem is
not {\em poised} if the points used project, into the fitting frame,
onto an algebraic curve of degree $d$. More generally, the problem is
ill poised if the condition number is too large.
\end{itemize}
%%
In these cases, the estimation may not be relevant. To inform the user
of these issues, we provide the PCA results and the condition number
of the fitting in the class \ccc{Monge_info}.
@ -216,10 +268,10 @@ of the fitting in the class \ccc{Monge_info}.
In this section, we detail the mathematics involved, in order to
justify the design choices made.
Note that there are 3 relevant direct orthonormal basis: world-basis
$(w_x,w_y,w_z)$, fitting-basis $(f_x,f_y,f_z)$, monge-basis
$(d_1,d_2,n)$.
%%
To do so, observe that there are three relevant direct orthonormal
basis: the world-basis $(w_x,w_y,w_z)$, the fitting-basis
$(f_x,f_y,f_z)$, the monge-basis $(d_1,d_2,n)$.
\begin{figure}[h!]
\begin{ccTexOnly}
@ -228,7 +280,7 @@ $(d_1,d_2,n)$.
\end{ccTexOnly}
\label{fig:jet_fitting_basis}
\caption{The three basis envolved in the estimation.}
\caption{The three basis involved in the estimation.}
\begin{ccHtmlOnly}
<CENTER>
@ -237,49 +289,61 @@ $(d_1,d_2,n)$.
\end{ccHtmlOnly}
\end{figure}
\subsection{Compute a basis for the fitting}
\subsection{Computing a basis for the fitting}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\bf input : samples\\ output : fitting-basis}
{\bf Input : samples\\ Output : fitting-basis}
Perform a Principal Component Analysis, this means we need a linear
algebra method to perform an eigen analysis of a symmetric matrix :
{\tt eigen\_symm\_algo}. This analysis gives an orthonormal basis
whose $z$-axis is provided by the eigenvector associated to the
smallest eigenvalue \footnote{Another possibility is to choose as
z-axis the axis of the world-basis with the least angle with the axis
determined with the PCA. Then the change of basis reduces to a
permutation of axis.}. Note one may have to swap the sense of a vector
to get a direct basis.
%, and we
%shall assume a function {\tt eigen\_symm\_algo} doing so is available.
%
%a Principal Component Analysis, this means we need a linear
%algebra method to perform an eigen analysis of a symmetric matrix :
%{\tt eigen\_symm\_algo}.
%
Performing a PCA requires diagonalizing a symmetric matrix. This
analysis gives an orthonormal basis whose $z$-axis is provided by the
eigenvector associated to the smallest eigenvalue
\footnote{Another possibility is to choose as z-axis the axis of the
world-basis with the least angle with the axis determined with the
PCA. Then the change of basis reduces to a permutation of axis.}. Note
one may have to swap the orientation of a vector to get a direct
basis.
Let's note $P_{W\rightarrow F}$ the matrix to change coordinates from the
world-basis $(w_x,w_y,w_z)$ to the fitting-basis $(f_x,f_y,f_z)$. The
rows of $P_{W\rightarrow F}$ are the coordinates of the vectors
$(f_x,f_y,f_z)$ in the world-basis. This matrix represents a
orthogonal transformation hence its inverse is its tranpose. To obtain
orthogonal transformation hence its inverse is its transpose. To obtain
the coordinates of a point in the fitting-basis from the coordinates
in the world-basis, one has to multiply by $ P_{W\rightarrow F}$.
Possible feedback is the eigenvalues, a good sampling is characterized
by a small eigenvalue and two similar bigger ones.
As mentioned above, the eigenvalues are returned, from which the
sampling quality can be assessed by checking one eigenvalue is
significantly smaller.
\paragraph{Implementation details.}
Given a symmetric matrix $M$, we assume the following function:
%%
\begin{verbatim}
void eigen_symm_algo(const LAMatrix& M, LAVector& eigen_vals, LAMatrix& eigen_vecs)
\end{verbatim}
This function computes the eigenvalues and eigenvectors of the real
symmetric matrix M. The eigenvalues are stored in the vector
eigen\_vals and are in decreasing order. The corresponding eigenvectors are
stored in the columns of the matrix eigen\_vecs. For example, the
eigenvector in the first column corresponds to the first (and largest)
eigenvalue. The eigenvectors are guaranteed to be mutually orthogonal
and normalised to unit magnitude.
%%
This function computes the eigenvalues and eigenvectors of matrix $M$.
The eigenvalues are stored in the vector eigen\_vals and are in
decreasing order. The corresponding eigenvectors are stored in the
columns of the matrix eigen\_vecs. For example, the eigenvector in the
first column corresponds to the first (and largest) eigenvalue. The
eigenvectors are guaranteed to be mutually orthogonal and normalised
to unit magnitude.
\subsection{Solving the interpolation / approximation problem}
\label{sec:solving}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\bf input : samples, fitting-basis \\ output : coeff $A_{i,j}$ of the
{\bf Input : samples, fitting-basis \\ Output : coefficients $A_{i,j}$ of the
bivariate fitted polynomial in the fitting-basis }
Computations are done in the fitting-basis and the origin is the point
@ -289,12 +353,13 @@ translation ($-p$) and multiplication by $ P_{W\rightarrow F}$.
We solve the system $MA=Z$, in the least square sense for
approximation, with a function {\tt solve\_ls\_svd}. There is a
preconditionning of the matrix $M$ so as to improve the condition
preconditioning of the matrix $M$ so as to improve the condition
number. Assuming the $\{x_i\}$, $\{y_i\}$ are of order $h$, the
pre-conditioning consists of performing a column scaling by dividing
each monomial $x_i^ky_i^l$ by $h^{k+l}$. The parameter $h$ is chosen
as the mean value of the $\{x_i\}$ and $\{y_i\}$. In other words, the
new system is $M'Y=(MD^{-1}(DA)=Z$ with $D$ the diagonal matrix
each monomial $x_i^ky_i^l$ by $h^{k+l}$ ---refer to
Eq. (\ref{eq:fit-linalg}). Practically, the parameter $h$ is chosen as
the mean value of the $\{x_i\}$ and $\{y_i\}$. In other words, the new
system is $M'Y=(MD^{-1}(DA)=Z$ with $D$ the diagonal matrix
$D=(1,h,h,h^2,\ldots,h^d,h^d)$, so that the solution $A$ of the
original system is $A=D^{-1}Y$.
@ -324,17 +389,19 @@ D_r^{-1} & 0_{N_d-r,\ r}\\
\right)
U^TZ.
\end{equation}
%%
One can provide the condition number of the matrix $M$ (after
preconditioning) which is the ratio of the maximal and the minimal
singular values. It is infinite if the system is under constrained,
that is the smallest singular value is zero. Then, an exception is
raised.
One can provide the condition number
of the matrix $M$ (after preconditionning) which is the ratio of the
maximal and the minimal singular values. It is infinite if the system
is under constrained, that is the smallest singular value is
zero. Then we should provide an exception.
\paragraph{Implementation details.}
We assume function:
\begin{verbatim}
void solve_ls_svd_algo(const LAMatrix& M, const LAVector& B, Vector& X, double& cond_nb)
\end{verbatim}
%%
This function first factorizes the m-by-n matrix M into the singular
value decomposition $M = U S V^T$ for $m \geq n$. Then it solves the
system $MX = B$ in the least square sense using the singular value
@ -359,11 +426,13 @@ condition number of the system. For more on these alternatives, see
\subsection{Principal curvature / directions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\bf input : coeff of the fit $A_{i,j}$,
{\bf Input : coefficients of the fit $A_{i,j}$,
fitting-basis \\
output : monge-basis wrt fitting-basis and world-basis
Output : monge-basis wrt fitting-basis and world-basis
}
In the fitting basis, we have determined a height function expressed
by Eq. (\refs{eq-answer}).
Computations are done in the fitting-basis. The partial derivatives,
evaluated at $(x,y)=(0,0)$, of the fitted polynomial $J_{A,d}(x,y)$ are
$A_{i,j}=\frac{\partial^{i+j}J_{A,d}}{\partial^ix
@ -375,13 +444,18 @@ A_{0,0}+A_{1,0}x+A_{0,1}y+\frac{1}{2}(A_{2,0}x^2+2A_{1,1}xy+A_{0,2}y^2)
\end{eqnarray}
The origin, that is the point of the fitted surface where the
estimation is performed, is $(0,0,A_{0,0})$. \\ The normal is
$n=(-A_{1,0},-A_{0,1},1)/\sqrt{A_{1,0}^2+A_{0,1}^2+1}$. The Weingarten
---The origin, that is the point of the fitted surface where the
estimation is performed, is $(0,0,A_{0,0})$. \\
%%
---The normal is
$n=(-A_{1,0},-A_{0,1},1)/\sqrt{A_{1,0}^2+A_{0,1}^2+1}$.\\
%%
---Curvature related properties are retrieved resorting to standard
differential calculus \cite{co-carmo}. More precisely, the Weingarten
operator $W=-I^{-1}II$ is first computed in the basis of the tangent
plane $\{ (1,0,A_{1,0}), (0,1,A_{0,1}) \}$. We compute an orthonormal basis of the
tangent plane using the Gram-Schimdt algorithm, and then we compute
Weingarten in this basis (apply a change of basis matrix
plane $\{ (1,0,A_{1,0}), (0,1,A_{0,1}) \}$. We compute an orthonormal
basis of the tangent plane using the Gram-Schimdt algorithm, and then
we compute Weingarten in this basis (apply a change of basis matrix
$W'=P^{-1}WP$). It is then symmetric, we can apply the eigensystem
function for a symmetric matrix:
\verb+eigen_symm_algo+.
@ -392,33 +466,33 @@ orthonormal direct basis $(d_1,d_2,n)$. Let's note $P_{F
fitting-basis to the monge-basis. Its rows are the coordinates of the
vectors $(d_1,d_2,n)$ in the fitting-basis. It is an orthogonal matrix
$P_{F \rightarrow M}^{-1}=P_{F \rightarrow M}^T$. The monge-basis
expressed in the world-basis is obtained by multipling the coordinates
expressed in the world-basis is obtained by multiplying the coordinates
of $(d_1,d_2,n)$ in the fitting-basis by $P_{W\rightarrow F}^{-1}$,
(the same holds for the origin point which has in addition to be
translated by $p$, i.e. the coordinates of the origin point are
$P_{W\rightarrow F}^{-1} (0,0,A_{0,0}) +p$.
\subsection{Computation of higher order Monge coeff}
\subsection{Computing higher order Monge coefficients}
{\bf input : coeff of the fit, monge-basis wrt fitting-basis ($P_{F
{\bf Input : coefficients of the fit, monge-basis wrt fitting-basis ($P_{F
\rightarrow M}$)\\
output : third and fourth order coeff of Monge}
Output : third and fourth order coefficients of Monge}
We use explicite formula. The implicit equation of the fitted
We use explicit formula. The implicit equation of the fitted
polynomial surface in the fitting-basis with origin the point
$(0,0,A_{0,0})$ is $Q=0$ with
\begin{equation}
Q=-w-A_{0,0} +\sum_{i,j}\frac{A_{i,j}u^iv^j}{i!j!}.
\end{equation}
%%
The equation in the monge-basis is obtained by substituting $(u,v,w)$
by $P^T_{F\rightarrow M}(x,y,z)$, let's denote $f(x,y,z)$ this implicit
by $P^T_{F\rightarrow M}(x,y,z)$. Denote $f(x,y,z)=0$ this implicit
equation. By definition of the monge-basis, we have locally (at
$(0,0,0)$)
\begin{equation}
f(x,y,z)=0 \Leftrightarrow z=g(x,y)
\end{equation}
and the taylor expansion of $g$ at $(0,0)$ are the Monge coefficients
and the Taylor expansion of $g$ at $(0,0)$ are the Monge coefficients
sought.
%
Let's denote the partial derivatives evaluated at the origin of $f$
@ -459,27 +533,31 @@ f_{0,0,1} ^{2}}}
\section{Software Design}
%%%%%%%%%%%%%%%%%%%%%%%
following \cgal's best practices in geometric software development,
the package developed in fully generic in the C++ sense ---and is
templated by three classes.
\subsection{Options and interface specifications}
%%%%%%%%%%%%%%%%%%%%%
Using the fitting strategy requires specifying two degrees: the degree
$d$ of the fitted polynomial ($d \geq 1$), and the degree $d'$ of the
monge coeff one wants to compute, with $d' \leq d $. In the sequel,
monge coefficients one wants to compute, with $d' \leq d $. In the sequel,
we also assume users are satisfied with Monge coefficients of order
four, that is, we assume $d' \leq 4$.
\medskip
Regarding interpolation versus approximation, we provide a single
function {\tt Monge\_via\_jet\_fitting} with parameters $d,d'$ and a
range iterator. If $size()==N_d$ then interpolation is performed, else
$size() > N_d$ and approximation is used. If $size()< N_d$ we provide
range iterator. If $N==N_d$ then interpolation is performed, else
$N > N_d$ and approximation is used. If $N < N_d$ we provide
an exception.
\subsection{Template parameters}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The package developed in fully generic in the C++ sense, and is
templated by three classes. In addition, we assume implicit cast
In addition, we assume implicit cast
between Data, Local and Linalg are well defined.
\subsubsection{\tt Data\_Kernel}