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Merge pull request #3525 from sloriot/Bounding_volume-doc_html_fix 

Bounding volume doc html fix
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Laurent Rineau 2018-12-19 16:08:08 +01:00
parent d4a13b219e 37d05cd835
commit ed4611c1d5
1 changed files with 15 additions and 15 deletions
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30
Bounding_volumes/doc/Bounding_volumes/CGAL/Approximate_min_ellipsoid_d.h
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@ -13,28 +13,28 @@ x\in\E^d \mid x^T E x + x^T e + \eta\leq 0 \}\f$, where \f$ E\f$ is some
positive definite matrix from the set \f$ \mathbb{R}^{d\times d}\f$, \f$ e\f$ is some positive definite matrix from the set \f$ \mathbb{R}^{d\times d}\f$, \f$ e\f$ is some
real \f$ d\f$-vector, and \f$ \eta\in\mathbb{R}\f$. A pointset \f$ P\subseteq \E^d\f$ is real \f$ d\f$-vector, and \f$ \eta\in\mathbb{R}\f$. A pointset \f$ P\subseteq \E^d\f$ is
called <I>full-dimensional</I> if its affine hull has dimension \f$ d\f$. called <I>full-dimensional</I> if its affine hull has dimension \f$ d\f$.
For a finite, full-dimensional pointset \f$ P\f$ we denote by \f$ \mel(P)\f$ the For a finite, full-dimensional pointset \f$ P\f$ we denote by \f$ (P)\f$ the
smallest ellipsoid that contains all points of \f$ P\f$; this ellipsoid smallest ellipsoid that contains all points of \f$ P\f$; this ellipsoid
exists and is unique. exists and is unique.
For a given finite and full-dimensional pointset \f$ P\subset \E^d\f$ and a For a given finite and full-dimensional pointset \f$ P\subset \E^d\f$ and a
real number \f$ \epsilon\ge 0\f$, we say that an ellipsoid \f$ {\cal real number \f$ \epsilon\ge 0\f$, we say that an ellipsoid \f$ {\cal
E}\subset\E^d\f$ is an <I>\f$ (1+\epsilon)\f$-appoximation</I> to \f$ \mel(P)\f$ if E}\subset\E^d\f$ is an <I>\f$ (1+\epsilon)\f$-appoximation</I> to \f$ (P)\f$ if
\f$ P\subset {\cal E}\f$ and \f$ \vol({\cal E}) \leq (1+\epsilon) \f$ P\subset {\cal E}\f$ and \f$ ({\cal E}) \leq (1+\epsilon)
\vol(\mel(P))\f$. In other words, an \f$ (1+\epsilon)\f$-approximation to ((P))\f$. In other words, an \f$ (1+\epsilon)\f$-approximation to
\f$ \mel(P)\f$ is an enclosing ellipsoid whose volume is by at most a \f$ (P)\f$ is an enclosing ellipsoid whose volume is by at most a
factor of \f$ 1+\epsilon\f$ larger than the volume of the smallest factor of \f$ 1+\epsilon\f$ larger than the volume of the smallest
enclosing ellipsoid of \f$ P\f$. enclosing ellipsoid of \f$ P\f$.
Given this notation, an object of class `Approximate_min_ellipsoid_d` represents an Given this notation, an object of class `Approximate_min_ellipsoid_d` represents an
\f$ (1+\epsilon)\f$-approximation to \f$ \mel(P)\f$ for a given finite and \f$ (1+\epsilon)\f$-approximation to \f$ (P)\f$ for a given finite and
full-dimensional multiset of points \f$ P\subset\E^d\f$ and a real constant full-dimensional multiset of points \f$ P\subset\E^d\f$ and a real constant
\f$ \epsilon>0\f$.\cgalFootnote{A <I>multiset</I> is a set where elements may have multiplicity greater than \f$ 1\f$.} When an \f$ \epsilon>0\f$.\cgalFootnote{A <I>multiset</I> is a set where elements may have multiplicity greater than \f$ 1\f$.} When an
`Approximate_min_ellipsoid_d<Traits>` object is constructed, an `Approximate_min_ellipsoid_d<Traits>` object is constructed, an
iterator over the points \f$ P\f$ and the number \f$ \epsilon\f$ have to be iterator over the points \f$ P\f$ and the number \f$ \epsilon\f$ have to be
specified; the number \f$ \epsilon\f$ defines the <I>desired specified; the number \f$ \epsilon\f$ defines the <I>desired
approximation ratio</I> \f$ 1+\epsilon\f$. The underlying algorithm will then approximation ratio</I> \f$ 1+\epsilon\f$. The underlying algorithm will then
try to compute an \f$ (1+\epsilon)\f$-approximation to \f$ \mel(P)\f$, and one of try to compute an \f$ (1+\epsilon)\f$-approximation to \f$ (P)\f$, and one of
the following two cases takes place. the following two cases takes place.
<UL> <UL>
<LI>The algorithm determines that \f$ P\f$ is not full-dimensional (see <LI>The algorithm determines that \f$ P\f$ is not full-dimensional (see
@ -44,7 +44,7 @@ the following two cases takes place.
in all cases decide correctly whether \f$ P\f$ is full-dimensional or in all cases decide correctly whether \f$ P\f$ is full-dimensional or
not. If `is_full_dimensional()` returns `false`, the points not. If `is_full_dimensional()` returns `false`, the points
lie in such a "thin" subspace of \f$ \E^d\f$ that the algorithm is lie in such a "thin" subspace of \f$ \E^d\f$ that the algorithm is
incapable of computing an approximation to \f$ \mel(P)\f$. More incapable of computing an approximation to \f$ (P)\f$. More
precisely, if `is_full_dimensional()` returns `false`, there precisely, if `is_full_dimensional()` returns `false`, there
exist two parallel hyperplanes in \f$ \E^d\f$ with the points \f$ P\f$ in exist two parallel hyperplanes in \f$ \E^d\f$ with the points \f$ P\f$ in
between so that the distance \f$ \delta\f$ between the hyperplanes is between so that the distance \f$ \delta\f$ between the hyperplanes is
@ -55,7 +55,7 @@ If \f$ P\f$ is not full-dimensional, linear algebra techniques should be
used to determine an affine subspace \f$ S\f$ of \f$ \E^d\f$ that contains the used to determine an affine subspace \f$ S\f$ of \f$ \E^d\f$ that contains the
points \f$ P\f$ as a (w.r.t.\ \f$ S\f$) full-dimensional pointset; once \f$ S\f$ is points \f$ P\f$ as a (w.r.t.\ \f$ S\f$) full-dimensional pointset; once \f$ S\f$ is
determined, the algorithm can be invoked again to compute an determined, the algorithm can be invoked again to compute an
approximation to (the lower-dimensional) \f$ \mel(P)\f$ in \f$ S\f$. Since approximation to (the lower-dimensional) \f$ (P)\f$ in \f$ S\f$. Since
`is_full_dimensional()` might (due to rounding errors, see `is_full_dimensional()` might (due to rounding errors, see
above) return `false` even though \f$ P\f$ is full-dimensional, the above) return `false` even though \f$ P\f$ is full-dimensional, the
lower-dimensional subspace \f$ S\f$ containing \f$ P\f$ need not exist. lower-dimensional subspace \f$ S\f$ containing \f$ P\f$ need not exist.
@ -66,7 +66,7 @@ ellipsoid of the projected points within \f$ H\f$; the fitting can be
done for instance using the `linear_least_squares_fitting()` done for instance using the `linear_least_squares_fitting()`
function from the \cgal package `Principal_component_analysis`. function from the \cgal package `Principal_component_analysis`.
<LI>The algorithm determines that \f$ P\f$ is full-dimensional. In this <LI>The algorithm determines that \f$ P\f$ is full-dimensional. In this
case, it provides an approximation \f$ {\cal E}\f$ to \f$ \mel(P)\f$, but case, it provides an approximation \f$ {\cal E}\f$ to \f$ (P)\f$, but
depending on the input problem (i.e., on the pair \f$ (P,\epsilon)\f$), depending on the input problem (i.e., on the pair \f$ (P,\epsilon)\f$),
it may not have achieved the desired approximation ratio but merely it may not have achieved the desired approximation ratio but merely
some <I>worse</I> approximation ratio \f$ 1+\epsilon'>1+\epsilon\f$. The some <I>worse</I> approximation ratio \f$ 1+\epsilon'>1+\epsilon\f$. The
@ -126,7 +126,7 @@ Cholesky-decomposition. The algorithm's running time is
To illustrate the usage of `Approximate_min_ellipsoid_d` we give two examples in 2D. The To illustrate the usage of `Approximate_min_ellipsoid_d` we give two examples in 2D. The
first program generates a random set \f$ P\subset\E^2\f$ and outputs the first program generates a random set \f$ P\subset\E^2\f$ and outputs the
points and a \f$ 1.01\f$-approximation of \f$ \mel(P)\f$ as an EPS-file, which points and a \f$ 1.01\f$-approximation of \f$ (P)\f$ as an EPS-file, which
you can view using <TT>gv</TT>, for instance. (In both examples you can you can view using <TT>gv</TT>, for instance. (In both examples you can
change the variables `n` and `d` to experiment with the code.) change the variables `n` and `d` to experiment with the code.)
@ -204,7 +204,7 @@ typedef unspecified_type Axis_direction_iterator;
/*! /*!
initializes `ame` to an \f$ (1+\epsilon)\f$-approximation of initializes `ame` to an \f$ (1+\epsilon)\f$-approximation of
\f$ \mel(P)\f$ with \f$ P\f$ being the set of points in the range \f$ (P)\f$ with \f$ P\f$ being the set of points in the range
[`first`,`last`). The number \f$ \epsilon\f$ in this will [`first`,`last`). The number \f$ \epsilon\f$ in this will
be at most `eps`, if possible. However, due to the be at most `eps`, if possible. However, due to the
limited precision in the algorithm's underlying arithmetic, it limited precision in the algorithm's underlying arithmetic, it
@ -260,7 +260,7 @@ unsigned int number_of_points( ) const;
returns a number returns a number
\f$ \epsilon'\f$ such that the computed approximation is (under exact \f$ \epsilon'\f$ such that the computed approximation is (under exact
arithmetic) guaranteed to be an \f$ (1+\epsilon')\f$-approximation to arithmetic) guaranteed to be an \f$ (1+\epsilon')\f$-approximation to
\f$ \mel(P)\f$. \f$ (P)\f$.
\pre `ame.is_full_dimensional() == true`. \pre `ame.is_full_dimensional() == true`.
\post \f$ \epsilon'>0\f$. \post \f$ \epsilon'>0\f$.
*/ */
@ -404,7 +404,7 @@ bool is_full_dimensional( ) const;
/// An object `ame` is valid iff <UL> <LI>`ame` contains all points of /// An object `ame` is valid iff <UL> <LI>`ame` contains all points of
/// its defining set \f$ P\f$, <LI>`ame` is an \f$ /// its defining set \f$ P\f$, <LI>`ame` is an \f$
/// (1+\epsilon')\f$-approximation to the smallest ellipsoid \f$ /// (1+\epsilon')\f$-approximation to the smallest ellipsoid \f$
/// \mel(P)\f$ of \f$ P\f$, <LI>The ellipsoid represented by `ame` /// (P)\f$ of \f$ P\f$, <LI>The ellipsoid represented by `ame`
/// fulfills the inclusion ( \ref eqapproximate_min_ellipsoid_incl /// fulfills the inclusion ( \ref eqapproximate_min_ellipsoid_incl
/// ). </UL> /// ). </UL>
/// @{ /// @{
@ -426,7 +426,7 @@ bool is_valid( bool verbose = false) const;
/*! /*!
Writes the points \f$ P\f$ and the computed approximation to Writes the points \f$ P\f$ and the computed approximation to
\f$ \mel(P)\f$ as an EPS-file under pathname `name`. \pre The dimension of points \f$ P\f$ must be \f$ 2\f$. \f$ (P)\f$ as an EPS-file under pathname `name`. \pre The dimension of points \f$ P\f$ must be \f$ 2\f$.
<I>Note:</I> this <I>Note:</I> this
routine is provided as a debugging routine; future version of routine is provided as a debugging routine; future version of
\cgal might not provide it anymore. \cgal might not provide it anymore.