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Merge pull request #3525 from sloriot/Bounding_volume-doc_html_fix
Bounding volume doc html fix
This commit is contained in:
Laurent Rineau
commit
ed4611c1d5
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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
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positive definite matrix from the set \f$ \mathbb{R}^{d\times d}\f$, \f$ e\f$ is some
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real \f$ d\f$-vector, and \f$ \eta\in\mathbb{R}\f$. A pointset \f$ P\subseteq \E^d\f$ is
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called <I>full-dimensional</I> if its affine hull has dimension \f$ d\f$.
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For a finite, full-dimensional pointset \f$ P\f$ we denote by \f$ \mel(P)\f$ the
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For a finite, full-dimensional pointset \f$ P\f$ we denote by \f$ (P)\f$ the
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smallest ellipsoid that contains all points of \f$ P\f$; this ellipsoid
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exists and is unique.
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For a given finite and full-dimensional pointset \f$ P\subset \E^d\f$ and a
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real number \f$ \epsilon\ge 0\f$, we say that an ellipsoid \f$ {\cal
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E}\subset\E^d\f$ is an <I>\f$ (1+\epsilon)\f$-appoximation</I> to \f$ \mel(P)\f$ if
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\f$ P\subset {\cal E}\f$ and \f$ \vol({\cal E}) \leq (1+\epsilon)
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\vol(\mel(P))\f$. In other words, an \f$ (1+\epsilon)\f$-approximation to
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\f$ \mel(P)\f$ is an enclosing ellipsoid whose volume is by at most a
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E}\subset\E^d\f$ is an <I>\f$ (1+\epsilon)\f$-appoximation</I> to \f$ (P)\f$ if
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\f$ P\subset {\cal E}\f$ and \f$ ({\cal E}) \leq (1+\epsilon)
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((P))\f$. In other words, an \f$ (1+\epsilon)\f$-approximation to
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\f$ (P)\f$ is an enclosing ellipsoid whose volume is by at most a
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factor of \f$ 1+\epsilon\f$ larger than the volume of the smallest
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enclosing ellipsoid of \f$ P\f$.
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Given this notation, an object of class `Approximate_min_ellipsoid_d` represents an
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\f$ (1+\epsilon)\f$-approximation to \f$ \mel(P)\f$ for a given finite and
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\f$ (1+\epsilon)\f$-approximation to \f$ (P)\f$ for a given finite and
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full-dimensional multiset of points \f$ P\subset\E^d\f$ and a real constant
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\f$ \epsilon>0\f$.\cgalFootnote{A <I>multiset</I> is a set where elements may have multiplicity greater than \f$ 1\f$.} When an
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`Approximate_min_ellipsoid_d<Traits>` object is constructed, an
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iterator over the points \f$ P\f$ and the number \f$ \epsilon\f$ have to be
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specified; the number \f$ \epsilon\f$ defines the <I>desired
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approximation ratio</I> \f$ 1+\epsilon\f$. The underlying algorithm will then
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try to compute an \f$ (1+\epsilon)\f$-approximation to \f$ \mel(P)\f$, and one of
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try to compute an \f$ (1+\epsilon)\f$-approximation to \f$ (P)\f$, and one of
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the following two cases takes place.
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<UL>
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<LI>The algorithm determines that \f$ P\f$ is not full-dimensional (see
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@ -44,7 +44,7 @@ the following two cases takes place.
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in all cases decide correctly whether \f$ P\f$ is full-dimensional or
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not. If `is_full_dimensional()` returns `false`, the points
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lie in such a "thin" subspace of \f$ \E^d\f$ that the algorithm is
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incapable of computing an approximation to \f$ \mel(P)\f$. More
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incapable of computing an approximation to \f$ (P)\f$. More
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precisely, if `is_full_dimensional()` returns `false`, there
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exist two parallel hyperplanes in \f$ \E^d\f$ with the points \f$ P\f$ in
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between so that the distance \f$ \delta\f$ between the hyperplanes is
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@ -55,7 +55,7 @@ If \f$ P\f$ is not full-dimensional, linear algebra techniques should be
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used to determine an affine subspace \f$ S\f$ of \f$ \E^d\f$ that contains the
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points \f$ P\f$ as a (w.r.t.\ \f$ S\f$) full-dimensional pointset; once \f$ S\f$ is
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determined, the algorithm can be invoked again to compute an
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approximation to (the lower-dimensional) \f$ \mel(P)\f$ in \f$ S\f$. Since
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approximation to (the lower-dimensional) \f$ (P)\f$ in \f$ S\f$. Since
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`is_full_dimensional()` might (due to rounding errors, see
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above) return `false` even though \f$ P\f$ is full-dimensional, the
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lower-dimensional subspace \f$ S\f$ containing \f$ P\f$ need not exist.
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@ -66,7 +66,7 @@ ellipsoid of the projected points within \f$ H\f$; the fitting can be
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done for instance using the `linear_least_squares_fitting()`
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function from the \cgal package `Principal_component_analysis`.
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<LI>The algorithm determines that \f$ P\f$ is full-dimensional. In this
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case, it provides an approximation \f$ {\cal E}\f$ to \f$ \mel(P)\f$, but
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case, it provides an approximation \f$ {\cal E}\f$ to \f$ (P)\f$, but
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depending on the input problem (i.e., on the pair \f$ (P,\epsilon)\f$),
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it may not have achieved the desired approximation ratio but merely
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some <I>worse</I> approximation ratio \f$ 1+\epsilon'>1+\epsilon\f$. The
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@ -126,7 +126,7 @@ Cholesky-decomposition. The algorithm's running time is
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To illustrate the usage of `Approximate_min_ellipsoid_d` we give two examples in 2D. The
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first program generates a random set \f$ P\subset\E^2\f$ and outputs the
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points and a \f$ 1.01\f$-approximation of \f$ \mel(P)\f$ as an EPS-file, which
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points and a \f$ 1.01\f$-approximation of \f$ (P)\f$ as an EPS-file, which
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you can view using <TT>gv</TT>, for instance. (In both examples you can
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change the variables `n` and `d` to experiment with the code.)
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@ -204,7 +204,7 @@ typedef unspecified_type Axis_direction_iterator;
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/*!
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initializes `ame` to an \f$ (1+\epsilon)\f$-approximation of
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\f$ \mel(P)\f$ with \f$ P\f$ being the set of points in the range
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\f$ (P)\f$ with \f$ P\f$ being the set of points in the range
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[`first`,`last`). The number \f$ \epsilon\f$ in this will
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be at most `eps`, if possible. However, due to the
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limited precision in the algorithm's underlying arithmetic, it
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@ -260,7 +260,7 @@ unsigned int number_of_points( ) const;
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returns a number
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\f$ \epsilon'\f$ such that the computed approximation is (under exact
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arithmetic) guaranteed to be an \f$ (1+\epsilon')\f$-approximation to
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\f$ \mel(P)\f$.
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\f$ (P)\f$.
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\pre `ame.is_full_dimensional() == true`.
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\post \f$ \epsilon'>0\f$.
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*/
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@ -404,7 +404,7 @@ bool is_full_dimensional( ) const;
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/// An object `ame` is valid iff <UL> <LI>`ame` contains all points of
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/// its defining set \f$ P\f$, <LI>`ame` is an \f$
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/// (1+\epsilon')\f$-approximation to the smallest ellipsoid \f$
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/// \mel(P)\f$ of \f$ P\f$, <LI>The ellipsoid represented by `ame`
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/// (P)\f$ of \f$ P\f$, <LI>The ellipsoid represented by `ame`
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/// fulfills the inclusion ( \ref eqapproximate_min_ellipsoid_incl
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/// ). </UL>
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/// @{
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@ -426,7 +426,7 @@ bool is_valid( bool verbose = false) const;
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/*!
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Writes the points \f$ P\f$ and the computed approximation to
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\f$ \mel(P)\f$ as an EPS-file under pathname `name`. \pre The dimension of points \f$ P\f$ must be \f$ 2\f$.
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\f$ (P)\f$ as an EPS-file under pathname `name`. \pre The dimension of points \f$ P\f$ must be \f$ 2\f$.
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<I>Note:</I> this
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routine is provided as a debugging routine; future version of
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\cgal might not provide it anymore.
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