diff --git a/Bounding_volumes/doc/Bounding_volumes/CGAL/Approximate_min_ellipsoid_d.h b/Bounding_volumes/doc/Bounding_volumes/CGAL/Approximate_min_ellipsoid_d.h index c53940face3..535dc70d085 100644 --- a/Bounding_volumes/doc/Bounding_volumes/CGAL/Approximate_min_ellipsoid_d.h +++ b/Bounding_volumes/doc/Bounding_volumes/CGAL/Approximate_min_ellipsoid_d.h @@ -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 real \f$ d\f$-vector, and \f$ \eta\in\mathbb{R}\f$. A pointset \f$ P\subseteq \E^d\f$ is called full-dimensional 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$; this ellipsoid exists and is unique. For a given finite and full-dimensional pointset \f$ P\subset \E^d\f$ and a real number \f$ \epsilon\ge 0\f$, we say that an ellipsoid \f$ {\cal -E}\subset\E^d\f$ is an \f$ (1+\epsilon)\f$-appoximation to \f$ \mel(P)\f$ if -\f$ P\subset {\cal E}\f$ and \f$ \vol({\cal E}) \leq (1+\epsilon) -\vol(\mel(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 +E}\subset\E^d\f$ is an \f$ (1+\epsilon)\f$-appoximation to \f$ (P)\f$ if +\f$ P\subset {\cal E}\f$ and \f$ ({\cal E}) \leq (1+\epsilon) +((P))\f$. In other words, an \f$ (1+\epsilon)\f$-approximation to +\f$ (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 enclosing ellipsoid of \f$ P\f$. Given this notation, an object of class `Approximate_min_ellipsoid_d` represents an -\f$ (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 \f$ \epsilon>0\f$.\cgalFootnote{A multiset is a set where elements may have multiplicity greater than \f$ 1\f$.} When an `Approximate_min_ellipsoid_d` object is constructed, an iterator over the points \f$ P\f$ and the number \f$ \epsilon\f$ have to be specified; the number \f$ \epsilon\f$ defines the desired approximation ratio \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.