%% cross references
\ccIndexSubitemSeeAlso{minimum enclosing}{smallest enclosing}
\ccIndexSubitemSeeAlso{minimum spanning}{smallest enclosing}
\ccIndexSubitemSeeAlso{concentric spheres}{annulus}

%% -----------------------------------------------------------------------------
\ccRefChapter{Bounded Volumes}
\ccChapterAuthor{Kaspar Fischer \and Bernd G{\"a}rtner \and Thomas Herrmann \and Michael Hoffmann
   \and Sven Sch{\"o}nherr}


This chapter describes concepts, classes, and functions for bounding areas and volumes.

These are Smallest enclosing circle and ellipse (2D), smallest enclosing rectangle,
parallelogram, and strip (2D), rectangular $p$-center (2D), smallest
enclosing sphere and annulus (dD), approximate
minimum-volume enclosing ellipsoid with user-specified
approximation ratio (dD).

\section*{Assertions}
The optimization code uses infix \ccc{OPTIMISATION} in the assertions,
e.g.\ defining the compiler flag
\ccc{CGAL_OPTIMISATION_NO_PRECONDITIONS} switches precondition
checking off, cf.~Section~\ref{sec:checks}.

\section{Classified References Pages}

%% -----------------------------------------------------------------------------
\subsection*{Bounding Areas and Volumes}

\ccRefIdfierPage{CGAL::Min_circle_2<Traits>}\\[1ex]
\ccRefIdfierPage{CGAL::Min_circle_2_traits_2<K>}\\[1ex]
\ccRefConceptPage{MinCircle2Traits}

\smallskip

\ccRefIdfierPage{CGAL::Min_ellipse_2<Traits>}\\[1ex]
\ccRefIdfierPage{CGAL::Min_ellipse_2_traits_2<K>}\\[1ex]
\ccRefConceptPage{MinEllipse2Traits}

\smallskip

\ccRefIdfierPage{CGAL::Approximate_min_ellipsoid_d<Traits>}\\
\ccRefConceptPage{ApproximateMinEllipsoid_d_Traits_d}

\smallskip

\ccRefIdfierPage{CGAL::min_rectangle_2}\\
\ccRefIdfierPage{CGAL::min_parallelogram_2}\\
\ccRefIdfierPage{CGAL::min_strip_2}\\[1ex]
\ccRefIdfierPage{CGAL::Min_quadrilateral_default_traits_2<K>}\\[1ex]
\ccRefConceptPage{MinQuadrilateralTraits_2}

\smallskip

\ccRefIdfierPage{CGAL::rectangular_p_center_2}\\[1ex]
\ccRefIdfierPage{CGAL::Rectangular_p_center_default_traits_2<K>}\\[1ex]
\ccRefConceptPage{RectangularPCenterTraits_2}

\smallskip

\ccRefIdfierPage{CGAL::Min_sphere_d<Traits>}\\
\ccRefIdfierPage{CGAL::Min_annulus_d<Traits>}\\[1ex]
\ccRefIdfierPage{CGAL::Min_sphere_annulus_d_traits_2<K,ET,NT>}\\
\ccRefIdfierPage{CGAL::Min_sphere_annulus_d_traits_3<K,ET,NT>}\\
\ccRefIdfierPage{CGAL::Min_sphere_annulus_d_traits_d<K,ET,NT>}\\[1ex]
\ccRefConceptPage{MinSphereAnnulusDTraits}

\smallskip

\ccRefIdfierPage{CGAL::Min_sphere_of_spheres_d<Traits>}\\
\ccRefConceptPage{MinSphereOfSpheresTraits}



\input{Bounding_volumes_ref/main_Min_circle_2.tex}
\input{Bounding_volumes_ref/main_Min_ellipse_2.tex}
\input{Bounding_volumes_ref/main_Min_quadrilateral_2.tex}
\input{Bounding_volumes_ref/main_Rectangular_p_centers.tex}

\input{Bounding_volumes_ref/main_Min_sphere_d.tex}
\input{Bounding_volumes_ref/main_Min_annulus_d.tex}

\input{Bounding_volumes_ref/main_Min_sphere_of_spheres_d.tex}
\input{Bounding_volumes_ref/main_Approximate_min_ellipsoid_d.tex}

\input{Bounding_volumes_ref/Min_sphere_annulus_d_traits_2.tex}
\input{Bounding_volumes_ref/Min_sphere_annulus_d_traits_3.tex}
\input{Bounding_volumes_ref/Min_sphere_annulus_d_traits_d.tex}
\input{Bounding_volumes_ref/MinSphereAnnulusDTraits.tex}