mirror of https://github.com/CGAL/cgal
90 lines
3.3 KiB
Plaintext
90 lines
3.3 KiB
Plaintext
/// \defgroup PkgConvexHull3 3D Convex Hulls Reference
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/// \defgroup PkgConvexHull3Concepts Concepts
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/// \ingroup PkgConvexHull3
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/// \defgroup PkgConvexHull3Traits Traits Classes
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/// \ingroup PkgConvexHull3
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/*! \defgroup PkgConvexHull3Functions Convex Hull Functions
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\ingroup PkgConvexHull3
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The function `convex_hull_3()` computes the convex hull of a given set of
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three-dimensional points.
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Two versions of this function are available. The first can be used when it
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is known that the result will be a polyhedron and the second when a
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degenerate hull may also be possible.
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*/
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/// \defgroup PkgConvexityChecking Convexity Checking
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/// \ingroup PkgConvexHull3
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/*!
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\addtogroup PkgConvexHull3
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\todo fix or keep the `Default_traits`
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\cgalPkgDescriptionBegin{3D Convex Hulls,PkgConvexHull3Summary}
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\cgalPkgPicture{Convex_hull_3/fig/bunny.png}
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\cgalPkgSummaryBegin
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\cgalPkgAuthors{Susan Hert and Stefan Schirra}
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\cgalPkgDesc{This package provides functions for computing convex hulls in three dimensions as well as functions for checking if sets of points are strongly convex or not. One can compute the convex hull of a set of points in three dimensions in one of three ways: using a static algorithm, using an incremental construction algorithm, or using a triangulation to get a fully dynamic computation.}
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\cgalPkgManuals{Chapter_3D_Convex_Hulls,PkgConvexHull3}
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\cgalPkgSummaryEnd
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\cgalPkgShortInfoBegin
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\cgalPkgSince{1.1}
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\cgalPkgDependsOn{All algorithms produce as output a \ref PkgPolyhedronSummary "Polyhedron". The dynamic algorithms depend on \ref PkgTriangulation3Summary "3D Triangulations".}
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\cgalPkgBib{cgal:hs-ch3}
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\cgalPkgLicense{\ref licensesGPL "GPL"}
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\cgalPkgDemo{See Polyhedral Surface,polyhedron_3.zip}
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\cgalPkgShortInfoEnd
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\cgalPkgDescriptionEnd
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A subset \f$S \subseteq \mathbb{R}^3 \f$ is convex if for any two points `p` and `q`
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in the set the line segment with endpoints `p` and `q` is contained
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in \f$S\f$. The convex hull of a set \f$ S \f$ is the smallest convex set containing
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\f$ S \f$. The convex hull of a set of points `P` is a convex
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polytope with vertices in `P`. A point in `P` is an extreme point
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(with respect to `P`) if it is a vertex of the convex hull of `P`.
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\cgal provides functions for computing convex hulls in two, three
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and arbitrary dimensions as well as functions for testing if a given set of
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points in is strongly convex or not. This chapter describes the functions
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available for three dimensions.
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## Assertions ##
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The assertion flags for the convex hull and extreme point algorithms
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use `CH` in their names (<I>e.g.</I>, `CGAL_CH_NO_POSTCONDITIONS`).
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For the convex hull algorithms, the postcondition
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check tests only convexity (if not disabled), but not containment of the
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input points in the polygon or polyhedron defined by the output points.
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The latter is considered an expensive checking and can be enabled by
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defining `CGAL_CH_CHECK_EXPENSIVE`.
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\cgalClassifedRefPages
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## Concepts ##
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- `ConvexHullPolyhedron_3`
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- `ConvexHullPolyhedronFacet_3`
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- `ConvexHullPolyhedronHalfedge_3`
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- `ConvexHullPolyhedronVertex_3`
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- `ConvexHullTraits_3`
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- `IsStronglyConvexTraits_3`
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## Traits Classes ##
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- `CGAL::Convex_hull_traits_3<R>`
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## Convex Hull Functions ##
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- `CGAL::convex_hull_3`
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- `CGAL::convex_hull_incremental_3`
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- `CGAL::convex_hull_3_to_polyhedron_3`
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## Convexity Checking Function ##
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- `CGAL::is_strongly_convex_3`
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*/
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