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@ -32,13 +32,13 @@ triangulation to get a fully dynamic computation.
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\section secconvex_hull_3 Static Convex Hull Construction
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The function
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`convex_hull_3`C provides an
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`::convex_hull_3` provides an
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implementation of the quickhull algorithm \cite bdh-qach-96 for three
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dimensionsquickhull, 3D. There are two versions of this
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function available, one that can be used when it is known that the output
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will be a polyhedron (<I>i.e.</I>, there are more than three points and
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they are not all collinear) and one that handles all degenerate cases
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and returns a `CGAL::Object`, which may be a point, a segment, a
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and returns a `::Object`, which may be a point, a segment, a
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triangle, or a polyhedron. Both versions accept a range of input
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iterators defining the set of points whose convex hull is to be computed
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and a traits class defining the geometric types and predicates used in
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@ -46,7 +46,7 @@ computing the hull.
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## Traits Class ##
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The function `convex_hull_3` is parameterized by a traits class,
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The function `::convex_hull_3` is parameterized by a traits class,
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which specifies the types and geometric primitives to be used in the
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computation. If input points from a kernel with exact predicates
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and non-exact constructions are used, and a certified result is expected,
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@ -56,11 +56,11 @@ account.
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## Convexity Checking ##
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The function `is_strongly_convex_3`C
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The function `::is_strongly_convex_3`
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implements the algorithm of Mehlhorn <I>et al.</I> \cite mnssssu-cgpvg-96
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to determine if the vertices of a given polytope constitute a strongly convex
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point set or not. This function is used in postcondition testing for
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`convex_hull_3`.
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`::convex_hull_3`.
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## Example ##
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@ -75,15 +75,15 @@ of the convex hull.
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# Incremental Convex Hull Construction #
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The function `convex_hull_incremental_3` C provides an
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interface similar to `convex_hull_3` for the \f$ d\f$-dimensional
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The function `::convex_hull_incremental_3` C provides an
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interface similar to `::convex_hull_3` for the \f$ d\f$-dimensional
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incremental construction algorithm \cite cms-frric-93
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implemented by the class `CGAL::Convex_hull_d<R>` that is specialized
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implemented by the class `::Convex_hull_d<R>` that is specialized
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to three dimensions. This function accepts an iterator range over a set of
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input points and returns a polyhedron, but it does not have a traits class
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in its interface. It uses the kernel
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class `Kernel` used in the polyhedron type to define an instance of the
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adapter traits class `CGAL::Convex_hull_d_traits_3<Kernel>`.
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adapter traits class `::Convex_hull_d_traits_3<Kernel>`.
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In almost all cases, the static and the dynamic version will
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be faster than the incremental convex hull algorithm (mainly
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@ -93,10 +93,10 @@ completeness and educational purposes. You should use the dynamic
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version when you need an efficient incremental convex hull algorithm.
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To use the full functionality available with the \f$ d\f$-dimensional class
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`CGAL::Convex_hull_d<R>` in three dimensions (<I>e.g.</I>, the ability
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`::Convex_hull_d<R>` in three dimensions (<I>e.g.</I>, the ability
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to insert new points and to query if a point lies in the convex hull or not),
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you can instantiate the class `CGAL::Convex_hull_d<K>` with the adapter
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traits class `CGAL::Convex_hull_d_traits_3<K>`, as shown in the following
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you can instantiate the class `::Convex_hull_d<K>` with the adapter
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traits class `::Convex_hull_d_traits_3<K>`, as shown in the following
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example.
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## Example ##
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@ -106,7 +106,7 @@ example.
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# Dynamic Convex Hull Construction #
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Fully dynamic maintenance of a convex hull can be achieved by using the
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class `CGAL::Delaunay_triangulation_3`. This class supports insertion
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class `::Delaunay_triangulation_3`. This class supports insertion
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and removal of points (<I>i.e.</I>, vertices of the triangulation) and the
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convex hull edges are simply the finite edges of infinite faces.
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The following example illustrates the dynamic construction of a convex hull.
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@ -125,10 +125,10 @@ not all of them are vertices of the hull.
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# Performance #
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In the following, we compare the running times of the three approaches to compute 3D convex hulls.
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For the static version (using `CGAL::convex_hull_3`) and the dynamic version
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(using `CGAL::Delaunay_triangulation_3` and `CGAL::convex_hull_3_to_polyhedron_3`), the kernel
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used was `CGAL::Exact_predicates_inexact_constructions_kernel`. For the incremental version
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(using `CGAL::convex_hull_incremental_3`), the kernel used was `CGAL::Exact_predicates_exact_constructions_kernel`.
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For the static version (using `::convex_hull_3`) and the dynamic version
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(using `::Delaunay_triangulation_3` and `::convex_hull_3_to_polyhedron_3`), the kernel
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used was `::Exact_predicates_inexact_constructions_kernel`. For the incremental version
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(using `::convex_hull_incremental_3`), the kernel used was `::Exact_predicates_exact_constructions_kernel`.
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To compute the convex hull of a million of random points in a unit ball the static approach needed 1.63s, while
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the dynamic and incremental approaches needed 9.50s and 11.54s respectively.
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Andreas Fabri