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@ -695,14 +695,17 @@ as follows:
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<UL>
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<LI>You can obtain a pair of iterators of type \link
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ArrangementDcelFace::Hole_iterator `Hole_iterator`\endlink that
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Arrangement_2::Face::Hole_iterator `Hole_iterator`\endlink that
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define the range of holes inside a face \f$f\f$ by calling \link
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Arrangement_2::Face::holes_begin() `f->holes_begin()`\endlink and
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\link Arrangement_2::Face::holes_end() `f->holes_end()`\endlink. The
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value type of this iterator type is \link
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Arrangement_2::Ccb_halfedge_circulator
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`Ccb_halfedge_circulator`\endlink, defining the CCB that winds in a
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clockwise order around a hole.
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clockwise order around a hole. The call \link
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Arrangement_2::Face::number_of_holes()
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`f->number_of_holes()`\endlink return the number of holes in
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\f$f\f$.
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<LI>The calls \link Arrangement_2::Face::isolated_vertices_begin()
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`f->isolated_vertices_begin()`\endlink and \link
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@ -2434,27 +2437,26 @@ and \f$f\f$ be handles to a vertex of type
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provides the Boolean predicate \link
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Arrangement_2::Halfedge::is_fictitious `is_fictitious()`\endlink. If
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the call \link Arrangement_2::Halfedge::is_fictitious
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`e->is_fictitious()`\endlink predicate evaluates to `false`, you can
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access the \f$x\f$-monotone curve associated with
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\f$e\f$. Otherwise, \f$e\f$ is not associated with an \link
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Arrangement_2::X_monotone_curve_2 `X_monotone_curve_2`\endlink object.
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`e->is_fictitious()`\endlink evaluates to `false`, you can access
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the \f$x\f$-monotone curve associated with \f$e\f$. Otherwise,
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\f$e\f$ is not associated with an \link
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Arrangement_2::X_monotone_curve_2 `X_monotone_curve_2`\endlink
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object.
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<LI>The nested \link Arrangement_2::Face `Face`\endlink type provides
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the Boolean predicate \link Arrangement_2::Face::is_fictitious()
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`is_fictitious()`\endlink. The call \link
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Arrangement_2::Face::is_fictitious() `f->is_fictitious()`\endlink
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predicate evaluates to `true` only when \f$f\f$ lies outside the
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bounding rectangle. The method \link
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Arrangement_2::Face::outer_ccb() `outer_ccb()`\endlink (see Section
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\ref arr_sssectr_face must not be invoked for the fictitious
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face. \cgalFootnote{Typically, the code is guarded against such
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malicious calls with adequate preconditions. However, these
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preconditions are suppressed when the code is compiled with maximum
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optimization.}
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Note that a valid unbounded face (of an arrangement
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that supports unbounded curves) has a valid outer CCB, although the
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CCB comprises only fictitious halfedges if the arrangement is
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induced only by bounded curves.
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evaluates to `true` only when \f$f\f$ lies outside the bounding
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rectangle. The method \link Arrangement_2::Face::outer_ccb()
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`outer_ccb()`\endlink (see Section \ref arr_sssectr_face must not be
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invoked for the fictitious face. \cgalFootnote{Typically, the code
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is guarded against such malicious calls with adequate
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preconditions. However, these preconditions are suppressed when the
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code is compiled with maximum optimization.} Note that a valid
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unbounded face (of an arrangement that supports unbounded curves)
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has a valid outer CCB, although the CCB comprises only fictitious
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halfedges if the arrangement is induced only by bounded curves.
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</UL>
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@ -2876,27 +2878,29 @@ can be used to represent a 2D arrangement embedded in a 3D
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surface. The template is parameterized by template parameters
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<em>geometry traits</em> and <em>topology traits</em>. The
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topology-traits type deals with the topology of the parametric
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surface. As explained in the previous section, a parametric surface
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\f$S \f$ is given by a continuous function \f$\phi_S: \Phi \rightarrow
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\mathbb{R}^3\f$, where the domain \f$\Phi = X \times Y\f$ is a
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rectangular two-dimensional parameter space; \f$S =
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\phi_S(\Phi)\f$. \f$X\f$ and \f$Y\f$ are open, half-open, or closed
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intervals with endpoints in \f$\overline{\mathbb{R}}\f$.
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surface; see Section \ref aos_sec-topol_traits. As explained in the
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previous section, a parametric surface \f$S \f$ is given by a
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continuous function \f$\phi_S: \Phi \rightarrow \mathbb{R}^3\f$, where
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the domain \f$\Phi = X \times Y\f$ is a rectangular two-dimensional
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parameter space; \f$S = \phi_S(\Phi)\f$. \f$X\f$ and \f$Y\f$ are open,
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half-open, or closed intervals with endpoints in
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\f$\overline{\mathbb{R}}\f$.
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The geometry-traits type introduces the type names of the basic
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geometric objects (i.e., point, curve, and \f$u \f$-monotone curve)
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and the set of operations on objects of these types required to
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construct and maintain the arrangement and to operate on it. The
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traits-concept hierarchy described in Section \ref aos_sec-geom_traits
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accurately defines the requirements imposed on a model of a
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geometry traits class. Recall that requirements apply to the parameter
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space \f$\Phi = X \times Y\f$; thus, they are defined in terms of
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\f$x\f$ and \f$y\f$ coordinates. Most requirements apply to all type
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of arrangements regardless of the embedding surface. However, several
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accurately defines the requirements imposed on a model of a geometry
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traits class. Recall that requirements apply to the parameter space
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\f$\Phi = X \times Y\f$; thus, they are defined in terms of \f$x\f$
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and \f$y\f$ coordinates. Most requirements apply to all type of
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arrangements regardless of the embedding surface. However, several
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refined concepts apply only to arrangements on surfaces that are not
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the plane, that is, modeled with \f$\phi_S\f$ not being the identity
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mapping. They differ according to the type of the side boundaries of
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the parameter space being open, closed, contracted, or identified.
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in the plane, that is, modeled with \f$\phi_S\f$ not being the
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identity mapping. They differ according to the type of the side
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boundaries of the parameter space being open, closed, contracted, or
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identified.
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At this point only one type of arrangements on non planar surfaces is
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supported, namely, arrangements embedded in the sphere and induced by
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@ -2931,6 +2935,127 @@ Sites are drawn in red and Voronoi edges are drawn in blue.
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(d) A degenerate power diagram of 14 sites on the sphere.
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\cgalFigureCaptionEnd
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<!----------------------------------------------------------------------------->
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\subsection aos_ssec-curved_surfaces-basic Basic Manipulation and Traversal Methods
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<!----------------------------------------------------------------------------->
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The types \link Arrangement_on_surface_2::Vertex `Vertex`\endlink,
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\link Arrangement_on_surface_2::Halfedge `Halfedge`\endlink, and \link
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Arrangement_on_surface_2::Face `Face`\endlink nested in the
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`Arrangement_on_surface_2` class template support the methods listed
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in Section \ref arr_ssectraverse and in Section \ref
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arr_sssecunb_basic. Let \f$v\f$, \f$e\f$, and \f$f\f$ be handles to a
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vertex of type \link Arrangement_on_surface_2::Vertex
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`Vertex`\endlink, a halfedge of type \link
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Arrangement_on_surface_2::Halfedge `Halfedge`\endlink, and a face of
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type \link Arrangement_on_surface_2::Face `Face`\endlink,
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respectively.
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<UL>
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<LI>The calls \link
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Arrangement_on_surface_2::Vertex::parameter_space_in_x()
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`v->parameter_space_in_x()`\endlink and \link
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Arrangement_on_surface_2::Vertex::parameter_space_in_y()
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`v->parameter_space_in_y()`\endlink determine the location of the
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geometric embedding of the vertex \f$v\f$. The call \link
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Arrangement_on_surface_2::Vertex::parameter_space_in_x()
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`v->parameter_space_in_x()`\endlink returns `CGAL::ARR_INTERIOR` if
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the geometric embedding of \f$v\f$ is a point \f$p\f$ that does not
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lie on the left nor on the right sides of the boundary of the
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parameter space. If the left and right sides are not identified,
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then `CGAL::ARR_LEFT_BOUNDARY` or `CGAL::ARR_RIGHT_BOUNDARY` are
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returned if \f$p\f$ lies on the left or the right side,
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respectively. If the left and right sides are identified, the
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return value is non-deterministic. Similarly, the call \link
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Arrangement_2::Vertex::parameter_space_in_y()
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`v->parameter_space_in_y()`\endlink returns `CGAL::ARR_INTERIOR` if
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the geometric embedding of \f$v\f$ is a point \f$q\f$ that does not
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lie on the bottom nor on the top sides of the boundary of the
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parameter space. If the bottom and top sides are not identified,
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then `CGAL::ARR_BOTTOM_BOUNDARY` or `CGAL::ARR_TOP_BOUNDARY` are
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returned if \f$q\f$ lies on the bottom or the top side,
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respectively. If the bottom and top sides are identified, the
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return value is non-deterministic.
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If you want to determine whether a point \f$p\f$ lies on identified
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sides, you need to exploit one of the two traits functors \link
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ArrangementIdentifiedVerticalTraits_2::Is_on_y_identification_2
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`Is_on_y_identification_2`\endlink or \link
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ArrangementIdentifiedVerticalTraits_2::Is_on_y_identification_2
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`Is_on_y_identification_2`\endlink; see Section \ref
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aos_ssec-traits-curved. If \f$p\f$ is incident to an
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\f$x\f$-monotone curve \f$c\f$ and \f$c\f$ is not vertical (thus,
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does not lie on the identified side as well), you can determine the
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boundary side of the end of \f$c\f$ associated with \f$p\f$
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exploiting either the \link
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ArrangementVerticalSideTraits_2::Parameter_space_in_x_2
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`Parameter_space_in_x_2`\endlink functor or the \link
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ArrangementHorizontalSideTraits_2::Parameter_space_in_y_2
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`Parameter_space_in_x_2`\endlink functor; see Section \ref
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aos_ssec-traits-curved.
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<LI>A face in a planar arrangement has at most one outer CCB. However,
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a face in an arrangement embedded on a surface, such as a torus, may
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have two outer CCBs; to this end, the methods \link
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Arrangement_on_surface_2::Face::outer_ccbs_begin()
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`outer_ccbs_begin()`\endlink and \link
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Arrangement_on_surface_2::Face::outer_ccbs_end()
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`outer_ccbs_end()`\endlink return a pair of iterators of type \link
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Arrangement_on_surface_2::Face::Outer_ccb_iterator
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`Outer_ccb_iterator`\endlink that define a range of outer CCBs
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inside the face \f$f\f$. Similarly, the methods \link
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Arrangement_on_surface_2::Face::inner_ccbs_begin()
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`inner_ccbs_begin()`\endlink and \link
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Arrangement_on_surface_2::Face::inner_ccbs_end()
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`inner_ccbs_end()`\endlink return a pair of iterators of type \link
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Arrangement_on_surface_2::Face::Inner_ccb_iterator
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`Inner_ccb_iterator`\endlink that define a range of inner CCBs
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inside the face \f$f\f$. (The latter pair of member functions and
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the iterator type are equivalent to the methods \link
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Arrangement_on_surface_2::Face::holes_begin()
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`holes_begin()`\endlink, \link
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Arrangement_on_surface_2::Face::holes_end() `holes()`\endlink, and
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\link Arrangement_on_surface_2::Face::Hole_iterator
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`Hole_iterator`\endlink; see Section arr_sssectr_face.) The calls
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\link Arrangement_on_surface_2::Face::number_of_outer_ccbs()
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`f->number_of_outer_ccbs()`\endlink and \link
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Arrangement_on_surface_2::Face::number_of_inner_ccbs()
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`f->number_of_inner_ccbs()`\endlink return the number of outer CCBs
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and the number of inner CCBs in \f$f\f$, respectively.
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</UL>
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The template function `is_in_x_range()` listed below, and defined in the
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file `is_in_x_range.h`, checks whether a given point \f$p\f$ is in the
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$x$-range of the curve associated with a given halfedge \f$e\f$. The
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function template also exemplifies how some of the above functions can
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be used.
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\code
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bool is_in_x_range(const X_monotone_curve_2& xcv, const Point_2& point) const {
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typedef Arr_geodesic_arc_on_sphere_traits_2<Kernel, atan_x, atan_y> Traits;
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Direction_2 p = project_xy(point);
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if (xcv.is_vertical()) {
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const Direction_3& normal = xcv.normal();
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Direction_2 q = (xcv.is_directed_right()) ?
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Direction_2(-(normal.dy()), normal.dx()) :
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Direction_2(normal.dy(), -(normal.dx()));
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const Kernel& kernel = *this;
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return kernel.equal_2_object()(p, q);
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}
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// The curve is not vertical:
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Direction_2 r = project_xy(xcv.right());
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const Kernel& kernel = *this;
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if (kernel.equal_2_object()(p, r)) return true;
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Direction_2 l = Traits::project_xy(xcv.left());
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if (kernel.equal_2_object()(p, l)) return true;
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return kernel.counterclockwise_in_between_2_object()(p, l, r);
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}
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\endcode
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<!--=========================================================================-->
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\section aos_sec-geom_traits The Geometry Traits
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<!--=========================================================================-->
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|
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@ -5250,9 +5375,17 @@ instantiate the curve-data traits:
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<!--=========================================================================-->
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\section aos_sec-topol_traits The Topology Traits
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<!--=========================================================================-->
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At this point we do not expose the topology traits concept. The
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package contains one topology traits, namely,
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A topology traits class encapsulates the definitions of the
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topological entities and the implementation of the functions that
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handle these topological entities, used by the
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`Arrangement_on_surface_2<GeometryTraits_2, TopologyTraits>` class
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template and by the peripheral modules. Every topology traits class
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must model the basic concept `ArrangementBasicTopologyTraits`. A model
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of this basic concept holds the (\sc{Dcel}) data structure used to
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represent the arrangement cells (i.e., vertices, edges, and facets)
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and the incident * relations between them. At this point we do not
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expose the concepts that refine the basic concept. The package
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contains one topology traits, namely,
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`Arr_spherical_topology_traits_2`. It can serve as a topology traits
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for an arrangement embedded on a sphere. More precisely, for an
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arrangement embedded on a sphere defined over a parameter space the
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Efi Fogel