mirror of https://github.com/CGAL/cgal
counter-clockwise -> counterclockwise
This commit is contained in:
Andreas Fabri
parent
e2058c01c6
commit
2bdeff8687
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@ -116,7 +116,7 @@ In our context, a polygon must uphold the following conditions:
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<OL>
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<LI><I>Closed Boundary</I> - the polygon's outer boundary must be a connected sequence of curves, that start and end at the same vertex.
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<LI><I>Simplicity</I> - the polygon must be simple.
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<LI><I>Orientation</I> - the polygon's outer boundary must be <I>counter-clockwise oriented</I>.
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<LI><I>Orientation</I> - the polygon's outer boundary must be <I>counterclockwise oriented</I>.
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</OL>
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\subsection bso_ssecpolygon_with_holes_validation Conditions for Valid Polygons with Holes
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@ -137,7 +137,7 @@ public:
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The direction of the first ray can be specified by the parameter
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initial_direction, which allows the first ray to start at any direction.
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The remaining directions are calculated in counter-clockwise order.
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The remaining directions are calculated in counterclockwise order.
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\param cone_number The number of cones
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\param initial_direction The direction of the first ray
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@ -169,7 +169,7 @@ protected:
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/* Construct edges in one cone bounded by two directions.
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\param cwBound The direction of the clockwise boundary of the cone.
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\param ccwBound The direction of the counter-clockwise boundary.
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\param ccwBound The direction of the counterclockwise boundary.
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\param g The Theta graph to be built.
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*/
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void add_edges_in_cone(const Direction_2& cwBound, const Direction_2& ccwBound, Graph_& g) {
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@ -164,7 +164,7 @@ protected:
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/* Construct edges in one cone bounded by two directions.
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\param cwBound The direction of the clockwise boundary of the cone.
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\param ccwBound The direction of the counter-clockwise boundary.
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\param ccwBound The direction of the counterclockwise boundary.
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\param g The Yao graph to be built.
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*/
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void add_edges_in_cone(const Direction_2& cwBound, const Direction_2& ccwBound, Graph_& g) {
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@ -149,7 +149,7 @@ Next, the gradient in a given triangle can be expressed as
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\f$\nabla u = \frac{1}{2 A_f} \sum_i u_i ( N \times e_i ) \f$
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where \f$A_f\f$ is the area of the triangle, \f$N\f$ is its outward unit normal, \f$e_i\f$ is the \f$i\f$th edge vector (oriented counter-clockwise), and \f$u_i\f$ is the value of \f$u\f$ at the opposing vertex. The integrated divergence associated with vertex \f$i\f$ can be written as
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where \f$A_f\f$ is the area of the triangle, \f$N\f$ is its outward unit normal, \f$e_i\f$ is the \f$i\f$th edge vector (oriented counterclockwise), and \f$u_i\f$ is the value of \f$u\f$ at the opposing vertex. The integrated divergence associated with vertex \f$i\f$ can be written as
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\f$\nabla \cdot X = \frac{1}{2} \sum_j cot\theta_1 (e_1 \cdot X_j) + cot \theta_2 (e_2 \cdot X_j)\f$
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@ -98,7 +98,7 @@ public:
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// set fh to the face at the right of [va,v]
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typename Tr::Face_circulator fc = tr.incident_faces(v, fh), fcbegin(fc);
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// circulators are counter-clockwise, so we start at the right of
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// circulators are counterclockwise, so we start at the right of
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// [va,v]
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do {
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if( !tr.is_infinite(fc) )
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@ -95,12 +95,12 @@ bool is_forward(Halfedge_const_handle e) const
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void check_order_preserving_embedding(Vertex_const_handle v) const;
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/*{\Mop checks if the embedding of the targets of the edges in
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the adjacency list |A(v)| is counter-clockwise order-preserving with
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the adjacency list |A(v)| is counterclockwise order-preserving with
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respect to the order of the edges in |A(v)|.}*/
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void check_order_preserving_embedding() const;
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/*{\Mop checks if the embedding of all vertices of |P| is
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counter-clockwise order-preserving with respect to the adjacency
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counterclockwise order-preserving with respect to the adjacency
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list ordering of all vertices.}*/
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void check_forward_prefix_condition(Vertex_const_handle v) const;
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@ -79,12 +79,12 @@ bool is_forward(Halfedge_const_handle e) const
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void check_order_preserving_embedding(Vertex_const_handle v) const;
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/*{\Mop checks if the embedding of the targets of the edges in
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the adjacency list |A(v)| is counter-clockwise order-preserving with
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the adjacency list |A(v)| is counterclockwise order-preserving with
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respect to the order of the edges in |A(v)|.}*/
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void check_order_preserving_embedding() const;
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/*{\Mop checks if the embedding of all vertices of |P| is
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counter-clockwise order-preserving with respect to the adjacency
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counterclockwise order-preserving with respect to the adjacency
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list ordering of all vertices.}*/
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void check_forward_prefix_condition(Vertex_const_handle v) const;
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@ -212,7 +212,7 @@ public:
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contains `p`. It outputs in the container pointed to by
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`eit` the boundary of the zone in conflict with `p`.
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The boundary edges of the conflict zone are output in
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counter-clockwise order and each edge is described through its
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counterclockwise order and each edge is described through its
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incident face which is not in conflict with `p`. The function
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returns in a std::pair the resulting output
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iterators. \pre `start` is in conflict with `p` and `p` lies in the original domain `domain`.
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@ -124,7 +124,7 @@ public:
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Comparison operator.
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Each translation \f$g\f$ of \f$\mathcal G\f$, when applied to the octagon \f$\mathcal D_O\f$,
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produces a copy of \f$\mathcal D_O\f$ labeled by the translation \f$g\f$. The copies of
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\f$\mathcal D_O\f$ incident to \f$\mathcal D_O\f$ are naturally ordered counter-clockwise around
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\f$\mathcal D_O\f$ incident to \f$\mathcal D_O\f$ are naturally ordered counterclockwise around
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\f$\mathcal D_O\f$. The comparison operator compares two translations based on the ordering
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of the copies of \f$\mathcal D_O\f$ that they produce. For more details, see Section
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\ref P4HT2_representation of the User manual.
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@ -11,7 +11,7 @@ A refinement of the concept `TriangulationFaceBase_2` that adds an interface for
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At the base level, a face stores handles to its incident vertices and to its neighboring faces.
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Compare with Section \ref Section_2D_Triangulations_Software_Design of the 2D Triangulations
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package. The vertices and neighbors are indexed counter-clockwise 0, 1, and 2. Neighbor `i` lies
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package. The vertices and neighbors are indexed counterclockwise 0, 1, and 2. Neighbor `i` lies
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opposite to vertex `i`.
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For periodic hyperbolic triangulations, the face base class needs to store three hyperbolic
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@ -54,7 +54,7 @@ Each face gives access to its three incident vertices and to its three adjacent
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to store additionally three hyperbolic translations. When applied to the three points stored in the vertices of
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the face, these translations produce the canonical representative of the face in the hyperbolic plane.
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The three vertices of a face are indexed with 0, 1, and 2 in positive (counter-clockwise) orientation. The
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The three vertices of a face are indexed with 0, 1, and 2 in positive (counterclockwise) orientation. The
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orientation of faces on the Bolza surface is defined as the orientation of their canonical representatives in
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the hyperbolic plane.
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@ -123,7 +123,7 @@ strict criteria:
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- Adjacent collinear edges touching at vertices of degree two are merged
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- The sequence of vertices representing a boundary starts from its
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lexicographically smallest vertex
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- Outer boundaries are oriented counter-clockwise and inner boundaries are
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- Outer boundaries are oriented counterclockwise and inner boundaries are
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oriented clockwise
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- The inner boundaries of a polygon with holes are stored in lexicographic
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order
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@ -21,7 +21,7 @@ reverses the action of the other.
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The join and split operations. Left to right: The vertex `v` is split
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into \f$ v_1\f$ and \f$ v_2\f$. The faces \f$ f\f$ and \f$ g\f$ are
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inserted after \f$ f_1\f$ and \f$ f_2\f$, respectively, in the
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counter-clockwise sense. The vertices \f$ v_1\f$, \f$ v_2\f$ and the
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counterclockwise sense. The vertices \f$ v_1\f$, \f$ v_2\f$ and the
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faces \f$ f\f$ and \f$ g\f$ are returned as a `boost::tuple` in that
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order. Right to left: The edge `(f,i)` is collapsed, and thus the
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vertices \f$ v_1\f$ and \f$ v_2\f$ are joined. The vertex `v` is
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@ -60,7 +60,7 @@ Vertex_handle join_vertices(Face_handle f, int i);
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/*!
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Splits the vertex `v` into two vertices `v1` and
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`v2`. The common faces `f` and `g` of `v1` and
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`v2` are created after (in the counter-clockwise sense) the
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`v2` are created after (in the counterclockwise sense) the
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faces `f1` and `f2`. The 4-tuple `(v1,v2,f,g)` is
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returned (see \cgalFigureRef{figsdgdssplitjoin}).
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*/
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@ -206,7 +206,7 @@ returns the sign of the distance of `q` from the Voronoi circle
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of `s1`, `s2`, `s3` (the Voronoi circle of three sites
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`s1`, `s2`, `s3` is a circle co-tangent to all three
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sites, that touches them in that order as we walk on its circumference
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in the counter-clockwise sense).
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in the counterclockwise sense).
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\pre the Voronoi circle of `s1`, `s2`, `s3` must exist.
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Must also provide `Sign operator()(Site_2 s1, Site_2 s2, Site_2 q)`, which returns the sign of the distance of
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@ -78,7 +78,7 @@ segment Voronoi diagram.
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/*!
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A constructor for the point \f$ v \f$ at which the three \f$ L_{\infty} \f$
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bisectors between the three given sites `s1`, `s2` and `s3` intersect
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and the regions of `s1`, `s2` and `s3` appear in the counter-clockwise order
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and the regions of `s1`, `s2` and `s3` appear in the counterclockwise order
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`s1`, `s2`, `s3` around \f$ v\f$.
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Point \f$ v \f$ is equidistant
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@ -116,7 +116,7 @@ of `s1`, `s2`, `s3`. The \f$ L_{\infty} \f$ Voronoi square
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of three sites
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`s1`, `s2`, `s3` is an axis-parallel square which is passing through all three
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sites and touches them in the `s1`, `s2`, `s3`
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order as we walk on the square in the counter-clockwise sense.
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order as we walk on the square in the counterclockwise sense.
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The center of the square is at the intersection of the three
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\f$ L_{\infty} \f$ bisectors of the three sites.
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@ -256,7 +256,7 @@ Since CGAL segments have also an orientation, we also orient
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For an <I>oriented</I> segment \f$ s \f$, we orient its
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\f$L_{\infty}\f$-perpendicular lines so that the lines'
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orientation is closest to the following orientation:
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the orientation of \f$ s \f$ rotated <I>counter-clockwise</I> by
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the orientation of \f$ s \f$ rotated <I>counterclockwise</I> by
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\f$ \pi/2 \f$.
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Let `s` be a segment and `p` a point contained in its interior.
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@ -70,7 +70,7 @@ For any given input polygon, its offset polygons at a certain distance are compo
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This method returns all such contours in an unspecified order and with no parental relationship
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between them (that is why it is called `construct_offset_contours()` and not `construct_offset_polygons()`).
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Those offset contours in the resulting sequence which are oriented counter-clockwise are outer contours
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Those offset contours in the resulting sequence which are oriented counterclockwise are outer contours
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and those oriented clockwise are holes. It is up to the user to match each hole to its parent in order
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to reconstruct the parent-hole relationship of the conceptual output. It is sufficient to test each hole
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against each parent as there won't be a hole inside a hole, a parent inside any other contour,
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@ -96,10 +96,10 @@ of the straight skeleton HDS, such border halfedges are oriented such that their
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outwards the polygon. Therefore, the opposite halfedge of any border halfedge is oriented such that
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its left side faces inward the polygon.
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This \cgal package requires the input polygon (with holes) to be weakly simple and oriented counter-clockwise.
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This \cgal package requires the input polygon (with holes) to be weakly simple and oriented counterclockwise.
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The skeleton halfedges are oriented such that their <I>left</I> side faces inward the region they bound.
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That is, the vertices (both contour and skeleton) of a face are circulated in counter-clockwise order.
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That is, the vertices (both contour and skeleton) of a face are circulated in counterclockwise order.
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There is one and only one contour halfedge incident upon any face.
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The contours of the input polygon are traced by the border halfedges of the HDS (those facing outward),
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@ -115,7 +115,7 @@ Thus, the 2 opposite halfedges of a skeleton edge link the edge to its 2 definin
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Starting from any border contour halfedge, circulating the structure walks through border counter
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halfedges and traces the vertices of the polygon's contours (in opposite order).
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Starting from any non-border but contour halfedge, circulating the structure walks counter-clockwise
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Starting from any non-border but contour halfedge, circulating the structure walks counterclockwise
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around the face corresponding to that contour halfedge. The vertices around a face always describe
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a non-convex weakly simple polygon.
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@ -169,7 +169,7 @@ whose <I>straight skeleton</I> is to be built.
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Each contour must be input in turn starting with the <I>outer contour</I> and following with the holes (if any).
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The order of the holes is unimportant but the outer contour must be entered first.
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The outer contour must be oriented counter-clockwise while holes must be oriented clockwise.
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The outer contour must be oriented counterclockwise while holes must be oriented clockwise.
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It is an error to enter more than one outer contour, to enter a hole which is not inside
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the outer contour, to enter a hole that is inside another hole, or to enter a contour which crosses
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@ -175,7 +175,7 @@ for each boundary that does not contain the defining contour edge of its inciden
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</center>
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\cgalFigureCaptionBegin{SLSArtificial}
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A polygon with four holes (black) and its straight skeleton (red).
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Three artificial bisectors (green) are added in a post-processing step to recover the simply-connected
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Three artificial bisectors (green) are added in a postprocessing step to recover the simply-connected
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property for the straight skeleton face (gray) of the rightmost vertical contour edge.
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\cgalFigureCaptionEnd
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@ -454,7 +454,7 @@ To construct the straight skeleton of a polygon the user must:
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<li>Instantiate the straight skeleton builder.</li>
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<li> Enter one contour at a time, starting from the outer contour, via the method
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`Straight_skeleton_builder_2::enter_contour()`. The input polygon must be
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weakly simple and counter-clockwise oriented (see the definitions at the beginning
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weakly simple and counterclockwise oriented (see the definitions at the beginning
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of this chapter). Collinear edges are allowed. The insertion order of each hole is
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unimportant but the outer contour must be entered first.</li>
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<li>Call `Straight_skeleton_builder_2::construct_skeleton()` once <I>all</I> the contours
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@ -480,18 +480,18 @@ The resulting sequence of offset contours can contain both outer and inner conto
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However, this algorithm returns a sequence of contours in arbitrary order and there is no
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indication whatsoever of the parental relationship between inner and outer contours.
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On the other hand, each outer contour is counter-clockwise oriented while each hole is clockwise-oriented.
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On the other hand, each outer contour is counterclockwise oriented while each hole is clockwise-oriented.
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And since offset contours do form simple polygons, it is guaranteed that no hole will be
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inside another hole, no outer contour will be inside any other contour, and each hole will be inside
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exactly 1 outer contour.
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Parental relationships are <I>not</I> automatically reconstructed by this algorithm because
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this relation is not directly given by the input polygon and must be done as a post-processing
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this relation is not directly given by the input polygon and must be done as a postprocessing
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step. The function `CGAL::arrange_offset_polygons_2()` can be used to do that efficiently.
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A user can reconstruct the parental relationships as a post processing operation by testing
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each inner contour (which is identified by being clockwise) against each outer contour (identified
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as being counter-clockwise) for insideness.
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as being counterclockwise) for insideness.
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This algorithm requires exact predicates but not exact constructions.
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Therefore, the `Exact_predicates_inexact_constructions_kernel` should be used.
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@ -106,7 +106,7 @@ which offers combinatorial repairing while reading bad inputs.
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\section IOStreamOBJ Wavefront Advanced Visualizer Object Format (OBJ)
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The `OBJ` file format, using the file extension `.obj`, is a simple \ascii data format that represents 3D geometry.
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Vertices are stored in a counter-clockwise order by default, making explicit declaration of face normals unnecessary.
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Vertices are stored in a counterclockwise order by default, making explicit declaration of face normals unnecessary.
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A precise specification of the format is available <a href="https://www.martinreddy.net/gfx/3d/OBJ.spec">here</a>.
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@ -134,7 +134,7 @@ Vertex_handle join_vertices(Edges_circulator ec);
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/*!
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splits the vertex `v` into two vertices `v1` and `v2`.
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The common faces `f` and `g` of `v1` and `v2` are created after (in the counter-clockwise sense) the
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The common faces `f` and `g` of `v1` and `v2` are created after (in the counterclockwise sense) the
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faces `f1` and `f2`. The 4-tuple `(v1,v2,f,g)` is returned (see Fig. \ref figtdssplitjoin).
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\pre `dimension()` must be equal to `2`, `f1` and `f2` must be different from `Face_handle()` and `v` must be a vertex of both `f1` and `f2`.
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@ -305,7 +305,7 @@ outputs the boundary edges of the conflict zone of point `p` into an output iter
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This functions outputs in the container pointed to by `eit`,
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the boundary of the zone in conflict with `p`. The boundary edges
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of the conflict zone are output in counter-clockwise order
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of the conflict zone are output in counterclockwise order
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and each edge is described through the incident face
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which is not in conflict with `p`.
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The function returns the resulting output iterator.
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@ -221,7 +221,7 @@ i. e., the faces whose circumcircle contains `p`.
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It outputs in the container pointed to by `eit` the
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the boundary of the zone in conflict with `p`.
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The boundary edges
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of the conflict zone are output in counter-clockwise order
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of the conflict zone are output in counterclockwise order
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and each edge is described through its incident face
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which is not in conflict with `p`.
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The function returns in a `std::pair` the resulting output iterators.
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@ -295,7 +295,7 @@ have negative power wrt. `p`. It outputs in the container
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pointed to by `eit` the boundary of the zone in conflict
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with `p`. It inserts the vertices that would be hidden by `p`
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into the container pointed to by `vit`. The boundary edges of
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the conflict zone are output in counter-clockwise order and each edge
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the conflict zone are output in counterclockwise order and each edge
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is described through its incident face which is not in conflict with
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`p`. The function returns in a `CGAL::Triple` the resulting output
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iterators.
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@ -60,9 +60,9 @@ In order to perform fast and exact computations with a fundamental domain, every
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\subsection Subsection_Hyperbolic_Surface_Triangulations_DS_Domains Data Structure for Domains
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We represent every fundamental domain as a polygon in the Poincaré disk, given by the list of its vertices in counter-clockwise order and by the list of its side pairings.
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We represent every fundamental domain as a polygon in the Poincaré disk, given by the list of its vertices in counterclockwise order and by the list of its side pairings.
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This package can generate a random convex fundamental domain \f$ P \f$ of a surface of genus two, with eight vertices \f$ z_0, \dots, z_7 \in \mathbb{C} \f$.
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The vertices and the sides are in counter-clockwise order, the side between \f$ z_0 \f$ and \f$ z_1 \f$ is \f$ A \f$, the side between \f$ z_4 \f$ and \f$ z_5 \f$ is \f$ \overline{A} \f$ and so on as on \cgalFigureRef{THS2-octagon}. The side pairings are \f$ A \f$ with \f$\overline{A} \f$ , \f$ B \f$ with \f$ \overline{B} \f$ , \f$ C \f$ with \f$ \overline{C} \f$ and \f$ D \f$ with \f$ \overline{D} \f$.
|
||||
The vertices and the sides are in counterclockwise order, the side between \f$ z_0 \f$ and \f$ z_1 \f$ is \f$ A \f$, the side between \f$ z_4 \f$ and \f$ z_5 \f$ is \f$ \overline{A} \f$ and so on as on \cgalFigureRef{THS2-octagon}. The side pairings are \f$ A \f$ with \f$\overline{A} \f$ , \f$ B \f$ with \f$ \overline{B} \f$ , \f$ C \f$ with \f$ \overline{C} \f$ and \f$ D \f$ with \f$ \overline{D} \f$.
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These octagons are symmetric, i.e. \f$ z_i = -z_{i+4} \f$ for every \f$ i \f$, where indices are modulo eight.
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Such octagons are described in \cgalCite{aigon2005hyperbolic}.
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@ -35,7 +35,7 @@ namespace CGAL {
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Represents a geodesic triangulation of a closed orientable hyperbolic surface.
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The triangulation is stored as combinatorial map decorated with one cross-ratio per edge.
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It is also possible to specify an anchor for the triangulation. An anchor consists in 1) a dart of the combinatorial map, belonging by definition to a vertex V and a triangle T, together with
|
||||
2) three points A,B,C in the hyperbolic plane. The points A,B,C are the three vertices in counter-clockwise order of a triangle. This triangle is a lift
|
||||
2) three points A,B,C in the hyperbolic plane. The points A,B,C are the three vertices in counterclockwise order of a triangle. This triangle is a lift
|
||||
of T, and A is a lift of V.
|
||||
*/
|
||||
template<class Traits>
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||||
|
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@ -157,7 +157,7 @@ public:
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|||
This function outputs in the container pointed to by `fit` the faces which are in conflict with point `p`,
|
||||
i. e., the faces whose circumcircle contains `p`. It outputs in the container pointed to by `eit`
|
||||
the boundary of the zone in conflict with `p`. The boundary edges of the conflict zone
|
||||
are output in counter-clockwise order and each edge is described through its incident face
|
||||
are output in counterclockwise order and each edge is described through its incident face
|
||||
which is not in conflict with `p`. The function returns in a `std::pair` the resulting output iterators.
|
||||
|
||||
\tparam OutItFaces is an output iterator with `Face_handle` as value type.
|
||||
|
|
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|
|
@ -806,7 +806,7 @@ private:
|
|||
}
|
||||
|
||||
//for face query: traverse the face to get all edges
|
||||
//and sort vertices in counter-clockwise order.
|
||||
//and sort vertices in counterclockwise order.
|
||||
void input_face (Face_const_handle fh) const
|
||||
{
|
||||
Ccb_halfedge_const_circulator curr = fh->outer_ccb();
|
||||
|
|
@ -840,7 +840,7 @@ private:
|
|||
}
|
||||
}
|
||||
//for vertex or edge query: traverse the face to get all edges
|
||||
//and sort vertices in counter-clockwise order.
|
||||
//and sort vertices in counterclockwise order.
|
||||
void input_face (Face_const_handle fh,
|
||||
EHs& good_edges,
|
||||
Arrangement_2& bbox) const
|
||||
|
|
|
|||
Loading…
Reference in New Issue