mirror of https://github.com/CGAL/cgal
Remove useless \ccTexHtml around math mode $$.
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@ -462,7 +462,7 @@ begins at a vertex of the face which
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is given
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as an optional argument or at an arbitrary vertex of the triangulation
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if no optional argument is given. It takes
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time \ccTexHtml{$O(n)$}{O(n)} in the worst case, but only \ccTexHtml{$O(\sqrt{n})$}{O(sqrt(n))}
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time $O(n)$ in the worst case, but only $O(\sqrt{n})$
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on average if the vertices are distributed uniformly at random.
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The class \ccc{Triangulation_hierarchy_2<Traits,Tds>},
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described in section~\ref{Section_2D_Triangulations_Hierarchy},
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@ -473,14 +473,14 @@ Insertion of a point is done by locating a face that contains the
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point, and splitting this face into three new faces.
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If the point falls outside the convex hull, the triangulation
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is restored by flips. Apart from the location, insertion takes a
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time \ccTexHtml{$O(1)$}{O(1)}. This bound is only an amortized bound
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time $O(1)$. This bound is only an amortized bound
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for points located outside the convex hull.
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Removal of a vertex is done by removing all adjacent triangles, and
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re-triangulating the hole. Removal takes a time at most proportional to
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\ccTexHtml{$d^2$}{d^2}, where
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\ccTexHtml{$d$}{d} is the degree of the removed vertex,
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which is \ccTexHtml{$O(1)$}{O(1)} for a random vertex.
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$d^2$, where
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$d$ is the degree of the removed vertex,
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which is $O(1)$ for a random vertex.
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The face, edge, and vertex iterators on finite features
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are derived from their counterparts visiting all (finite and infinite)
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@ -624,23 +624,23 @@ The insertion of a new point in the Delaunay triangulation
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is performed using first the insertion member function
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of the basic triangulation and second
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performing a sequence of flips to restore the Delaunay property.
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The number of flips that have to be performed is \ccTexHtml{$O(d)$}{O(d)}
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if the new vertex has degree \ccTexHtml{$d$}{d} in the updated
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The number of flips that have to be performed is $O(d)$
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if the new vertex has degree $d$ in the updated
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Delaunay triangulation. For
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points distributed uniformly at random,
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each insertion takes time \ccTexHtml{$O(1)$}{O(1)} on
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each insertion takes time $O(1)$ on
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average, once the point has been located in the triangulation.
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Removal calls the removal in the triangulation and then re-triangulates
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the hole created in such a way that the Delaunay criterion is
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satisfied. Removal of a
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vertex of degree \ccTexHtml{$d$}{d} takes time \ccTexHtml{$O(d^2)$}{O(d^2)}.
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The degree $d$ is \ccTexHtml{$O(1)$}{O(1)} for a random
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vertex of degree $d$ takes time $O(d^2)$.
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The degree $d$ is $O(1)$ for a random
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vertex in the triangulation.
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After having performed a point location, the
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nearest neighbor of a point is found in time \ccTexHtml{$O(n)$}{O(n)} in the
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worst case, but in time \ccTexHtml{$O(1)$}{O(1)}
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nearest neighbor of a point is found in time $O(n)$ in the
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worst case, but in time $O(1)$
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for vertices distributed uniformly at random and any query point.
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@ -265,20 +265,20 @@ and additionally tests the Delaunay property. This method is
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\ccHeading{Implementation}
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Insertion is implemented by inserting in the triangulation, then
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performing a sequence of Delaunay flips. The number of flips is \ccTexHtml{$O(d)$}{O(d)}
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if the new vertex is of degree \ccTexHtml{$d$}{d} in the new triangulation. For
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points distributed uniformly at random, insertion takes time \ccTexHtml{$O(1)$}{O(1)} on
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performing a sequence of Delaunay flips. The number of flips is $O(d)$
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if the new vertex is of degree $d$ in the new triangulation. For
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points distributed uniformly at random, insertion takes time $O(1)$ on
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average.
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Removal calls the removal in the triangulation and then re-triangulates
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the hole in such a way that the Delaunay criterion is satisfied. Removal of a
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vertex of degree \ccTexHtml{$d$}{d} takes time \ccTexHtml{$O(d^2)$}{O(d^2)}.
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The degree \ccTexHtml{$d$}{d} is \ccTexHtml{$O(1)$}{O(1)} for a random
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vertex of degree $d$ takes time $O(d^2)$.
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The degree $d$ is $O(1)$ for a random
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vertex in the triangulation.
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After a point location step, the nearest neighbor
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is found in time \ccTexHtml{$O(n)$}{O(n)} in the
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worst case, but in time \ccTexHtml{$O(1)$}{O(1)}
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is found in time $O(n)$ in the
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worst case, but in time $O(1)$
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for vertices distributed uniformly at random and any query point.
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@ -976,20 +976,20 @@ See the \ccc{Qt_widget} class.
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Locate is implemented by a line walk from a vertex of the face given
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as optional parameter (or from a finite vertex of
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\ccStyle{infinite_face()} if no optional parameter is given). It takes
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time \ccTexHtml{$O(n)$}{O(n)} in the worst case, but only \ccTexHtml{$O(\sqrt{n})$}{O(sqrt(n))}
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time $O(n)$ in the worst case, but only $O(\sqrt{n})$
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on average if the vertices are distributed uniformly at random.
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Insertion of a point is done by locating a face that contains the
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point, and then splitting this face.
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If the point falls outside the convex hull, the triangulation
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is restored by flips. Apart from the location, insertion takes a time
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time \ccTexHtml{$O(1)$}{O(1)}. This bound is only an amortized bound
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time $O(1)$. This bound is only an amortized bound
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for points located outside the convex hull.
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Removal of a vertex is done by removing all adjacent triangles, and
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re-triangulating the hole. Removal takes time \ccTexHtml{$O(d^2)$}{O(d^2)} in the worst
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case, if \ccTexHtml{$d$}{d} is the degree of the removed vertex,
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which is \ccTexHtml{$O(1)$}{O(1)} for a random vertex.
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re-triangulating the hole. Removal takes time $O(d^2)$ in the worst
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case, if $d$ is the degree of the removed vertex,
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which is $O(1)$ for a random vertex.
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The face, edge, and vertex iterators on finite features
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are derived from their counterparts visiting all (finite and infinite)
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