Remove useless \ccTexHtml around math mode $$.

This commit is contained in:
Sylvain Pion 2009-01-03 11:38:41 +00:00
parent 993add6e80
commit 49cc30ead7
3 changed files with 24 additions and 24 deletions

View File

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

View File

@ -976,20 +976,20 @@ See the \ccc{Qt_widget} class.
Locate is implemented by a line walk from a vertex of the face given
as optional parameter (or from a finite vertex of
\ccStyle{infinite_face()} if no optional parameter is given). It takes
time \ccTexHtml{$O(n)$}{O(n)} in the worst case, but only \ccTexHtml{$O(\sqrt{n})$}{O(sqrt(n))}
time $O(n)$ in the worst case, but only $O(\sqrt{n})$
on average if the vertices are distributed uniformly at random.
Insertion of a point is done by locating a face that contains the
point, and then splitting this face.
If the point falls outside the convex hull, the triangulation
is restored by flips. Apart from the location, insertion takes a time
time \ccTexHtml{$O(1)$}{O(1)}. This bound is only an amortized bound
time $O(1)$. This bound is only an amortized bound
for points located outside the convex hull.
Removal of a vertex is done by removing all adjacent triangles, and
re-triangulating the hole. Removal takes time \ccTexHtml{$O(d^2)$}{O(d^2)} in the worst
case, if \ccTexHtml{$d$}{d} is the degree of the removed vertex,
which is \ccTexHtml{$O(1)$}{O(1)} for a random vertex.
re-triangulating the hole. Removal takes time $O(d^2)$ in the worst
case, if $d$ is the degree of the removed vertex,
which is $O(1)$ for a random vertex.
The face, edge, and vertex iterators on finite features
are derived from their counterparts visiting all (finite and infinite)