Abbreviations trigger end of \brief description

Fixes bug #15482:
https://gforge.inria.fr/tracker/index.php?func=detail&aid=15482&group_id=52&atid=13845
This commit is contained in:
Alexandros Konstantinakis-Karmis 2013-03-13 11:58:56 +01:00
parent e991fe72ad
commit 3b1d281efb
91 changed files with 158 additions and 159 deletions

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@ -13,8 +13,7 @@ i.e.\ commutative ring with 0, 1, +, * and unity free of zero divisors.
It refines `Assignable`, `CopyConstructible`, `DefaultConstructible`
and `FromIntConstructible`.
It refines `EqualityComparable`, where equality is defined w.r.t.
the ring element being represented.
It refines `EqualityComparable`, where equality is defined w.r.t.\ the ring element being represented.
The operators unary and binary plus +, unary and binary minus -,
multiplication * and their compound forms +=, -=, *= are required and

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@ -124,7 +124,7 @@ and it is SINGULAR if it is a boundary simplex that is not included in a \f$ k+1
In REGULARIZED mode, for \f$ k=(0,1,2)\f$
each k-dimensional simplex of the triangulation
can be classified as EXTERIOR, REGULAR or INTERIOR, i.e.
can be classified as EXTERIOR, REGULAR or INTERIOR, i.e.\
there is no singular faces.
A \f$ k\f$ simplex is REGULAR if it is on the boundary of alpha complex
and belongs to a tetrahedral cell of the complex.

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@ -82,7 +82,7 @@ returns the isolated vertex-information record.
Isolated_vertex* isolated_vertex();
/*!
returns whether the vertex is not associated with a valid point (i.e. it
returns whether the vertex is not associated with a valid point (i.e.\ it
lies at infinity).
*/
bool has_null_point () const;

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@ -53,7 +53,7 @@ full-dimensional.)
If \f$ P\f$ is not full-dimensional, linear algebra techniques should be
used to determine an affine subspace \f$ S\f$ of \f$ \E^d\f$ that contains the
points \f$ P\f$ as a (w.r.t. \f$ S\f$) full-dimensional pointset; once \f$ S\f$ is
points \f$ P\f$ as a (w.r.t.\ \f$ S\f$) full-dimensional pointset; once \f$ S\f$ is
determined, the algorithm can be invoked again to compute an
approximation to (the lower-dimensional) \f$ \mel(P)\f$ in \f$ S\f$. Since
`is_full_dimensional()` might (due to rounding errors, see
@ -315,7 +315,7 @@ returns an iterator pointing to the first of the \f$ d\f$ Cartesian
coordinates of the computed ellipsoid's center.
The returned point is a floating-point approximation to the
ellipsoid's exact center; no guarantee is given w.r.t. the involved
ellipsoid's exact center; no guarantee is given w.r.t.\ the involved
relative error. \pre `ame.is_full_dimensional() == true`.
*/
Center_coordinate_iterator center_cartesian_begin();
@ -334,7 +334,7 @@ returns an iterator pointing to the first of the \f$ d\f$ descendantly
sorted lengths of the computed ellipsoid's axes. The \f$ d\f$ returned
numbers are floating-point approximations to the exact
axes-lengths of the computed ellipsoid; no guarantee is given
w.r.t. the involved relative error. (See also method
w.r.t.\ the involved relative error. (See also method
`axes_direction_cartesian_begin()`.) \pre `ame.is_full_dimensional() == true`, and \f$ d\in\{2,3\}\f$.
*/
Axes_lengths_iterator axes_lengths_begin();
@ -353,7 +353,7 @@ computed ellipsoid's \f$ i\f$th axis direction (i.e., unit vector in
direction of the ellipsoid's \f$ i\f$th axis). The direction described
by this iterator is a floating-point approximation to the exact
axis direction of the computed ellipsoid; no guarantee is given
w.r.t. the involved relative error. An approximation to the
w.r.t.\ the involved relative error. An approximation to the
length of axis \f$ i\f$ is given by the \f$ i\f$th entry of
`axes_lengths_begin()`.
\pre `ame.is_full_dimensional() == true`, and \f$ d\in\{2,3\}\f$, and \f$ 0\leq i < d\f$.

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@ -10,7 +10,7 @@ finite set of points in \f$ d\f$-dimensional Euclidean space \f$ \E^d\f$, where
difference \f$ R^2-r^2\f$ is minimal. For a point set \f$ P\f$ we denote by \f$ ma(P)\f$
the smallest annulus that contains all points of \f$ P\f$. Note that \f$ ma(P)\f$
can be degenerate,
i.e. \f$ ma(P)=\emptyset\f$ if
i.e.\ \f$ ma(P)=\emptyset\f$ if
\f$ P=\emptyset\f$ and \f$ ma(P)=\{p\}\f$ if
\f$ P=\{p\}\f$.
@ -159,13 +159,13 @@ int ambient_dimension( ) const;
/*!
returns the number of points of `min_annulus`, i.e. \f$ |P|\f$.
returns the number of points of `min_annulus`, i.e.\ \f$ |P|\f$.
*/
int number_of_points( ) const;
/*!
returns the number of support points of `min_annulus`, i.e. \f$ |S|\f$.
returns the number of support points of `min_annulus`, i.e.\ \f$ |S|\f$.
*/
int number_of_support_points( ) const;
@ -270,7 +270,7 @@ of the center of `min_annulus`.
\note
The coordinates have a rational
representation, i.e. the first \f$ d\f$ elements of the iterator
representation, i.e.\ the first \f$ d\f$ elements of the iterator
range are the numerators and the \f$ (d\!+\!1)\f$-st element is the
common denominator.
*/
@ -308,7 +308,7 @@ ET squared_radii_denominator( ) const;
/// The bounded area of the smallest enclosing annulus lies between
/// the inner and the outer sphere. The boundary is the union of both
/// spheres. By definition, an empty annulus has no boundary and no
/// bounded side, i.e. its unbounded side equals the whole space \f$
/// bounded side, i.e.\ its unbounded side equals the whole space \f$
/// \E^d\f$.
/// @{
@ -353,7 +353,7 @@ bool is_empty( ) const;
/*!
returns `true`, iff `min_annulus` is degenerate, i.e. if `min_annulus` is empty or equal to a single point.
returns `true`, iff `min_annulus` is degenerate, i.e.\ if `min_annulus` is empty or equal to a single point.
*/
bool is_degenerate( ) const;
@ -403,7 +403,7 @@ InputIterator last );
/// An object `min_annulus` is valid, iff <UL> <LI>`min_annulus`
/// contains all points of its defining set \f$ P\f$,
/// <LI>`min_annulus` is the smallest annulus containing its support
/// set \f$ S\f$, and <LI>\f$ S\f$ is minimal, i.e. no support point
/// set \f$ S\f$, and <LI>\f$ S\f$ is minimal, i.e.\ no support point
/// is redundant. </UL> <I>Note:</I> In this release only the first
/// item is considered by the validity check.
/// @{

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@ -9,7 +9,7 @@ enclosing a finite (multi)set of points in two-dimensional Euclidean
space \f$ \E^2\f$. For a point set \f$ P\f$ we denote by \f$ mc(P)\f$ the smallest circle
that contains all points of \f$ P\f$. Note that \f$ mc(P)\f$ can be
degenerate,
i.e. \f$ mc(P)=\emptyset\f$ if
i.e.\ \f$ mc(P)=\emptyset\f$ if
\f$ P=\emptyset\f$ and \f$ mc(P)=\{p\}\f$ if
\f$ P=\{p\}\f$.
@ -174,13 +174,13 @@ const Traits& traits = Traits());
/*!
returns the number of points of `min_circle`, i.e. \f$ |P|\f$.
returns the number of points of `min_circle`, i.e.\ \f$ |P|\f$.
*/
int number_of_points( ) const;
/*!
returns the number of support points of `min_circle`, i.e. \f$ |S|\f$.
returns the number of support points of `min_circle`, i.e.\ \f$ |S|\f$.
*/
int number_of_support_points( ) const;
@ -228,7 +228,7 @@ const Circle& circle( ) const;
/// \name Predicates
/// By definition, an empty `Min_circle_2` has no boundary and no
/// bounded side, i.e. its unbounded side equals the whole space \f$
/// bounded side, i.e.\ its unbounded side equals the whole space \f$
/// \E^2\f$.
/// @{
@ -271,7 +271,7 @@ bool is_empty( ) const;
/*!
returns `true`, iff `min_circle` is degenerate,
i.e. if `min_circle` is empty or equal to a single point, equivalently
i.e.\ if `min_circle` is empty or equal to a single point, equivalently
if the number of support points is less than 2.
*/
bool is_degenerate( ) const;
@ -318,7 +318,7 @@ void clear( );
/// An object `min_circle` is valid, iff <UL> <LI>`min_circle`
/// contains all points of its defining set \f$ P\f$, <LI>`min_circle`
/// is the smallest circle spanned by its support set \f$ S\f$, and
/// <LI>\f$ S\f$ is minimal, i.e. no support point is redundant. </UL>
/// <LI>\f$ S\f$ is minimal, i.e.\ no support point is redundant. </UL>
/// @{
/*!

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@ -9,7 +9,7 @@ enclosing a finite (multi)set of points in two-dimensional euclidean
space \f$ \E^2\f$. For a point set \f$ P\f$ we denote by \f$ me(P)\f$ the smallest
ellipse that contains all points of \f$ P\f$. Note that \f$ me(P)\f$ can be
degenerate,
i.e. \f$ me(P)=\emptyset\f$ if
i.e.\ \f$ me(P)=\emptyset\f$ if
\f$ P=\emptyset\f$, \f$ me(P)=\{p\}\f$ if \f$ P=\{p\}\f$,
and <span class="mbox">\f$ me(P) = \{ (1-\lambda)p + \lambda q \mid 0 \leq \lambda \leq 1 \}\f$</span> if \f$ P=\{p,q\}\f$.
@ -186,13 +186,13 @@ const Traits& traits = Traits());
/*!
returns the number of points of `min_ellipse`, i.e. \f$ |P|\f$.
returns the number of points of `min_ellipse`, i.e.\ \f$ |P|\f$.
*/
int number_of_points( ) const;
/*!
returns the number of support points of `min_ellipse`, i.e. \f$ |S|\f$.
returns the number of support points of `min_ellipse`, i.e.\ \f$ |S|\f$.
*/
int number_of_support_points( ) const;
@ -240,7 +240,7 @@ const Ellipse& ellipse( ) const;
/// \name Predicates
/// By definition, an empty `Min_ellipse_2` has no boundary and no
/// bounded side, i.e. its unbounded side equals the whole space \f$
/// bounded side, i.e.\ its unbounded side equals the whole space \f$
/// \E^2\f$.
/// @{
@ -283,7 +283,7 @@ bool is_empty( ) const;
/*!
returns `true`, iff `min_ellipse` is degenerate,
i.e. if `min_ellipse` is empty, equal to a single point or equal to a
i.e.\ if `min_ellipse` is empty, equal to a single point or equal to a
segment, equivalently if the number of support points is less
than 3.
*/
@ -331,7 +331,7 @@ void clear( );
/// An object `min_ellipse` is valid, iff <UL> <LI>`min_ellipse`
/// contains all points of its defining set \f$ P\f$,
/// <LI>`min_ellipse` is the smallest ellipse spanned by its support
/// set \f$ S\f$, and <LI>\f$ S\f$ is minimal, i.e. no support point
/// set \f$ S\f$, and <LI>\f$ S\f$ is minimal, i.e.\ no support point
/// is redundant. </UL> <I>Note:</I> In this release only the first
/// item is considered by the validity check.
/// @{

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@ -8,7 +8,7 @@ An object of the class `Min_sphere_d` is the unique sphere of
smallest volume enclosing a finite (multi)set of points in \f$ d\f$-dimensional
Euclidean space \f$ \E^d\f$. For a set \f$ P\f$ we denote by \f$ ms(P)\f$ the
smallest sphere that contains all points of \f$ P\f$. \f$ ms(P)\f$ can
be degenerate, i.e. \f$ ms(P)=\emptyset\f$
be degenerate, i.e.\ \f$ ms(P)=\emptyset\f$
if \f$ P=\emptyset\f$ and \f$ ms(P)=\{p\}\f$ if
\f$ P=\{p\}\f$.
@ -135,13 +135,13 @@ const Traits& traits = Traits());
/*!
returns the number of points of `min_sphere`, i.e. \f$ |P|\f$.
returns the number of points of `min_sphere`, i.e.\ \f$ |P|\f$.
*/
int number_of_points( ) const;
/*!
returns the number of support points of `min_sphere`, i.e. \f$ |S|\f$.
returns the number of support points of `min_sphere`, i.e.\ \f$ |S|\f$.
*/
int number_of_support_points( ) const;
@ -193,7 +193,7 @@ FT squared_radius( ) const;
/// \name Predicates
/// By definition, an empty `Min_sphere_d` has no boundary and no
/// bounded side, i.e. its unbounded side equals the whole space \f$
/// bounded side, i.e.\ its unbounded side equals the whole space \f$
/// \E^d\f$.
/// @{
@ -238,7 +238,7 @@ bool is_empty( ) const;
/*!
returns `true`, iff `min_sphere` is degenerate, i.e. if
returns `true`, iff `min_sphere` is degenerate, i.e.\ if
`min_sphere` is empty or equal to a single point, equivalently if
the number of support points is less than 2.
*/
@ -294,7 +294,7 @@ InputIterator last );
/// An object `min_sphere` is valid, iff <UL> <LI>`min_sphere`
/// contains all points of its defining set \f$ P\f$, <LI>`min_sphere`
/// is the smallest sphere containing its support set \f$ S\f$, and
/// <LI>\f$ S\f$ is minimal, i.e. no support point is redundant. </UL>
/// <LI>\f$ S\f$ is minimal, i.e.\ no support point is redundant. </UL>
///
/// \note Under inexact arithmetic, the result of the
/// validation is not realiable, because the checker itself can suffer

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@ -37,7 +37,7 @@ public:
/// @{
/*!
is the constant 2, i.e. the dimension of \f$ \R^2\f$.
is the constant 2, i.e.\ the dimension of \f$ \R^2\f$.
*/
typedef Hidden_type D;

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@ -37,7 +37,7 @@ public:
/// @{
/*!
is the constant 3, i.e. the dimension of \f$ \R^3\f$.
is the constant 3, i.e.\ the dimension of \f$ \R^3\f$.
*/
typedef Hidden_type D;

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@ -246,7 +246,7 @@ const;
/*!
returns `true`, iff
`minsphere` is empty, i.e. iff \f$ ms(S)=\emptyset\f$.
`minsphere` is empty, i.e.\ iff \f$ ms(S)=\emptyset\f$.
*/
bool is_empty( ) const;

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@ -40,7 +40,7 @@ public:
/// @{
/*!
is the constant 2, i.e. the dimension of \f$ \R^2\f$.
is the constant 2, i.e.\ the dimension of \f$ \R^2\f$.
*/
typedef Hidden_type D;

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@ -33,7 +33,7 @@ public:
/// @{
/*!
is the constant 3, i.e. the dimension of \f$ \R^3\f$.
is the constant 3, i.e.\ the dimension of \f$ \R^3\f$.
*/
typedef Hidden_type D;

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@ -5,7 +5,7 @@
An object of the class `Circle` is a circle in two-dimensional
Euclidean plane \f$ \E^2\f$. Its boundary splits the plane into a bounded
and an unbounded side. By definition, an empty `#1` has no
boundary and no bounded side, i.e. its unbounded side equals the
boundary and no bounded side, i.e.\ its unbounded side equals the
whole plane \f$ \E^2\f$. A `#1` containing exactly one point \f$ p\f$
has no bounded side, its boundary is \f$ \{p\}\f$, and its unbounded side
equals \f$ \E^2 \setminus \{p\}\f$.
@ -110,7 +110,7 @@ bool is_empty( ) const;
/*!
returns `true`, iff `circle` is degenerate, i.e. if `circle` is empty
returns `true`, iff `circle` is degenerate, i.e.\ if `circle` is empty
or equal to a single point.
\note Only needed, if the corresponding predicate of `Min_circle_2` is used.

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@ -5,7 +5,7 @@
An object `ellipse` of the class `Ellipse` is an ellipse in two-dimensional
Euclidean plane \f$ \E^2\f$. Its boundary splits the plane into a bounded
and an unbounded side. By definition, an empty `ellipse` has no
boundary and no bounded side, i.e. its unbounded side equals the
boundary and no bounded side, i.e.\ its unbounded side equals the
whole plane \f$ \E^2\f$.
*/
class Ellipse {

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@ -131,7 +131,7 @@ bool is_y_monotone();
/*!
Test for equality. Two arcs are equal, iff their non-oriented
supporting circles are equal (i.e. they have same center and same
supporting circles are equal (i.e.\ they have same center and same
squared radius) and their endpoints are equal.
\relates Circular_arc_2
*/

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@ -65,7 +65,7 @@ Test for nonequality.
bool operator!=(const Circular_arc_point_2<CircularKernel> &p, const Circular_arc_point_2<CircularKernel> &q);
/*!
Returns true iff`p` is lexicographically smaller than `q`, i.e. either if `p.x() < q.x()`
Returns true iff`p` is lexicographically smaller than `q`, i.e.\ either if `p.x() < q.x()`
or if `p.x() == q.x()` and `p.y() < q.y()`.
\relates Circular_arc_point_2
*/

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@ -96,7 +96,7 @@ bool is_vertical();
/*!
Test for equality. Two arcs are equal, iff their non-oriented
supporting lines are equal (i.e. they contain the same set of
supporting lines are equal (i.e.\ they contain the same set of
points) and their endpoints are equal.
\relates Line_arc_2
*/

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@ -71,7 +71,7 @@ Test for nonequality.
bool operator!=(const Circular_arc_point_3<SphericalKernel> &p, const Circular_arc_point_3<SphericalKernel> &q);
/*!
Returns true iff `p` is lexicographically smaller than `q`, i.e. either
Returns true iff `p` is lexicographically smaller than `q`, i.e.\ either
if `p.x() < q.x()` or if `p.x() == q.x()` and `p.y() < q.y()`
or if `p.x() == q.x()` and `p.y() == q.y()` and `p.z() < q.z()`.
\relates Circular_arc_point_3

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@ -81,7 +81,7 @@ bool is_vertical();
/*!
Test for equality. Two segments are equal, iff their non-oriented
supporting lines are equal (i.e. they define the same set of
supporting lines are equal (i.e.\ they define the same set of
points), and their endpoints are the same.
*/
bool operator==(const Line_arc_3<SphericalKernel> &s1, const Line_arc_3<SphericalKernel> &s2);

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@ -10,7 +10,7 @@ seen from the side of the plane of the circle pointed by its <I>positive</I> nor
vectors.
In this definition, we say that a normal vector \f$ (a,b,c)\f$ is <I>positive</I> if
\f$ (a,b,c)>(0,0,0)\f$ (i.e. \f$ (a>0) || (a==0) \&\& (b>0) || (a==0)\&\&(b==0)\&\&(c>0)\f$).
\f$ (a,b,c)>(0,0,0)\f$ (i.e.\ \f$ (a>0) || (a==0) \&\& (b>0) || (a==0)\&\&(b==0)\&\&(c>0)\f$).
*/

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@ -41,7 +41,7 @@ const SphericalKernel::Circular_arc_3 &a1);
/*!
For two segments. Two segments are equal, iff their non-oriented
supporting lines are equal (i.e. they define the same set of
supporting lines are equal (i.e.\ they define the same set of
points), and their endpoints are the same.
*/
bool operator()

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@ -16,7 +16,7 @@ public:
/// @{
/*!
Tests whether the arc `a` is \f$ \theta\f$-monotone, i.e. the intersection of
Tests whether the arc `a` is \f$ \theta\f$-monotone, i.e.\ the intersection of
any meridian anchored at the poles of the context sphere used by the function `SphericalKernel::is_theta_monotone_3_object`
and the arc `a` is reduced to at most one point in general, and two points if a pole of that sphere is
an endpoint of `a`. Note that a bipolar circle has no such arcs.

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@ -324,7 +324,7 @@ equal to the set of all the darts of the combinatorial map.
A last important property of cells is that for all dimensions <I>i</I>
the set of <I>i</I>-cells forms a partition of the set of darts
<I>D</I>, i.e. for any <I>i</I>, the union of the sets of darts of all
<I>D</I>, i.e.\ for any <I>i</I>, the union of the sets of darts of all
the <I>i</I>-cells is equal to <I>D</I>, and the sets of darts of two
different <I>i</I>-cells are disjoint.
@ -722,9 +722,9 @@ Similarly, the functor `OnSplit` is called
when one attribute is split in two, because its corresponding cell is
split in two during some operation, unless it is equal to
`Null_functor`. In any high level operation, `OnMerge` is called
before to start the operation (i.e. before modifying the combinatorial
before to start the operation (i.e.\ before modifying the combinatorial
map), and `OnSplit` is called when the operation is finished
(i.e. after all the modifications were made).
(i.e.\ after all the modifications were made).
What we said for the dart also holds for the cell attribute. The
combinatorial map can be used with any user defined model of the

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@ -45,7 +45,7 @@ returning a Boolean value, but also primitives returning a value of
some enumeration type, e.g. `CGAL::Sign`. So the value computed
by a predicate does not involve any numerical data. Basic constructions
construct new primitive geometric objects that may involve newly computed
numerical data, i.e. that is not part of the input to the constructions.
numerical data, i.e.\ that is not part of the input to the constructions.
An example of such a basic constructions is computing the midpoint of
the straight line segment between two given points.
A special kind of constructions is selections. For selections, all the data

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@ -440,11 +440,11 @@ highly specialized libraries and software for non-geometric issues,
for instance, for numeric solvers, or visualization. We first list software
that is essential to build (all) libraries of \cgal, that is,
this software must be found during the configuration of \cgal for an
actived library of \cgal (i.e. <TT>WITH_<library>=ON</TT>);
actived library of \cgal (i.e.\ <TT>WITH_<library>=ON</TT>);
see \ref sec3partysoftwareconfig to specify the location of 3rd
party software.
The libraries \stl (shipped with any compiler) and \sc{Boost} are essential to all components (i.e. libCGAL,
The libraries \stl (shipped with any compiler) and \sc{Boost} are essential to all components (i.e.\ libCGAL,
libCGAL_Core, libCGAL_ImageIO, libCGAL_Qt3 and libCGAL_Qt4).
\subsection thirdpartystl Standard Template Library (STL)
@ -1074,7 +1074,7 @@ If the parameter is not given, the script creates <B>one executable for each giv
source file</B>.
<DT><B>`-c com1:com2:...`</B><DD> Lists components ("com1",
"com2") of \cgal to which the executable(s) should be linked. Valid components are \cgal's
libraries (i.e. "Core", "ImageIO", "Qt3" and "Qt4"; note
libraries (i.e.\ "Core", "ImageIO", "Qt3" and "Qt4"; note
that it only make sense to either pick "Qt3" or "Qt4") and all
preconfigured 3rd party software, such as "MPFI", "RS3"). An example is `-c Core:GMP:RS3:MPFI`

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@ -7,7 +7,7 @@ It can be used as the random number generating function object in the
\stl algorithm `std::random_shuffle`.
Instances of `Random` can be seen as input streams. Different
streams are <I>independent</I> of each other, i.e. the sequence of
streams are <I>independent</I> of each other, i.e.\ the sequence of
numbers from one stream does <I>not</I> depend upon how many numbers
were extracted from the other streams. At each time, an instance has
a <I>state</I> that uniquely determines the subsequent numbers being

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@ -88,7 +88,7 @@ typedef const Point_3& reference;
/*!
Creates an input iterator `g` generating points of type `Point_3` uniformly
distributed in the half-open cube with side length \f$ 2 a\f$, centered
at the origin, i.e. \f$ \forall p = *g: -a \le p.x(),p.y(),p.z() < a\f$ .
at the origin, i.e.\ \f$ \forall p = *g: -a \le p.x(),p.y(),p.z() < a\f$ .
Three random numbers are needed from `rnd` for each point.
*/
@ -155,7 +155,7 @@ typedef const Point_3& reference;
/*!
creates an input iterator `g` generating points of type `Point_3` uniformly
distributed in the open sphere with radius \f$ r\f$,
i.e. \f$ |*g| < r\f$ . Three random numbers are needed from
i.e.\ \f$ |*g| < r\f$ . Three random numbers are needed from
`rnd` for each point.
*/
@ -225,7 +225,7 @@ typedef const Point_3& reference;
/*!
creates an input iterator `g` generating points of type `Point_3` uniformly
distributed on the boundary of a sphere with radius \f$ r\f$,
i.e. \f$ |*g| == r\f$ . Two random numbers are needed from
i.e.\ \f$ |*g| == r\f$ . Two random numbers are needed from
`rnd` for each point.
*/

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@ -66,7 +66,7 @@ Note that
way that its vertices form a subset of the vertex set of \f$ P\f$ and
<LI>the vertices of a maximum area `k`-gon, where the `k` vertices
are to be drawn from a planar point set \f$ S\f$, lie on the convex
hull of \f$ S\f$ i.e. a convex polygon.
hull of \f$ S\f$ i.e.\ a convex polygon.
</UL>
\pre the - at least three - points denoted by the range
@ -131,7 +131,7 @@ Note that
way that its vertices form a subset of the vertex set of \f$ P\f$ and
<LI>the vertices of a maximum perimeter `k`-gon, where the `k`
vertices are to be drawn from a planar point set \f$ S\f$, lie on the
convex hull of \f$ S\f$ i.e. a convex polygon.
convex hull of \f$ S\f$ i.e.\ a convex polygon.
</UL>

View File

@ -22,7 +22,7 @@ meet the requirements for the traits class of the
`polygon_area_2()` function. A model of this traits class is
`Regular_triangulation_euclidean_traits_2<K, Weight>`.
<LI>The value type of `OutputIterator` is equivalent to
`std::pair<Rt::Weighted_point, Rt::Geom_traits::FT>`, i.e. a pair
`std::pair<Rt::Weighted_point, Rt::Geom_traits::FT>`, i.e.\ a pair
associating a point and its regular neighbor coordinate.
</OL>

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@ -17,7 +17,7 @@ the sample points onto the tangent plane at `p`.
The functions `surface_neighbor_coordinates_certified_3()` return, in
addition, a second Boolean value (the fourth value of the quadruple)
that certifies whether or not, the Voronoi cell of `p` can be affected
by points that lie outside the input range, i.e. outside the ball
by points that lie outside the input range, i.e.\ outside the ball
centered on `p` passing through the furthest sample point from `p` in
the range `[first, beyond)`. If the sample points are collected by a
`k`-nearest neighbor or a range search query, this permits to check
@ -29,7 +29,7 @@ whether the neighborhood which has been considered is large enough.
<LI>`Dt` is equivalent to the class
`Delaunay_triangulation_3`.
<LI>The value type of `OutputIterator` is equivalent to
`std::pair<Dt::Point_3, Dt::Geom_traits::FT>`, i.e. a pair
`std::pair<Dt::Point_3, Dt::Geom_traits::FT>`, i.e.\ a pair
associating a point and its natural neighbor coordinate.
<LI>`ITraits` is equivalent to the class `Voronoi_intersection_2_traits_3<K>`.
</OL>
@ -68,7 +68,7 @@ starting at `out`. The function returns a triple with an
iterator that is placed past-the-end of the resulting sequence of
point/coordinate pairs, the normalization factor of the coordinates
and a Boolean value which is set to true iff the coordinate
computation was successful, i.e. if `p` lies inside the convex
computation was successful, i.e.\ if `p` lies inside the convex
hull of the projection of the points \f$ \mathcal{P}\f$ onto the tangent
plane.
*/

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@ -18,7 +18,7 @@ surface at `p`.
The functions \c surface_neighbors_certified_3() also return, in
addition, a Boolean value that certifies whether or not, the Voronoi
cell of `p` can be affected by points that lie outside the input
range, i.e. outside the ball centered on `p` passing through the
range, i.e.\ outside the ball centered on `p` passing through the
furthest sample point from `p` in the range `[first, beyond)`. If the sample
points are collected by a k-nearest neighbor or a range search
query, this permits to verify that a large enough neighborhood has
@ -30,7 +30,7 @@ been considered.
<LI>`Dt` is equivalent to the class
`Delaunay_triangulation_3`.
<LI>`OutputIterator::value_type` is equivalent to
`Dt::Point_3`, i.e. a point type.
`Dt::Point_3`, i.e.\ a point type.
<LI>`ITraits` is equivalent to the class `Voronoi_intersection_2_traits_3<K>`.
</OL>

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@ -110,7 +110,7 @@ Constructor object for
`Aff_transformation_d`. Provides :
`Aff_transformation_d operator()(Vector v)` which returns the
outer product, i.e. the quadratic matrix `v`\f$ ^t\f$`v`.
outer product, i.e.\ the quadratic matrix `v`\f$ ^t\f$`v`.
*/
typedef Hidden_type Construct_outer_product_d;

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@ -25,7 +25,7 @@ to interpolate this function on an arbitrary query point.
More formally, let \f$ \mathcal{P}=\{\mathbf{p_1},\ldots ,\mathbf{p_n}\}\f$ be a set of
\f$ n\f$ points in \f$ \mathbb{R}^2\f$ or \f$ \mathbb{R}^3\f$ and \f$ \Phi\f$ be a scalar
function defined on the convex hull of \f$ \mathcal{P}\f$. We assume that
the function values are known at the points of \f$ \mathcal{P}\f$, i.e. to
the function values are known at the points of \f$ \mathcal{P}\f$, i.e.\ to
each \f$ \mathbf{p_i} \in \mathcal{P}\f$, we associate \f$ z_i =
\Phi(\mathbf{p_i})\f$. Sometimes, the gradient of \f$ \Phi\f$ is also known
at \f$ \mathbf{p_i}\f$. It is denoted \f$ \mathbf{g_i}= \nabla
@ -93,7 +93,7 @@ continuously differentiable except on the data points \f$ \mathcal{P}\f$.<BR>
The interpolation package of \cgal provides functions to compute
natural neighbor coordinates for \f$ 2D\f$ and \f$ 3D\f$ points with respect
to Voronoi diagrams as well as with respect to power diagrams (only
\f$ 2D\f$), i.e. for weighted points. Refer to the reference pages
\f$ 2D\f$), i.e.\ for weighted points. Refer to the reference pages
`natural_neighbor_coordinates_2()`,
`sibson_natural_neighbor_coordinates_3()`
`laplace_natural_neighbor_coordinates_3()` and
@ -151,7 +151,7 @@ plane \f$ \mathcal{T}_x\f$ of the surface \f$ \mathcal{S}\f$ at the point
\f$ \mathbf{x} \in \mathcal{S}\f$ approximates \f$ \mathcal{S}\f$ in the
neighborhood of \f$ \mathbf{x}\f$. It has been shown in \cite bf-lcss-02
that, if the surface \f$ \mathcal{S}\f$ is well sampled with respect to the
curvature and the local thickness of \f$ \mathcal{S}\f$, i.e. it is an \f$ \epsilon\f$-sample, the intersection
curvature and the local thickness of \f$ \mathcal{S}\f$, i.e.\ it is an \f$ \epsilon\f$-sample, the intersection
of the tangent plane \f$ \mathcal{T}_x\f$ with the Voronoi cell of
\f$ \mathbf{x}\f$ in the Voronoi diagram of \f$ \mathcal{P} \cup
\{\mathbf{x}\}\f$ has a small diameter. Consequently, inside this
@ -211,7 +211,7 @@ neighbors are necessarily a subset of the natural neighbors of the
query point in this triangulation. \cgal provides a function that
encapsulates the filtering based on the \f$ 3D\f$ Delaunay triangulation.
For input points filtered by distance, functions are provided that
indicate whether or not points that lie outside the input range (i.e.
indicate whether or not points that lie outside the input range (i.e.\
points that are further from \f$ \mathbf{x}\f$ than the furthest input
point) can still influence the result. This allows to iteratively
enlarge the set of input points until the range is sufficient to
@ -332,7 +332,7 @@ of \f$ \mathbf{p_i}\f$ with respect to \f$ \mathbf{p_i}\f$ associated to
\cgal provides functions to approximate the gradients of all data
points that are inside the convex hull. There is one function for each
type of natural neighbor coordinate (i.e. `natural_neighbor_coordinates_2()`, `regular_neighbor_coordinates_2()`).
type of natural neighbor coordinate (i.e.\ `natural_neighbor_coordinates_2()`, `regular_neighbor_coordinates_2()`).
\subsection subsecinterpol_examples Example for Linear Interpolation

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@ -198,7 +198,7 @@ of the fitting basis.
direct, and the maximal, minimal curvatures are defined wrt this
basis. If the user has a predefined normal \f$ n_0\f$ (e.g. the sample
points come from an oriented mesh) then if \f$ n_0 . n >0\f$ then max-min
is correct; if not, i.e. \f$ n_0 . n <0\f$, the user should switch to the
is correct; if not, i.e.\ \f$ n_0 . n <0\f$, the user should switch to the
orthonormal direct basis \f$ (d_1',d_2',n')=(d_2,d_1,-n)\f$ with the
maximal curvature \f$ k_1'=-k_2\f$ and the minimal curvature
\f$ k_2'=-k_1\f$. If \f$ n_0 . n =0\f$ or is small, the orientation of the
@ -366,7 +366,7 @@ given by
\f}
The equations for interpolation become \f$ MA=Z\f$. For approximation, the
system \f$ MA=Z\f$ is solved in the least square sense, i.e. one seeks the
system \f$ MA=Z\f$ is solved in the least square sense, i.e.\ one seeks the
vector \f$ A\f$ such that \f$ A = \arg \min_A ||MA-Z||_2\f$.
In any case, there is a preconditioning of the matrix \f$ M\f$ so as to
@ -476,7 +476,7 @@ M}^{-1}=P_{F \rightarrow M}^T\f$. The Monge basis expressed in the
world-basis is obtained by multiplying the coordinates of
\f$ (d_1,d_2,n)\f$ in the fitting-basis by \f$ P_{W\rightarrow F}^{-1}\f$,
(the same holds for the origin point which has in addition to be
translated by \f$ p\f$, i.e. the coordinates of the origin point are
translated by \f$ p\f$, i.e.\ the coordinates of the origin point are
\f$ P_{W\rightarrow F}^{-1} (0,0,A_{0,0}) +p\f$.
</UL>

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@ -177,7 +177,7 @@ Segment_d<Kernel> operator+(const Vector_d<Kernel>& v) ;
/*!
returns true if `s` is
degenerate i.e. `s.source()=s.target()`.
degenerate i.e.\ `s.source()=s.target()`.
*/
bool is_degenerate() ;

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@ -197,7 +197,7 @@ operations, for example computing squared distances or returning a
the number type parameter of `Homogeneous` low, the number type
`Quotient<RingNumberType>` is used instead. This number type
turns a ring type into a field type. It maintains numbers as
quotients, i.e. a numerator and a denominator. Thereby, divisions are
quotients, i.e.\ a numerator and a denominator. Thereby, divisions are
circumvented. With `Homogeneous_d<RingNumberType>`,
`Homogeneous_d<RingNumberType>::%FT` is equal to
`Quotient<RingNumberType>` while

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@ -11,7 +11,7 @@ namespace CGAL {
Lets say you want to maintain a sorted list of items (each item is
associate with a real number key). You can imagine placing each of the
items on the point on the real line corresponding to its key. Now, let
the key for each item change continuously (i.e. no jumps are allowed).
the key for each item change continuously (i.e.\ no jumps are allowed).
As long as no two (consecutive) items cross, the sorted order is
intact. When two items cross, they need to be exchanged in the list and then
the sorted order is once again correct. This is a trivial example of a
@ -99,7 +99,7 @@ must be updated, as well as the set of certificate functions that
verify it. We call such occurrences <I>events</I>.
Maintaining a kinetic data structure is then a matter of determining
which certificate function changes sign next, i.e. determining which
which certificate function changes sign next, i.e.\ determining which
certificate function has the first real root that is greater than the
current time, and then updating the structure and the set of
certificate functions. In addition, the trajectories of primitives are

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@ -93,7 +93,7 @@ Data at(Key key) const;
/*!
Set the editing state of
the object. A notification is sent when the editing state is set to
false after it has been true, i.e. the editing session is finished.
false after it has been true, i.e.\ the editing session is finished.
This allows changes to be batched together.
*/
void set_is_editing(bool is_editing);

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@ -99,7 +99,7 @@ the traits, items, and for the allocator classes, and by default
A linear cell complex is valid, if it is a valid combinatorial map
where each dart is associated with an attribute containing a point
(i.e. an instance of a model of the `CellAttributeWithPoint`
(i.e.\ an instance of a model of the `CellAttributeWithPoint`
concept). Note that there are no validity constraints on the geometry
(test on self intersection, planarity of 2-cells...).
We can see two examples of `Linear_cell_complex` in

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@ -23,7 +23,7 @@ More exactly, a matrix \f$ M = (m_{i j}) \in S^{r \times l}\f$
\f}
Now let \f$ \mathcal{M}\f$ be a set of \f$ n\f$ sorted matrices over \f$ S\f$
and \f$ f\f$ be a monotone predicate on \f$ S\f$, i.e.
and \f$ f\f$ be a monotone predicate on \f$ S\f$, i.e.\
\f[
f\: :\: S \longrightarrow\, \textit{bool} \quad{\rm with}\quad f(r)
\;\Longrightarrow\; \forall\, t \in S\,,\: t > r \; :\; f(t)\;.

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@ -70,7 +70,7 @@ void replace_column( int old, int new);
/*!
returns
a new Matrix consisting of all rows of `m` with even index,
(i.e. first row is row \f$ 0\f$ of `m`, second row is row \f$ 2\f$ of
(i.e.\ first row is row \f$ 0\f$ of `m`, second row is row \f$ 2\f$ of
`m` etc.). \pre `number_of_rows()` \f$ > 0\f$.
*/
Matrix* extract_all_even_rows() const;

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@ -7,7 +7,7 @@ The class `Mesh_constant_domain_field_3` is a model of concept `MeshDomainField_
a constant field accessible using queries on 3D-points.
The class `Mesh_constant_domain_field_3` can also be customized through `set_size()` operations to become
a piecewise constant field, i.e. a sizing field with a constant size on each subpart
a piecewise constant field, i.e.\ a sizing field with a constant size on each subpart
of the domain.
\tparam Gt is the geometric traits class. It must match the type `Triangulation::Geom_traits`,

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@ -59,7 +59,7 @@ The parameter `convergence` gives the threshold ratio.
- <b>`parameters::freeze_bound`</b>
is designed to reduce running time of each optimization iteration. Any vertex
that has
a displacement less than a given percentage of the length of its shortest incident edge, is frozen (i.e. is
a displacement less than a given percentage of the length of its shortest incident edge, is frozen (i.e.\ is
not relocated). The parameter `freeze_bound` gives the threshold ratio. At each iteration, any vertex that
moves, unfreezes all its incident vertices.

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@ -71,7 +71,7 @@ The type `Features` of this parameter is an internal undescribed type.
The library provides functions to construct appropriate values of that type.
<UL>
<LI>`parameters::features(domain)` sets `features` according to the domain,
i.e. 0 and 1-dimensional features are taken into account if `domain` is a
i.e.\ 0 and 1-dimensional features are taken into account if `domain` is a
`MeshDomainWithFeatures_3`. This is the default behavior
if parameter `features` is not specified.
<LI>`parameters::no_features()` prevents the representation

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@ -59,7 +59,7 @@ The parameter `convergence` gives the threshold ratio.
- <b>`parameters::freeze_bound`</b>
is designed to reduce running time of each optimization iteration.
Any vertex that has
a displacement less than a given percentage of the length of its shortest incident edge, is frozen (i.e. is
a displacement less than a given percentage of the length of its shortest incident edge, is frozen (i.e.\ is
not relocated). The parameter `freeze_bound` gives the threshold ratio. At each iteration, any vertex that
moves, unfreezes the neighboring vertices.

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@ -164,7 +164,7 @@ to the optimization function `exude_mesh_3()` through these mesh generation func
\cgalHeading{Parameters}
The parameters are named parameters. They are the same (i.e. they have the same
The parameters are named parameters. They are the same (i.e.\ they have the same
name and the same default values) as the parameters of `exude_mesh_3()`
function. See its manual page for further details.
@ -223,7 +223,7 @@ parameters to the optimization function
\cgalHeading{Parameters}
The parameters are named parameters. They are the same (i.e. they have the same
The parameters are named parameters. They are the same (i.e.\ they have the same
name and the same default values) as the parameters of the `lloyd_optimize_mesh_3()`
function. See its manual page for further details.
@ -377,7 +377,7 @@ allows the user to pass parameters to the optimization function
\cgalHeading{Parameters}
The parameters are named parameters. They are the same (i.e. they have the same
The parameters are named parameters. They are the same (i.e.\ they have the same
name and the same default values) as the parameters of `odt_optimize_mesh_3()`
function. See its manual page for further details.
@ -418,7 +418,7 @@ to the optimization function `perturb_mesh_3()` through these mesh generation fu
\cgalHeading{Parameters}
The parameters are named parameters. They are the same (i.e. they have the same
The parameters are named parameters. They are the same (i.e.\ they have the same
name and the same default values) as the parameters of `perturb_mesh_3()`
function. See its manual page for further details.

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@ -8,7 +8,7 @@ The concept `MeshComplexWithFeatures_3InTriangulation_3` refines the minimal con
`MeshComplex_3InTriangulation_3`, designed to represent
3D complexes having only faces with dimension 2 and 3.
Therefore, the concept `MeshComplexWithFeatures_3InTriangulation_3` may represent embedded complexes
including <I>features</I>, i.e. faces with dimension \f$ 0\f$ and \f$ 1\f$.
including <I>features</I>, i.e.\ faces with dimension \f$ 0\f$ and \f$ 1\f$.
The data structure includes a 3D triangulation which is itself a 3D complex.
To distinguish the faces of the embedded 3D complex from the

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@ -139,7 +139,7 @@ belong to
two corners incident on the curve segment.
If it is a cycle, then the same `Point_3` should be given twice and must be any
point on the cycle.
The `Index` values associated to the points are their indices w.r.t. their dimension.
The `Index` values associated to the points are their indices w.r.t.\ their dimension.
*/
template <typename OutputIterator>
OutputIterator

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@ -2,7 +2,7 @@
\ingroup PkgMesh_3SecondaryConcepts
\cgalConcept
The concept `MeshPolyline_3` implements a container of points designed to represent a polyline (i.e. a sequence of points).
The concept `MeshPolyline_3` implements a container of points designed to represent a polyline (i.e.\ a sequence of points).
Types and functions provided in this concept are such as standard template library containers
are natural models of this concept.

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@ -171,7 +171,7 @@ To some extend, the user may tune the Delaunay refinement
to a prescribed trade-off
between mesh quality and mesh density.
The mesh density refers to the number of mesh vertices and cells,
i.e. to the complexity of the mesh.
i.e.\ to the complexity of the mesh.
The mesh quality referred to here is measured by the radius edge
ratio of surface facets end mesh cells, where the radius edge ratio of
a simplex (triangle or tetrahedron) is the
@ -375,7 +375,7 @@ The type `Features` of this parameter is an internal undescribed type.
The library provides functions to construct appropriate values of that type.
<UL>
<LI>`parameters::features(domain)` sets `features` according to the domain,
i.e. 0 and 1-dimensional features are taken into account if `domain` is a
i.e.\ 0 and 1-dimensional features are taken into account if `domain` is a
`MeshDomainWithFeatures_3`
<LI>`parameters::no_features()` prevents the representation
of 0 and 1-dimensional features in the mesh. This is useful to get a smooth and rough approximation

View File

@ -58,7 +58,7 @@ quality of the decomposition.
Let us denote the vertices of the input polygons by
\f$ P = \left( p_0, \ldots, p_{m-1} \right)\f$ and
\f$ Q = \left( q_0, \ldots, q_{n-1} \right)\f$. We assume that both \f$ P\f$ and \f$ Q\f$
have positive orientations (i.e. their boundaries wind in a counterclockwise
have positive orientations (i.e.\ their boundaries wind in a counterclockwise
order around their interiors) and compute the convolution of the two polygon
boundaries. The <I>convolution</I> of these two polygons \cite grs-kfcg-83,
denoted \f$ P * Q\f$, is a collection of line segments of the form

View File

@ -10,7 +10,7 @@ either *running* or it is *stopped*. The state is controlled
with `Real_timer::start()` and `Real_timer::stop()`. The timer counts the
time elapsed since its creation or last reset. It counts only the time
where it is in the running state. The time information is given in seconds.
The timer counts also the number of intervals it was running, i.e. it
The timer counts also the number of intervals it was running, i.e.\ it
counts the number of calls of the `Real_timer::start()` member function since the
last reset. If the reset occures while the timer is running it counts as the
first interval.

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@ -10,7 +10,7 @@ either *running* or it is *stopped*. The state is controlled
with `Timer::start()` and `Timer::stop()`. The timer counts the
time elapsed since its creation or last reset. It counts only the time
where it is in the running state. The time information is given in seconds.
The timer counts also the number of intervals it was running, i.e. it
The timer counts also the number of intervals it was running, i.e.\ it
counts the number of calls of the `Timer::start()` member function since the
last reset. If the reset occures while the timer is running it counts as the
first interval.

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@ -21,7 +21,7 @@ either <I>running</I> or it is <I>stopped</I>. The state of an object
with `t.start()` and `t.stop()`. The timer counts the
time elapsed since its creation or last reset. It counts only the time
where it is in the running state. The time information is given in seconds.
The timer counts also the number of intervals it was running, i.e. it
The timer counts also the number of intervals it was running, i.e.\ it
counts the number of calls of the `start()` member function since the
last reset. If the reset occurs while the timer is running it counts as the
first interval.

View File

@ -877,7 +877,7 @@ public:
\ingroup PkgNef3
A volume is a full-dimensional connected point set in \f$ \mathbb{R}^3\f$. It is
bounded by several shells, i.e. a connected part of the boundary incident
bounded by several shells, i.e.\ a connected part of the boundary incident
to a single volume. An entry element to each shell is provided by the
iterator range (`shells_begin()`/`shells_end()`). A
`Shell_entry_iterator` is assignable to `SFace_handle`.
@ -1369,7 +1369,7 @@ public:
Nef_polyhedron_3<Traits> closure() const;
/*!
returns the regularization, i.e. the closure of the interior, of `N` .
returns the regularization, i.e.\ the closure of the interior, of `N` .
*/
Nef_polyhedron_3<Traits> regularization() const;

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@ -159,7 +159,7 @@ The local pyramids of each vertex are represented by
conceptually intersecting the local neighborhood with a small
\f$ \varepsilon\f$-sphere. This intersection forms a planar map on the
sphere (see next two figures), which together with the set-selection
mark for each item (i.e. vertices, edges, loops and faces)
mark for each item (i.e.\ vertices, edges, loops and faces)
forms a two-dimensional Nef polyhedron embedded in
the sphere. We add the set-selection mark for the vertex and call the
resulting structure the <I>sphere map</I> of the vertex.

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@ -5,7 +5,7 @@ namespace CORE {
\ingroup nt_core
The class `CORE::BigFloat` is a variable precision floating-point type.
Rounding mode and precision (i.e. mantissa length) of
Rounding mode and precision (i.e.\ mantissa length) of
`CORE::BigFloat` can be set.
Since it also carries the error of a computed value.

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@ -90,7 +90,7 @@ Gmpq(const std::string& str, int base);
/// defined. It is guaranteed that `q.numerator()` and
/// `q.denominator()` return values `nt_num` and `nt_den` such that `q
/// = nt_num/nt_den`, only if `q.numerator()` and `q.denominator()`
/// are called consecutively wrt. `q`, i.e. `q` is not involved in any
/// are called consecutively wrt. `q`, i.e.\ `q` is not involved in any
/// other operation between these calls.
/// @{

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@ -29,7 +29,7 @@ It is guaranteed that `q.numerator()` and
`q.denominator()` return values `nt_num` and
`nt_den` such that `q = nt_num/nt_den`, only
if `q.numerator()` and `q.denominator()` are called
consecutively wrt `q`, i.e. `q` is not involved in
consecutively wrt `q`, i.e.\ `q` is not involved in
any other operation between these calls.
The stream operations are available as well.

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@ -7,7 +7,7 @@
The class `leda_bigfloat` is a wrapper class that provides the functions
needed to use the number type `bigfloat`.
`bigfloat`
Rounding mode and precision (i.e. mantissa length) of
Rounding mode and precision (i.e.\ mantissa length) of
`leda_bigfloat` can be set.
For more details on the number types of \leda we refer to the \leda manual \cite cgal:mnsu-lum.

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@ -131,7 +131,7 @@ computation as long as all intermediate results are rational. For the
same kind of problems, Cartesian representation with number type
`leda_rational` leads to exact computation as well.
The number type `leda_bigfloat` in \leda is a variable precision
floating-point type. Rounding mode and precision (i.e. mantissa length) of
floating-point type. Rounding mode and precision (i.e.\ mantissa length) of
`leda_bigfloat` can be set.
The most sophisticated number type in \leda is the number type called

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@ -189,7 +189,7 @@ typedef Triangulation_data_structure::Vertex_iterator Vertex_iterator;
/*!
iterator over the vertices whose
corresponding points lie in the original domain, i.e. for each set
corresponding points lie in the original domain, i.e.\ for each set
of periodic copies the `Unique_vertex_iterator` iterates over
exactly one representative.
*/
@ -396,7 +396,7 @@ Triangulation_data_structure & tds();
/*!
The current triangulation remains a triangulation in the 1-sheeted
covering space even after adding points if this method returns
`true`. This test relies on a heuristic, i.e. if it answers
`true`. This test relies on a heuristic, i.e.\ if it answers
`false` nothing is known. This test runs in constant-time when
not computing in the 1-sheeted covering space. (This test uses the length
of the longest edge in the triangulation as a
@ -406,7 +406,7 @@ bool is_extensible_triangulation_in_1_sheet_h1() const;
/*!
The same as `is_extensible_triangulation_in_1_sheet_h1()` but with
a more precise heuristic, i.e. it might answer `true` in cases in which
a more precise heuristic, i.e.\ it might answer `true` in cases in which
`is_extensible_triangulation_in_1_sheet_h1()` would not. However, it is
much less time efficient when not computing in the 1-sheeted covering
space. (This test uses the diameter of the largest empty ball in the
@ -828,7 +828,7 @@ that allow one to traverse it.
\name Cell, Face, Edge and Vertex Iterators
The following iterators allow the user to visit cells, facets, edges
and vertices of the stored triangulation, i.e. in case of computing in
and vertices of the stored triangulation, i.e.\ in case of computing in
a multiply sheeted covering space all stored periodic copies of each
item are returned. These iterators are non-mutable, bidirectional and
their value types are respectively `Cell`, `Facet`, `Edge` and
@ -884,7 +884,7 @@ Cell_iterator cells_end() const;
/*!
Starts at an arbitrary vertex. Iterates over all vertices whose
corresponding points lie in the original domain, i.e. for each set
corresponding points lie in the original domain, i.e.\ for each set
of periodic copies the `Unique_vertex_iterator` iterates over
exactly one representative. Returns `unique_vertices_end()` if
`t`.`number_of_vertices()` \f$ =0\f$.

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@ -63,7 +63,7 @@ typedef Hidden_type Iso_cuboid_3;
/// The following three types represent geometric primitives in \f$
/// \mathbb R^3\f$. They are required to provide functions converting
/// primitives from \f$ \mathbb T_c^3\f$ to \f$ \mathbb R^3\f$,
/// i.e. constructing representatives in \f$ \mathbb R^3\f$.
/// i.e.\ constructing representatives in \f$ \mathbb R^3\f$.
/// @{
/*!

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@ -14,7 +14,7 @@ namespace CGAL {
The periodic 3D-triangulation class of \cgal is designed to
represent the triangulations of a set of points in the
three-dimensional flat torus. The triangulation forms a partition of
the space it is computed in. It is a simplicial complex, i.e. it
the space it is computed in. It is a simplicial complex, i.e.\ it
contains all incident \f$ j\f$-simplices (\f$ j<k\f$) of any \f$ k\f$-simplex and two
\f$ k\f$-simplices either do not intersect or share a common \f$ j\f$-face,
\f$ j<k\f$. The occurring simplices of dimension up to three are called

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@ -1038,7 +1038,7 @@ public:
size_type size_of_vertices() const;
/*!
number of halfedges (incl. border halfedges).
number of halfedges (incl.\ border halfedges).
*/
size_type size_of_halfedges() const;
@ -1535,7 +1535,7 @@ n */
/// @{
/*!
reverses facet orientations (incl. plane equations if supported).
reverses facet orientations (incl.\ plane equations if supported).
*/
void inside_out();

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@ -12,11 +12,11 @@ The template argument `Coeff` must be at
least a model of `IntegralDomainWithoutDivision`.
For all operations naturally involving division, an `IntegralDomain`
is required.
`Polynomial` offers a full set of algebraic operators, i.e.
`Polynomial` offers a full set of algebraic operators, i.e.\
binary <TT>+</TT>, <TT>-</TT>, <TT>*</TT>, <TT>/</TT> as well as
<TT>+=</TT>, <TT>-=</TT>, <TT>*=</TT>, <TT>/=</TT>; not only for polynomials
but also for a polynomial and a number of the coefficient type.
(The <TT>/</TT> operator must only be used for integral divisions, i.e.
(The <TT>/</TT> operator must only be used for integral divisions, i.e.\
those with remainder zero.)
The operations are implemented naively: <TT>+</TT> and <TT>-</TT> need a number of `Coeff`
operations which is linear in the degree while * is quadratic.

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@ -10,7 +10,7 @@ sets in \f$ d\f$-dimensional Euclidean space \f$ \E^d\f$. For point sets \f$ P\f
we denote by \f$ pd(P,Q)\f$ the distance between the convex hulls of \f$ P\f$ and
\f$ Q\f$. Note that \f$ pd(P,Q)\f$ can be
degenerate,
i.e. \f$ pd(P,Q)=\infty\f$ if \f$ P\f$ or \f$ Q\f$ is empty.
i.e.\ \f$ pd(P,Q)=\infty\f$ if \f$ P\f$ or \f$ Q\f$ is empty.
Two inclusion-minimal subsets \f$ S_P\f$ of \f$ P\f$ and \f$ S_Q\f$ of \f$ Q\f$ with
\f$ pd(S_P,S_Q)=pd(P,Q)\f$ are called <I>pair of support
@ -19,7 +19,7 @@ points in \f$ S_P\f$ and \f$ S_Q\f$ are the <I>support points</I>. A pair of sup
sets has size at most \f$ d+2\f$ (by size we mean \f$ |S_P|+|S_Q|\f$). The distance
between the two polytopes is <I>realized</I> by a pair of points \f$ p\f$ and
\f$ q\f$ lying in the convex hull of \f$ S_P\f$ and \f$ S_Q\f$, repectively,
i.e. \f$ \sqrt{||p-q||}=pd(P,Q)\f$. In general, neither the support sets nor the
i.e.\ \f$ \sqrt{||p-q||}=pd(P,Q)\f$. In general, neither the support sets nor the
realizing points are necessarily unique.
The underlying algorithm can cope with all kinds of input, e.g. \f$ P\f$ and \f$ Q\f$

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@ -11,7 +11,7 @@ written to o]`} and returns the past-the-end iterator of this
sequence.
The function `all_furthest_neighbors_2()` computes all furthest
neighbors for the vertices of a convex polygon \f$ P\f$, i.e. for each
neighbors for the vertices of a convex polygon \f$ P\f$, i.e.\ for each
vertex \f$ v\f$ of \f$ P\f$ a vertex \f$ f_v\f$ of \f$ P\f$ such that the distance
between \f$ v\f$ and \f$ f_v\f$ is maximized.

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@ -77,7 +77,7 @@ Programs can be written to an output stream in MPSFormat, using one of the follo
\cgalPkgPicture{qp.png}
\cgalPkgSummaryBegin
\cgalPkgAuthors{Kaspar Fischer, Bernd G&auml;rtner, Sven Sch&ouml;nherr, and Frans Wessendorp}
\cgalPkgDesc{This package contains algorithms for minimizing linear and convex quadratic functions over polyhedral domains, described by linear equations and inequalities. The algorithms are exact, i.e. the solution is computed in terms of multiprecision rational numbers. The resulting solution is certified: along with the claims that the problem under consideration has an optimal solution, is infeasible, or is unbounded, the algorithms also deliver proofs for these facts. These proofs can easily (and independently from the algorithms) be checked for correctness. The solution algorithms are based on a generalization of the simplex method to quadratic objective functions. }
\cgalPkgDesc{This package contains algorithms for minimizing linear and convex quadratic functions over polyhedral domains, described by linear equations and inequalities. The algorithms are exact, i.e.\ the solution is computed in terms of multiprecision rational numbers. The resulting solution is certified: along with the claims that the problem under consideration has an optimal solution, is infeasible, or is unbounded, the algorithms also deliver proofs for these facts. These proofs can easily (and independently from the algorithms) be checked for correctness. The solution algorithms are based on a generalization of the simplex method to quadratic objective functions. }
\cgalPkgManuals{Chapter_Linear_and_Quadratic_Programming_Solver,PkgQPSolver}
\cgalPkgSummaryEnd
\cgalPkgShortInfoBegin

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@ -61,10 +61,10 @@ gradients of the principal curvatures. Ridges, illustrated in
A non umbilical point is called
<UL>
<LI>a max ridge point, if the <I>extremality coefficient</I> \f$ b_0=\langle
dk_1,d_1 \rangle\f$ vanishes, i.e. \f$ b_0=0\f$.
dk_1,d_1 \rangle\f$ vanishes, i.e.\ \f$ b_0=0\f$.
<LI>a min ridge point, if the <I>extremality coefficient</I>
\f$ b_3=\langle dk_2,d_2 \rangle\f$ vanishes, i.e. \f$ b_3=0\f$
\f$ b_3=\langle dk_2,d_2 \rangle\f$ vanishes, i.e.\ \f$ b_3=0\f$
\cgalFootnote{Notations \f$ b_0, b_3\f$ comes from Equation \ref eqmonge }.
</UL>
</EM>

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@ -70,7 +70,7 @@ the halfedge data structures.
It supports bidirectional iterators and allows a constant time amortized
`insert()` operation. You cannot specify where to insert new objects
(i.e. you don't know where they will end up in the iterator sequence,
(i.e.\ you don't know where they will end up in the iterator sequence,
although `insert()` returns an iterator pointing to the newly inserted
object). You can erase any element with a constant time complexity.
@ -79,7 +79,7 @@ memory since it doesn't store two additional pointers for the iterator needs.
It doesn't deallocate elements until the destruction or `clear()` of the
container. The iterator does not have constant amortized time complexity for
the increment and decrement operations in all cases, only when not too many
elements have not been freed (i.e. when the `size()` is close to the
elements have not been freed (i.e.\ when the `size()` is close to the
`capacity()`). Iterating from `begin()` to `end()` takes
`O(capacity())` time, not `size()`. In the case where the container
has a small `size()` compared to its `capacity()`, we advise to

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@ -95,7 +95,7 @@ namespace CGAL {
instead.
The function `predecessor` returns the previous iterator,
i.e. the result of `operator--` on a bidirectional iterator.
i.e.\ the result of `operator--` on a bidirectional iterator.
\sa `CGAL::successor()`

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@ -385,7 +385,7 @@ introduced by Arya and Mount \cite am-afvq-93. This technique
works only for neighbor queries with query items represented as points
and with a quadratic form distance, defined by \f$ d_A(x,y)=
(x-y)A(x-y)^T\f$, where the matrix \f$ A\f$ is positive definite,
i.e. \f$ d_A(x,y) \geq 0\f$. An important class of quadratic form
i.e.\ \f$ d_A(x,y) \geq 0\f$. An important class of quadratic form
distances are weighted Minkowski distances. Given a parameter \f$ p>0\f$
and parameters \f$ w_i \geq 0, 0 < i \leq d\f$, the weighted Minkowski
distance is defined by \f$ l_p(w)(r,q)= ({\Sigma_{i=1}^{i=d} \,

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@ -58,12 +58,12 @@ Halfedge_const_handle defining_contour_edge() const;
/// @{
/*!
Returns `true` iff this is a bisector (or skeleton) halfedge (i.e. is not a contour halfedge).
Returns `true` iff this is a bisector (or skeleton) halfedge (i.e.\ is not a contour halfedge).
*/
bool is_bisector() const;
/*!
Returns `true` iff this is a bisector and is inner (i.e. is not incident upon a contour vertex).
Returns `true` iff this is a bisector and is inner (i.e.\ is not incident upon a contour vertex).
*/
bool is_inner_bisector() const;

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@ -20,7 +20,7 @@ public:
/// @{
/*!
creates a color with rgb-value `(0,0,0)`, i.e. black.
creates a color with rgb-value `(0,0,0)`, i.e.\ black.
*/
Color();

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@ -133,7 +133,7 @@ Subdivision_method_3::CatmullClark_subdivision(P,d);
subdivision functions. `CatmullClark_subdivision(P,d)` computes the
Catmull-Clark subdivision surface of the polyhedron `P` after
`d` iterations of the refinements. The polyhedron `P` is
passed by reference, and is modified (i.e. subdivided) by the
passed by reference, and is modified (i.e.\ subdivided) by the
subdivision function.
This example shows how to subdivide a simple `Polyhedron_3`
@ -365,10 +365,10 @@ can not be used with `Subdivision_method_3`.
Although `Subdivision_method_3` does not require flags
to support the refinements and the stencils, it
still needs to know how to compute and store the geometry
data (i.e. the points). `Subdivision_method_3`
data (i.e.\ the points). `Subdivision_method_3`
expects that the typename `Point_3` is
defined in the geometry kernel of the polyhedron
(i.e. the `Polyhedron_3::Traits::Kernel`).
(i.e.\ the `Polyhedron_3::Traits::Kernel`).
A point of the type `Point_3` is returned by the geometry
policy and is then assigned to the new vertex.
The geometry policy is explained in next section.
@ -392,8 +392,8 @@ Each geometry mask receives a primitive handle
(e.g. `Halfedge_handle`) of the control mesh,
and returns a `Point_3` to the subdivided vertex.
The function collects the vertex neighbors of the primitive handle
(i.e. nodes on the stencil), and computes the new point
based on the neighbors and the mask (i.e. the stencil weights).
(i.e.\ nodes on the stencil), and computes the new point
based on the neighbors and the mask (i.e.\ the stencil weights).
\image html cc_mask.gif

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@ -55,7 +55,7 @@ typedef Hidden_type Error_code;
/*!
Assign to mesh's border vertices a 2D position (i.e. a `(u, v)` pair) on border's shape. Mark them as <I>parameterized</I>. Return false on error.
Assign to mesh's border vertices a 2D position (i.e.\ a `(u, v)` pair) on border's shape. Mark them as <I>parameterized</I>. Return false on error.
*/
Error_code parameterize_border(Adaptor& mesh);

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@ -7,7 +7,7 @@
A `ParameterizationMesh_3` surface consists of vertices, facets and an incidence relation on them. No notion of edge is requested. Vertices represent points in 3d-space. Facets are planar polygons without holes defined by the circular sequence of vertices along their border. The surface itself can have holes. The vertices along the border of a hole are called <I>border vertices</I>. A surface is <I>closed</I> if it contains no border vertices.
The surface must be an oriented 2-manifold with border vertices, i.e. the neighborhood of each point on the surface is either homeomorphic to a disc or to a half disc, except for vertices where many holes and surfaces with border can join.
The surface must be an oriented 2-manifold with border vertices, i.e.\ the neighborhood of each point on the surface is either homeomorphic to a disc or to a half disc, except for vertices where many holes and surfaces with border can join.
`ParameterizationMesh_3` defines the types, data and methods that a mesh must implement to allow surface parameterization. Among other things, this concept defines accessors to fields specific to parameterizations methods: index, `u`, `v`, `is_parameterized`.
@ -35,7 +35,7 @@ class ParameterizationMesh_3 {
public:
/// \name Types
/// The following mutable handles, iterators, and circulators have appropriate non-mutable counterparts, i.e. `const_handle`, `const_iterator`, and `const_circulator`. The mutable types are assignable to their non-mutable counterparts. Both circulators are assignable to the `Vertex_iterator`. The iterators are assignable to the respective handle types. Wherever the handles appear in function parameter lists, the corresponding iterators can be used as well.
/// The following mutable handles, iterators, and circulators have appropriate non-mutable counterparts, i.e.\ `const_handle`, `const_iterator`, and `const_circulator`. The mutable types are assignable to their non-mutable counterparts. Both circulators are assignable to the `Vertex_iterator`. The iterators are assignable to the respective handle types. Wherever the handles appear in function parameter lists, the corresponding iterators can be used as well.
/// @{
/*!
@ -174,7 +174,7 @@ typedef Hidden_type Vertex_around_vertex_const_circulator;
/// @}
/// \name Operations
/// The following mutable methods returning a handle, iterator, or circulator have appropriate non-mutable counterpart methods, i.e. `const`, returning a `const_handle`, `const_iterator`, or `const_circulator`.
/// The following mutable methods returning a handle, iterator, or circulator have appropriate non-mutable counterpart methods, i.e.\ `const`, returning a `const_handle`, `const_iterator`, or `const_circulator`.
/// @{
/*!

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@ -50,13 +50,13 @@ void set_vertex_seaming(Vertex_handle vertex, int seaming);
/*!
%Get oriented edge's seaming flag, i.e. position of the oriented edge w.r.t. to the UNIQUE main border.
%Get oriented edge's seaming flag, i.e.\ position of the oriented edge w.r.t.\ to the UNIQUE main border.
*/
int get_halfedge_seaming(Vertex_const_handle source, Vertex_const_handle target) const;
/*!
Set oriented edge's seaming flag, i.e. position of the oriented edge w.r.t. to the UNIQUE main border.
Set oriented edge's seaming flag, i.e.\ position of the oriented edge w.r.t.\ to the UNIQUE main border.
*/
void set_halfedge_seaming(Vertex_handle source, Vertex_handle target, int seaming);

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@ -120,7 +120,7 @@ The package provides an interface with `CGAL::Polyhedron_3<Traits>`:
## Output ##
A `(u,v)` pair is computed for each inner vertex (i.e. its halfedges share the same `(u,v)` pair), while a `(u,v)` pair is computed for each border halfedge. The user must iterate over the mesh halfedges to get the result.
A `(u,v)` pair is computed for each inner vertex (i.e.\ its halfedges share the same `(u,v)` pair), while a `(u,v)` pair is computed for each border halfedge. The user must iterate over the mesh halfedges to get the result.
## Sparse Linear Algebra ##

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@ -548,7 +548,7 @@ To support this duplication,
`Parameterization_polyhedron_adaptor_3<Polyhedron_3_>` stores
the result in the \f$ (u,v)\f$ fields of the input mesh halfedges.
A \f$ (u,v)\f$ pair is computed for
each inner vertex (i.e. its halfedges share the same \f$ (u,v)\f$ pair),
each inner vertex (i.e.\ its halfedges share the same \f$ (u,v)\f$ pair),
while a \f$ (u,v)\f$ pair is computed for each border halfedge.
The user has to iterate over the mesh halfedges to get the result.
Note that \f$ (u,v)\f$ fields do not exist in `Polyhedron_3<Traits>`,
@ -768,7 +768,7 @@ superclass of all surface parameterization classes.
\subsection Surface_mesh_parameterizationFixedborderparameterizer3 Fixed_border_parameterizer_3 Class
Linear fixed border parameterization algorithms are very close. They mainly
differ by the energy that they try to minimize, i.e. by the value of the \f$ w_{ij}\f$
differ by the energy that they try to minimize, i.e.\ by the value of the \f$ w_{ij}\f$
coefficient of the \f$ A\f$ matrix, for \f$ v_i\f$ and \f$ v_j\f$ neighbor vertices of the mesh
\cite cgal:fh-survey-05. One consequence is that most of the code of the fixed border methods is factorized in the
`Fixed_border_parameterizer_3<ParameterizationMesh_3, BorderParameterizer_3, SparseLinearAlgebraTraits_d>` class.

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@ -6,7 +6,7 @@
The concept `GetCost` describes the requirements for the <I>policy function object</I>
which gets the <I>collapse cost</I> of an edge.
The cost returned is a `boost::optional` value (i.e. it can be absent).
The cost returned is a `boost::optional` value (i.e.\ it can be absent).
An <I>absent</I> cost indicates that the edge should not be collapsed.
This could be the result of a computational limitation (such as overflow),
or can be intentionally returned to prevent the edge from being collapsed.

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@ -372,7 +372,7 @@ Poisson solve duration (in s)
\subsection SurfReconstPerfCont Contouring
The point set chosen for benchmarking the contouring stage is the Bimba con Nastrino point
set simplified to 100k points. We measure the contouring (i.e. the call to `make_surface_mesh()`)
set simplified to 100k points. We measure the contouring (i.e.\ the call to `make_surface_mesh()`)
duration and the reconstruction error for a range of approximation distances.
The reconstruction error is expressed as the average distance from input points to the reconstructed surface
in mm (the Bimba con Nastrino statue is 324 mm tall).

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@ -13,7 +13,7 @@ derived from the type `K::Point_2`.
Note that this template class is specialized for
`Exact_predicates_inexact_constructions_kernel`, so that it is as if
`Regular_triangulation_filtered_traits_2` was used, i.e. you get
`Regular_triangulation_filtered_traits_2` was used, i.e.\ you get
filtered predicates automatically.
\cgalModels `RegularTriangulationTraits_2`

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@ -10,7 +10,7 @@ The more general class `Projection_traits_xy_3` can be found in the 2D and 3D Li
\deprecated The class `Triangulation_euclidean_traits_xy_3` is a
geometric traits class which allows to triangulate a terrain. This
traits class is designed to build a two dimensional triangulation
embedded in 3D space, i.e. a triangulated surface, such that its on
embedded in 3D space, i.e.\ a triangulated surface, such that its on
the \f$ xy\f$ plane is a Delaunay triangulation. This is a usual
construction for GIS terrains. Instead of really projecting the 3D
points and maintaining a mapping between each point and its projection

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@ -1027,7 +1027,7 @@ constrained edge, the set of input constraints that overlap it.
The class `Constrained_triangulation_plus_2<Tr>`
is especially useful when the base constrained triangulation class
handles intersections of constraints and uses an exact number type,
i.e. when its intersection tag is `Exact_intersections_tag`.
i.e.\ when its intersection tag is `Exact_intersections_tag`.
Indeed in this case, the `Constrained_triangulation_plus_2<Tr>`
is specially designed to avoid cascading in the computations of
intersection points.

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@ -11,7 +11,7 @@ for a 3D-triangulation data structure, it is a model of the concept
Note that if the triangulation data structure is used as a parameter of a
geometric triangulation (Section \ref TDS3secdesign and
Chapter \ref chapterTriangulation3 "3D Triangulations"), then the vertex base class has to
fulfill additional geometric requirements, i.e. it has to be a model of the
fulfill additional geometric requirements, i.e.\ it has to be a model of the
concept `TriangulationVertexBase_3`.
This base class can be used directly or can serve as a base to derive

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@ -72,7 +72,7 @@ Thus, a 3D-triangulation data structure can store a triangulation of a
topological sphere \f$ S^d\f$ of \f$ \R^{d+1}\f$, for any \f$ d \in \{-1,0,1,2,3\}\f$.
Let us give, for each dimension, the example corresponding to the
triangulation data structure having a minimal number of vertices, i.e. a
triangulation data structure having a minimal number of vertices, i.e.\ a
simplex. These examples are illustrated by presenting their usual
geometric embedding.
<UL>
@ -88,7 +88,7 @@ being incident to an infinite cell. See \cgalFigureRef{TDS3figtoposimplex4}.
\cgalFigureEnd
<LI><I>dimension 2.</I> We have 4 vertices forming one 3-dimensional
simplex, i.e. the boundary of a tetrahedron. The geometric embedding in
simplex, i.e.\ the boundary of a tetrahedron. The geometric embedding in
the plane results from choosing one of these vertices to be infinite,
then the geometric triangulation has one finite triangle whose edges are
incident to the infinite triangles. See \cgalFigureRef{TDS3figtoposimplex3}.

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@ -226,11 +226,11 @@ between `p` and the power circle of the <I>finite</I> facet of `c`
is greater than \f$ \pi/2\f$.
- `ON_BOUNDARY` if p is orthogonal to the power sphere of `c`
i.e. \f$ \Pi({p}^{(w)}-{z(c)}^{(w)})=0\f$. For an infinite cell this means
i.e.\ \f$ \Pi({p}^{(w)}-{z(c)}^{(w)})=0\f$. For an infinite cell this means
that `p` is orthogonal to the power circle of its <I>finite</I> facet.
- `ON_UNBOUNDED_SIDE` if \f$ \Pi({p}^{(w)}-{z(c)}^{(w)})>0\f$
i.e. the angle between the weighted point `p` and the power sphere
i.e.\ the angle between the weighted point `p` and the power sphere
of `c` is less than \f$ \pi/2\f$ or if these two spheres do not
intersect. For an
infinite cell this means that `p` does not satisfy either of the

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@ -503,7 +503,7 @@ public:
/*!
Returns the in-degree of the vertex,
i.e. the number of halfedges that have `v` as their target.
i.e.\ the number of halfedges that have `v` as their target.
*/
size_type degree();