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- Undo 17-Jun change, but protect the indexing commands by \lcTex.

This commit is contained in:
Sylvain Pion 2003-09-16 13:51:20 +00:00
parent 2f7ca8366c
commit 5bc307bb48
12 changed files with 87 additions and 74 deletions
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3
Packages/Min_annulus_d/changes.txt
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@ -1,5 +1,8 @@
Changes done to the `Min_annulus_d' package
Version 1.1.8 on 16-Sep-2003
- Undo 17-Jun change, but protect the indexing commands by \lcTex.
Version 1.1.7 on 2-Sep-2003 [mh]
- Fix CGAL_NTS.

12
Packages/Min_annulus_d/doc_tex/Optimisation_ref/Min_annulus_d.tex
View File

@ -30,13 +30,13 @@ two concentric spheres with radii $r$ and $R$, $r \leq R$) enclosing a
finite set of points in $d$-dimensional Euclidean space $\E_d$, where the
difference $R^2-r^2$ is minimal. For a point set $P$ we denote by $ma(P)$
the smallest annulus that contains all points of $P$. Note that $ma(P)$
can be degenerate\ccIndexSubitem[t]{degeneracies}{\ccc{Min_annulus_d}},
can be degenerate\lcTex{\ccIndexSubitem[t]{degeneracies}{\ccFont Min_annulus_d}},
i.e.~$ma(P)=\mbox{\ccTexHtml{$\;\emptyset$}{Ø}}$ if
$P=\mbox{\ccTexHtml{$\;\emptyset$}{Ø}}$ and $ma(P)=\{p\}$ if
$P=\{p\}$.
An inclusion-minimal subset $S$ of $P$ with $ma(S)=ma(P)$ is called a
\emph{support set}\ccIndexSubitem[t]{support set}{\ccc{Min_annulus_d}},
\emph{support set}\lcTex{\ccIndexSubitem[t]{support set}{\ccFont Min_annulus_d}},
the points in $S$ are the \emph{support points}. A support set has size at
most $d+2$, and all its points lie on the boundary of $ma(P)$. In general,
the support set is not necessarily unique.
@ -142,7 +142,7 @@ two-, three-, and $d$-dimensional \cgal~kernel, respectively.
\ccMemberFunction{ int number_of_support_points( ) const;}{
returns the number of support points of \ccVar, i.e.~$|S|$.}
\ccIndexSubitem[t]{support set}{\ccc{Min_annulus_d}}
\lcTex{\ccIndexSubitem[t]{support set}{\ccFont Min_annulus_d}}
\ccMemberFunction{ int number_of_inner_support_points( ) const;}{
returns the number of support points of \ccVar{}
@ -277,7 +277,7 @@ unbounded side equals the whole space $\E_d$.
\ccMemberFunction{ bool is_degenerate( ) const;}{
returns \ccc{true}, iff \ccVar\ is degenerate, i.e.~if \ccVar\
is empty or equal to a single point.}
\ccIndexSubitem[t]{degeneracies}{\ccc{Min_annulus_d}}
\lcTex{\ccIndexSubitem[t]{degeneracies}{\ccFont Min_annulus_d}}
% -----------------------------------------------------------------------------
\ccModifiers
@ -312,7 +312,7 @@ unbounded side equals the whole space $\E_d$.
% -----------------------------------------------------------------------------
\ccHeading{Validity Check}
\ccIndexMemberFunctionGroup{validity check}
\ccIndexSubitem[t]{validity check}{\ccc{Min_annulus_d}}
\lcTex{\ccIndexSubitem[t]{validity check}{\ccFont Min_annulus_d}}
An object \ccVar\ is valid, iff
\begin{itemize}
@ -380,7 +380,7 @@ validity check.
The problem of finding the smallest enclosing annulus of a finite point set
can be formulated as an optimization problem with linear constraints and a
linear objective
function\ccIndexSubitem[t]{linear program}{\ccc{Min_annulus_d}}.
function\lcTex{\ccIndexSubitem[t]{linear program}{\ccFont Min_annulus_d}}.
The solution is obtained using our exact
solver for linear and quadratic programs~\cite{gs-eegqp-00}.

12
Packages/Min_annulus_d/doc_tex/basic/Optimisation_ref/Min_annulus_d.tex
View File

@ -30,13 +30,13 @@ two concentric spheres with radii $r$ and $R$, $r \leq R$) enclosing a
finite set of points in $d$-dimensional Euclidean space $\E_d$, where the
difference $R^2-r^2$ is minimal. For a point set $P$ we denote by $ma(P)$
the smallest annulus that contains all points of $P$. Note that $ma(P)$
can be degenerate\ccIndexSubitem[t]{degeneracies}{\ccc{Min_annulus_d}},
can be degenerate\lcTex{\ccIndexSubitem[t]{degeneracies}{\ccFont Min_annulus_d}},
i.e.~$ma(P)=\mbox{\ccTexHtml{$\;\emptyset$}{Ø}}$ if
$P=\mbox{\ccTexHtml{$\;\emptyset$}{Ø}}$ and $ma(P)=\{p\}$ if
$P=\{p\}$.
An inclusion-minimal subset $S$ of $P$ with $ma(S)=ma(P)$ is called a
\emph{support set}\ccIndexSubitem[t]{support set}{\ccc{Min_annulus_d}},
\emph{support set}\lcTex{\ccIndexSubitem[t]{support set}{\ccFont Min_annulus_d}},
the points in $S$ are the \emph{support points}. A support set has size at
most $d+2$, and all its points lie on the boundary of $ma(P)$. In general,
the support set is not necessarily unique.
@ -142,7 +142,7 @@ two-, three-, and $d$-dimensional \cgal~kernel, respectively.
\ccMemberFunction{ int number_of_support_points( ) const;}{
returns the number of support points of \ccVar, i.e.~$|S|$.}
\ccIndexSubitem[t]{support set}{\ccc{Min_annulus_d}}
\lcTex{\ccIndexSubitem[t]{support set}{\ccFont Min_annulus_d}}
\ccMemberFunction{ int number_of_inner_support_points( ) const;}{
returns the number of support points of \ccVar{}
@ -277,7 +277,7 @@ unbounded side equals the whole space $\E_d$.
\ccMemberFunction{ bool is_degenerate( ) const;}{
returns \ccc{true}, iff \ccVar\ is degenerate, i.e.~if \ccVar\
is empty or equal to a single point.}
\ccIndexSubitem[t]{degeneracies}{\ccc{Min_annulus_d}}
\lcTex{\ccIndexSubitem[t]{degeneracies}{\ccFont Min_annulus_d}}
% -----------------------------------------------------------------------------
\ccModifiers
@ -312,7 +312,7 @@ unbounded side equals the whole space $\E_d$.
% -----------------------------------------------------------------------------
\ccHeading{Validity Check}
\ccIndexMemberFunctionGroup{validity check}
\ccIndexSubitem[t]{validity check}{\ccc{Min_annulus_d}}
\lcTex{\ccIndexSubitem[t]{validity check}{\ccFont Min_annulus_d}}
An object \ccVar\ is valid, iff
\begin{itemize}
@ -380,7 +380,7 @@ validity check.
The problem of finding the smallest enclosing annulus of a finite point set
can be formulated as an optimization problem with linear constraints and a
linear objective
function\ccIndexSubitem[t]{linear program}{\ccc{Min_annulus_d}}.
function\lcTex{\ccIndexSubitem[t]{linear program}{\ccFont Min_annulus_d}}.
The solution is obtained using our exact
solver for linear and quadratic programs~\cite{gs-eegqp-00}.

4
Packages/Min_circle_2/changes.txt
View File

@ -1,4 +1,8 @@
Changes done to the `Min_circle_2' package
Version 3.23 on 16-Sep-2003
- Undo 17-Jun change, but protect the indexing commands by \lcTex.
Version 3.22 on 10-Sep-2003
- Qt_help_window class is in the CGAL namespace [rursu]

22
Packages/Min_circle_2/doc_tex/Optimisation_ref/Min_circle_2.tex
View File

@ -30,13 +30,13 @@ An object of the class \ccRefName\ is the unique circle of smallest area
enclosing a finite (multi)set of points in two-dimensional Euclidean
space~$\E_2$. For a point set $P$ we denote by $mc(P)$ the smallest circle
that contains all points of $P$. Note that $mc(P)$ can be
degenerate\ccIndexSubitem[t]{degeneracies}{\ccc{Min_circle_2}},
degenerate\lcTex{\ccIndexSubitem[t]{degeneracies}{\ccFont Min_circle_2}},
i.e.~$mc(P)=\mbox{\ccTexHtml{$\;\emptyset$}{Ø}}$ if
$P=\mbox{\ccTexHtml{$\;\emptyset$}{Ø}}$ and $mc(P)=\{p\}$ if
$P=\{p\}$.
An inclusion-minimal subset $S$ of $P$ with $mc(S)=mc(P)$ is called a
\emph{support set}\ccIndexSubitem[t]{support set}{\ccc{Min_circle_2}},
\emph{support set}\lcTex{\ccIndexSubitem[t]{support set}{\ccFont Min_circle_2}},
the points in $S$ are the \emph{support points}. A support set has size at
most three, and all its points lie on the boundary of $mc(P)$. In general,
neither the support set nor its size are necessarily unique.
@ -86,7 +86,7 @@ We provide the model \ccc{Min_circle_2_traits_2} using the two-dimensional
A \ccRefName\ object can be created from an arbitrary point set $P$ and by
specialized construction methods expecting no, one, two or three points as
arguments. The latter methods can be useful for reconstructing $mc(P)$ from
a given support~set\ccIndexSubitem[t]{support set}{\ccc{Min_circle_2}}
a given support~set\lcTex{\ccIndexSubitem[t]{support set}{\ccFont Min_circle_2}}
$S$ of $P$.
\ccConstructor{ template < class InputIterator >
@ -138,7 +138,7 @@ $S$ of $P$.
\ccMemberFunction{ int number_of_support_points( ) const;}{
returns the number of support points of \ccVar, i.e.\ $|S|$.}
\ccIndexSubitem[t]{support set}{\ccc{Min_circle_2}}
\lcTex{\ccIndexSubitem[t]{support set}{\ccFont Min_circle_2}}
\ccGlueBegin
\ccMemberFunction{ Point_iterator points_begin() const;}{
@ -198,7 +198,7 @@ bounded side, i.e.\ its unbounded side equals the whole space $\E_2$.
returns \ccc{true}, iff \ccVar\ is degenerate,
i.e.~if \ccVar\ is empty or equal to a single point, equivalently
if the number of support points is less than 2.}
\ccIndexSubitem[t]{degeneracies}{\ccc{Min_circle_2}}
\lcTex{\ccIndexSubitem[t]{degeneracies}{\ccFont Min_circle_2}}
% -----------------------------------------------------------------------------
\ccModifiers
@ -230,7 +230,7 @@ method is incremental itself.
% -----------------------------------------------------------------------------
\ccHeading{Validity Check}
\ccIndexMemberFunctionGroup{validity check}
\ccIndexSubitem[t]{validity check}{\ccc{Min_circle_2}}
\lcTex{\ccIndexSubitem[t]{validity check}{\ccFont Min_circle_2}}
An object \ccVar\ is valid, iff
\begin{itemize}
@ -303,13 +303,13 @@ An object \ccVar\ is valid, iff
\ccImplementation
\ccIndexImplementation
We implement the incremental algorithm\ccIndexSubitem[t]{incremental
algorithm}{\ccc{Min_circle_2}} of Welzl, with move-to-front
heuristic~\cite{w-sedbe-91a}\ccIndexSubitem[t]{move-to-front
heuristic}{\ccc{Min_circle_2}}. The whole implementation is described
We implement the incremental algorithm\lcTex{\ccIndexSubitem[t]{incremental
algorithm}{\ccFont Min_circle_2}} of Welzl, with move-to-front
heuristic~\cite{w-sedbe-91a}\lcTex{\ccIndexSubitem[t]{move-to-front
heuristic}{\ccFont Min_circle_2}}. The whole implementation is described
in~\cite{gs-seceg-98}.
If randomization\ccIndexSubitem[t]{randomization}{\ccc{Min_circle_2}} is
If randomization\lcTex{\ccIndexSubitem[t]{randomization}{\ccFont Min_circle_2}} is
chosen, the creation time is almost always linear in the number of points.
Access functions and predicates take constant time, inserting a point might
take up to linear time, but substantially less than computing the new

22
Packages/Min_circle_2/doc_tex/basic/Optimisation_ref/Min_circle_2.tex
View File

@ -30,13 +30,13 @@ An object of the class \ccRefName\ is the unique circle of smallest area
enclosing a finite (multi)set of points in two-dimensional Euclidean
space~$\E_2$. For a point set $P$ we denote by $mc(P)$ the smallest circle
that contains all points of $P$. Note that $mc(P)$ can be
degenerate\ccIndexSubitem[t]{degeneracies}{\ccc{Min_circle_2}},
degenerate\lcTex{\ccIndexSubitem[t]{degeneracies}{\ccFont Min_circle_2}},
i.e.~$mc(P)=\mbox{\ccTexHtml{$\;\emptyset$}{&Oslash;}}$ if
$P=\mbox{\ccTexHtml{$\;\emptyset$}{&Oslash;}}$ and $mc(P)=\{p\}$ if
$P=\{p\}$.
An inclusion-minimal subset $S$ of $P$ with $mc(S)=mc(P)$ is called a
\emph{support set}\ccIndexSubitem[t]{support set}{\ccc{Min_circle_2}},
\emph{support set}\lcTex{\ccIndexSubitem[t]{support set}{\ccFont Min_circle_2}},
the points in $S$ are the \emph{support points}. A support set has size at
most three, and all its points lie on the boundary of $mc(P)$. In general,
neither the support set nor its size are necessarily unique.
@ -86,7 +86,7 @@ We provide the model \ccc{Min_circle_2_traits_2} using the two-dimensional
A \ccRefName\ object can be created from an arbitrary point set $P$ and by
specialized construction methods expecting no, one, two or three points as
arguments. The latter methods can be useful for reconstructing $mc(P)$ from
a given support~set\ccIndexSubitem[t]{support set}{\ccc{Min_circle_2}}
a given support~set\lcTex{\ccIndexSubitem[t]{support set}{\ccFont Min_circle_2}}
$S$ of $P$.
\ccConstructor{ template < class InputIterator >
@ -138,7 +138,7 @@ $S$ of $P$.
\ccMemberFunction{ int number_of_support_points( ) const;}{
returns the number of support points of \ccVar, i.e.\ $|S|$.}
\ccIndexSubitem[t]{support set}{\ccc{Min_circle_2}}
\lcTex{\ccIndexSubitem[t]{support set}{\ccFont Min_circle_2}}
\ccGlueBegin
\ccMemberFunction{ Point_iterator points_begin() const;}{
@ -198,7 +198,7 @@ bounded side, i.e.\ its unbounded side equals the whole space $\E_2$.
returns \ccc{true}, iff \ccVar\ is degenerate,
i.e.~if \ccVar\ is empty or equal to a single point, equivalently
if the number of support points is less than 2.}
\ccIndexSubitem[t]{degeneracies}{\ccc{Min_circle_2}}
\lcTex{\ccIndexSubitem[t]{degeneracies}{\ccFont Min_circle_2}}
% -----------------------------------------------------------------------------
\ccModifiers
@ -230,7 +230,7 @@ method is incremental itself.
% -----------------------------------------------------------------------------
\ccHeading{Validity Check}
\ccIndexMemberFunctionGroup{validity check}
\ccIndexSubitem[t]{validity check}{\ccc{Min_circle_2}}
\lcTex{\ccIndexSubitem[t]{validity check}{\ccFont Min_circle_2}}
An object \ccVar\ is valid, iff
\begin{itemize}
@ -303,13 +303,13 @@ An object \ccVar\ is valid, iff
\ccImplementation
\ccIndexImplementation
We implement the incremental algorithm\ccIndexSubitem[t]{incremental
algorithm}{\ccc{Min_circle_2}} of Welzl, with move-to-front
heuristic~\cite{w-sedbe-91a}\ccIndexSubitem[t]{move-to-front
heuristic}{\ccc{Min_circle_2}}. The whole implementation is described
We implement the incremental algorithm\lcTex{\ccIndexSubitem[t]{incremental
algorithm}{\ccFont Min_circle_2}} of Welzl, with move-to-front
heuristic~\cite{w-sedbe-91a}\lcTex{\ccIndexSubitem[t]{move-to-front
heuristic}{\ccFont Min_circle_2}}. The whole implementation is described
in~\cite{gs-seceg-98}.
If randomization\ccIndexSubitem[t]{randomization}{\ccc{Min_circle_2}} is
If randomization\lcTex{\ccIndexSubitem[t]{randomization}{\ccFont Min_circle_2}} is
chosen, the creation time is almost always linear in the number of points.
Access functions and predicates take constant time, inserting a point might
take up to linear time, but substantially less than computing the new

3
Packages/Min_ellipse_2/changes.txt
View File

@ -1,5 +1,8 @@
Changes done to the `Min_ellipse_2' package.
Version 3.28 on 16-Sep-2003
- Undo 17-Jun change, but protect the indexing commands by \lcTex.
Version 3.27 on 10-Sep-2003 [rursu]
- Qt_help_window is in the CGAL namespace now.

30
Packages/Min_ellipse_2/doc_tex/Optimisation_ref/Min_ellipse_2.tex
View File

@ -29,14 +29,14 @@ An object of the class \ccRefName\ is the unique ellipse of smallest area
enclosing a finite (multi)set of points in two-dimensional euclidean
space~$\E_2$. For a point set $P$ we denote by $me(P)$ the smallest
ellipse that contains all points of $P$. Note that $me(P)$ can be
degenerate\ccIndexSubitem[t]{degeneracies}{\ccc{Min_ellipse_2}},
degenerate\lcTex{\ccIndexSubitem[t]{degeneracies}{\ccFont Min_ellipse_2}},
i.e.~$me(P)=\mbox{\ccTexHtml{$\;\emptyset$}{&Oslash;}}$ if
$P=\mbox{\ccTexHtml{$\;\emptyset$}{&Oslash;}}$, $me(P)=\{p\}$ if $P=\{p\}$,
and $me(P) = \{ \mbox{\ccTexHtml{$(1-\lambda)p + \lambda q \mid 0 \leq
\lambda \leq 1$}{(1-l)p + l q | 0 <= l <= 1}} \}$ if $P=\{p,q\}$.
An inclusion-minimal subset $S$ of $P$ with $me(S)=me(P)$ is called a
\emph{support set}\ccIndexSubitem[t]{support set}{\ccc{Min_ellipse_2}},
\emph{support set}\lcTex{\ccIndexSubitem[t]{support set}{\ccFont Min_ellipse_2}},
the points in $S$ are the \emph{support points}. A support set has size at
most five, and all its points lie on the boundary of $me(P)$. In general,
neither the support set nor its size are necessarily unique.
@ -86,8 +86,8 @@ We provide the model \ccc{Min_ellipse_2_traits_2} using the two-dimensional
A \ccRefName\ object can be created from an arbitrary point set $P$ and by
specialized construction methods expecting no, one, two, three, four or
five points as arguments. The latter methods can be useful for
reconstructing $me(P)$ from a given support set\ccIndexSubitem[t]{support
set}{\ccc{Min_ellipse_2}} $S$ of $P$.
reconstructing $me(P)$ from a given support set\lcTex{\ccIndexSubitem[t]{support
set}{\ccFont Min_ellipse_2}} $S$ of $P$.
\ccConstructor{ template < class InputIterator >
Min_Ellipse_2( InputIterator first,
@ -158,7 +158,7 @@ reconstructing $me(P)$ from a given support set\ccIndexSubitem[t]{support
\ccMemberFunction{ int number_of_support_points( ) const;}{
returns the number of support points of \ccVar, i.e.\ $|S|$.}
\ccIndexSubitem[t]{support set}{\ccc{Min_ellipse_2}}
\lcTex{\ccIndexSubitem[t]{support set}{\ccFont Min_ellipse_2}}
\ccGlueBegin
\ccMemberFunction{ Point_iterator points_begin() const;}{
@ -171,11 +171,11 @@ reconstructing $me(P)$ from a given support set\ccIndexSubitem[t]{support
\ccGlueBegin
\ccMemberFunction{ Support_point_iterator support_points_begin() const;}{
returns an iterator referring to the first support point of \ccVar.}
\ccIndexSubitem[t]{support set}{\ccc{Min_ellipse_2}}
\lcTex{\ccIndexSubitem[t]{support set}{\ccFont Min_ellipse_2}}
%
\ccMemberFunction{ Support_point_iterator support_points_end() const;}{
returns the corresponding past-the-end iterator.}
\ccIndexSubitem[t]{support set}{\ccc{Min_ellipse_2}}
\lcTex{\ccIndexSubitem[t]{support set}{\ccFont Min_ellipse_2}}
\ccGlueEnd
\ccMemberFunction{ const Point& support_point( int i) const;}{
@ -184,7 +184,7 @@ reconstructing $me(P)$ from a given support set\ccIndexSubitem[t]{support
\ccVar\ccc{.support_point(i)} with the same \ccc{i} returns
the same point.
\ccPrecond $0 \leq i< \mbox{\ccVar\ccc{.number_of_support_points()}}$.}
\ccIndexSubitem[t]{support set}{\ccc{Min_ellipse_2}}
\lcTex{\ccIndexSubitem[t]{support set}{\ccFont Min_ellipse_2}}
\ccMemberFunction{ const Ellipse& ellipse( ) const;}{
returns the current ellipse of \ccVar.}
@ -222,7 +222,7 @@ i.e.\ its unbounded side equals the whole space $\E_2$.
i.e.~if \ccVar\ is empty, equal to a single point or equal to a
segment, equivalently if the number of support points is less
than~3.}
\ccIndexSubitem[t]{degeneracies}{\ccc{Min_ellipse_2}}
\lcTex{\ccIndexSubitem[t]{degeneracies}{\ccFont Min_ellipse_2}}
% -----------------------------------------------------------------------------
\ccModifiers
@ -254,7 +254,7 @@ method is incremental itself.
% -----------------------------------------------------------------------------
\ccHeading{Validity Check}
\ccIndexMemberFunctionGroup{validity check}
\ccIndexSubitem[t]{validity check}{\ccc{Min_ellipse_2}}
\lcTex{\ccIndexSubitem[t]{validity check}{\ccFont Min_ellipse_2}}
An object \ccVar\ is valid, iff
\begin{itemize}
@ -329,14 +329,14 @@ validity check.
\ccImplementation
\ccIndexImplementation
We implement the incremental algorithm\ccIndexSubitem[t]{incremental
algorithm}{\ccc{Min_ellipse_2}} of Welzl, with move-to-front
heuristic~\cite{w-sedbe-91a}\ccIndexSubitem[t]{move-to-front
heuristic}{\ccc{Min_ellipse_2}}, using the primitives as described
We implement the incremental algorithm\lcTex{\ccIndexSubitem[t]{incremental
algorithm}{\ccFont Min_ellipse_2}} of Welzl, with move-to-front
heuristic~\cite{w-sedbe-91a}\lcTex{\ccIndexSubitem[t]{move-to-front
heuristic}{\ccFont Min_ellipse_2}}, using the primitives as described
in~\cite{gs-epsee-97,gs-seefe-97a}. The whole implementation is described
in~\cite{gs-seeeg-98}.
If randomization\ccIndexSubitem[t]{randomization}{\ccc{Min_ellipse_2}} is
If randomization\lcTex{\ccIndexSubitem[t]{randomization}{\ccFont Min_ellipse_2}} is
chosen, the creation time is almost always linear in the number of points.
Access functions and predicates take constant time, inserting a point might
take up to linear time, but substantially less than computing the new

30
Packages/Min_ellipse_2/doc_tex/basic/Optimisation_ref/Min_ellipse_2.tex
View File

@ -29,14 +29,14 @@ An object of the class \ccRefName\ is the unique ellipse of smallest area
enclosing a finite (multi)set of points in two-dimensional euclidean
space~$\E_2$. For a point set $P$ we denote by $me(P)$ the smallest
ellipse that contains all points of $P$. Note that $me(P)$ can be
degenerate\ccIndexSubitem[t]{degeneracies}{\ccc{Min_ellipse_2}},
degenerate\lcTex{\ccIndexSubitem[t]{degeneracies}{\ccFont Min_ellipse_2}},
i.e.~$me(P)=\mbox{\ccTexHtml{$\;\emptyset$}{&Oslash;}}$ if
$P=\mbox{\ccTexHtml{$\;\emptyset$}{&Oslash;}}$, $me(P)=\{p\}$ if $P=\{p\}$,
and $me(P) = \{ \mbox{\ccTexHtml{$(1-\lambda)p + \lambda q \mid 0 \leq
\lambda \leq 1$}{(1-l)p + l q | 0 <= l <= 1}} \}$ if $P=\{p,q\}$.
An inclusion-minimal subset $S$ of $P$ with $me(S)=me(P)$ is called a
\emph{support set}\ccIndexSubitem[t]{support set}{\ccc{Min_ellipse_2}},
\emph{support set}\lcTex{\ccIndexSubitem[t]{support set}{\ccFont Min_ellipse_2}},
the points in $S$ are the \emph{support points}. A support set has size at
most five, and all its points lie on the boundary of $me(P)$. In general,
neither the support set nor its size are necessarily unique.
@ -86,8 +86,8 @@ We provide the model \ccc{Min_ellipse_2_traits_2} using the two-dimensional
A \ccRefName\ object can be created from an arbitrary point set $P$ and by
specialized construction methods expecting no, one, two, three, four or
five points as arguments. The latter methods can be useful for
reconstructing $me(P)$ from a given support set\ccIndexSubitem[t]{support
set}{\ccc{Min_ellipse_2}} $S$ of $P$.
reconstructing $me(P)$ from a given support set\lcTex{\ccIndexSubitem[t]{support
set}{\ccFont Min_ellipse_2}} $S$ of $P$.
\ccConstructor{ template < class InputIterator >
Min_Ellipse_2( InputIterator first,
@ -158,7 +158,7 @@ reconstructing $me(P)$ from a given support set\ccIndexSubitem[t]{support
\ccMemberFunction{ int number_of_support_points( ) const;}{
returns the number of support points of \ccVar, i.e.\ $|S|$.}
\ccIndexSubitem[t]{support set}{\ccc{Min_ellipse_2}}
\lcTex{\ccIndexSubitem[t]{support set}{\ccFont Min_ellipse_2}}
\ccGlueBegin
\ccMemberFunction{ Point_iterator points_begin() const;}{
@ -171,11 +171,11 @@ reconstructing $me(P)$ from a given support set\ccIndexSubitem[t]{support
\ccGlueBegin
\ccMemberFunction{ Support_point_iterator support_points_begin() const;}{
returns an iterator referring to the first support point of \ccVar.}
\ccIndexSubitem[t]{support set}{\ccc{Min_ellipse_2}}
\lcTex{\ccIndexSubitem[t]{support set}{\ccFont Min_ellipse_2}}
%
\ccMemberFunction{ Support_point_iterator support_points_end() const;}{
returns the corresponding past-the-end iterator.}
\ccIndexSubitem[t]{support set}{\ccc{Min_ellipse_2}}
\lcTex{\ccIndexSubitem[t]{support set}{\ccFont Min_ellipse_2}}
\ccGlueEnd
\ccMemberFunction{ const Point& support_point( int i) const;}{
@ -184,7 +184,7 @@ reconstructing $me(P)$ from a given support set\ccIndexSubitem[t]{support
\ccVar\ccc{.support_point(i)} with the same \ccc{i} returns
the same point.
\ccPrecond $0 \leq i< \mbox{\ccVar\ccc{.number_of_support_points()}}$.}
\ccIndexSubitem[t]{support set}{\ccc{Min_ellipse_2}}
\lcTex{\ccIndexSubitem[t]{support set}{\ccFont Min_ellipse_2}}
\ccMemberFunction{ const Ellipse& ellipse( ) const;}{
returns the current ellipse of \ccVar.}
@ -222,7 +222,7 @@ i.e.\ its unbounded side equals the whole space $\E_2$.
i.e.~if \ccVar\ is empty, equal to a single point or equal to a
segment, equivalently if the number of support points is less
than~3.}
\ccIndexSubitem[t]{degeneracies}{\ccc{Min_ellipse_2}}
\lcTex{\ccIndexSubitem[t]{degeneracies}{\ccFont Min_ellipse_2}}
% -----------------------------------------------------------------------------
\ccModifiers
@ -254,7 +254,7 @@ method is incremental itself.
% -----------------------------------------------------------------------------
\ccHeading{Validity Check}
\ccIndexMemberFunctionGroup{validity check}
\ccIndexSubitem[t]{validity check}{\ccc{Min_ellipse_2}}
\lcTex{\ccIndexSubitem[t]{validity check}{\ccFont Min_ellipse_2}}
An object \ccVar\ is valid, iff
\begin{itemize}
@ -329,14 +329,14 @@ validity check.
\ccImplementation
\ccIndexImplementation
We implement the incremental algorithm\ccIndexSubitem[t]{incremental
algorithm}{\ccc{Min_ellipse_2}} of Welzl, with move-to-front
heuristic~\cite{w-sedbe-91a}\ccIndexSubitem[t]{move-to-front
heuristic}{\ccc{Min_ellipse_2}}, using the primitives as described
We implement the incremental algorithm\lcTex{\ccIndexSubitem[t]{incremental
algorithm}{\ccFont Min_ellipse_2}} of Welzl, with move-to-front
heuristic~\cite{w-sedbe-91a}\lcTex{\ccIndexSubitem[t]{move-to-front
heuristic}{\ccFont Min_ellipse_2}}, using the primitives as described
in~\cite{gs-epsee-97,gs-seefe-97a}. The whole implementation is described
in~\cite{gs-seeeg-98}.
If randomization\ccIndexSubitem[t]{randomization}{\ccc{Min_ellipse_2}} is
If randomization\lcTex{\ccIndexSubitem[t]{randomization}{\ccFont Min_ellipse_2}} is
chosen, the creation time is almost always linear in the number of points.
Access functions and predicates take constant time, inserting a point might
take up to linear time, but substantially less than computing the new

3
Packages/Min_sphere_d/changes.txt
View File

@ -124,3 +124,6 @@ Changes from 2.32
Changes from 2.33
- Replace \ccFont by \ccc for the HTML index.
Changes from 2.34
- Undo last change, but protect the indexing commands by \lcTex.

10
Packages/Min_sphere_d/doc_tex/Optimisation_ref/Min_sphere_d.tex
View File

@ -120,7 +120,7 @@ or clear operation.
\begin{ccIndexMemberFunctions}
\ccIndexMemberFunctionGroup{access}
\ccIndexSubitem[t]{support set}{\ccc{Min_sphere_d}}
\lcTex{\ccIndexSubitem[t]{support set}{\ccFont Min_sphere_d}}
\ccMemberFunction{ int number_of_points( ) const;}{
returns the number of points of \ccVar, i.e.\ $|P|$.}
@ -190,7 +190,7 @@ bounded side, i.e.\ its unbounded side equals the whole space $\E_d$.
returns \ccc{true}, iff \ccVar\ is empty (this implies
degeneracy).}
\ccIndexSubitem[t]{degeneracies}{\ccc{Min_sphere_d}}
\lcTex{\ccIndexSubitem[t]{degeneracies}{\ccFont Min_sphere_d}}
\ccMemberFunction{ bool is_degenerate( ) const;}{
returns \ccc{true}, iff \ccVar\ is degenerate, i.e.\ if
\ccVar\ is empty or equal to a single point, equivalently if
@ -237,7 +237,7 @@ bounded side, i.e.\ its unbounded side equals the whole space $\E_d$.
% -----------------------------------------------------------------------------
\ccHeading{Validity Check}
\ccIndexMemberFunctionGroup{validity check}
\ccIndexSubitem[t]{validity check}{\ccc{Min_sphere_d}}
\lcTex{\ccIndexSubitem[t]{validity check}{\ccFont Min_sphere_d}}
An object \ccVar\ is valid, iff
\begin{itemize}
\item \ccVar\ contains all points of its defining set $P$,
@ -302,8 +302,8 @@ bounded side, i.e.\ its unbounded side equals the whole space $\E_d$.
\ccImplementation
\ccIndexImplementation
\ccIndexSubitem[t]{incremental algorithm}{\ccc{Min_sphere_d}}
\ccIndexSubitem[t]{move-to-front heuristic}{\ccc{Min_sphere_d}}
\lcTex{\ccIndexSubitem[t]{incremental algorithm}{\ccFont Min_sphere_d}}
\lcTex{\ccIndexSubitem[t]{move-to-front heuristic}{\ccFont Min_sphere_d}}
We implement the algorithm of Welzl with move-to-front
heuristic~\cite{w-sedbe-91a} for small point sets, combined with a new
efficient method for large sets, which is particularly tuned for

10
Packages/Min_sphere_d/doc_tex/basic/Optimisation_ref/Min_sphere_d.tex
View File

@ -120,7 +120,7 @@ or clear operation.
\begin{ccIndexMemberFunctions}
\ccIndexMemberFunctionGroup{access}
\ccIndexSubitem[t]{support set}{\ccc{Min_sphere_d}}
\lcTex{\ccIndexSubitem[t]{support set}{\ccFont Min_sphere_d}}
\ccMemberFunction{ int number_of_points( ) const;}{
returns the number of points of \ccVar, i.e.\ $|P|$.}
@ -190,7 +190,7 @@ bounded side, i.e.\ its unbounded side equals the whole space $\E_d$.
returns \ccc{true}, iff \ccVar\ is empty (this implies
degeneracy).}
\ccIndexSubitem[t]{degeneracies}{\ccc{Min_sphere_d}}
\lcTex{\ccIndexSubitem[t]{degeneracies}{\ccFont Min_sphere_d}}
\ccMemberFunction{ bool is_degenerate( ) const;}{
returns \ccc{true}, iff \ccVar\ is degenerate, i.e.\ if
\ccVar\ is empty or equal to a single point, equivalently if
@ -237,7 +237,7 @@ bounded side, i.e.\ its unbounded side equals the whole space $\E_d$.
% -----------------------------------------------------------------------------
\ccHeading{Validity Check}
\ccIndexMemberFunctionGroup{validity check}
\ccIndexSubitem[t]{validity check}{\ccc{Min_sphere_d}}
\lcTex{\ccIndexSubitem[t]{validity check}{\ccFont Min_sphere_d}}
An object \ccVar\ is valid, iff
\begin{itemize}
\item \ccVar\ contains all points of its defining set $P$,
@ -302,8 +302,8 @@ bounded side, i.e.\ its unbounded side equals the whole space $\E_d$.
\ccImplementation
\ccIndexImplementation
\ccIndexSubitem[t]{incremental algorithm}{\ccc{Min_sphere_d}}
\ccIndexSubitem[t]{move-to-front heuristic}{\ccc{Min_sphere_d}}
\lcTex{\ccIndexSubitem[t]{incremental algorithm}{\ccFont Min_sphere_d}}
\lcTex{\ccIndexSubitem[t]{move-to-front heuristic}{\ccFont Min_sphere_d}}
We implement the algorithm of Welzl with move-to-front
heuristic~\cite{w-sedbe-91a} for small point sets, combined with a new
efficient method for large sets, which is particularly tuned for