mirror of https://github.com/CGAL/cgal
added: $CGAL_Chapter $
fixed: bug in `CGAL_Min_ellipse_2<R>::is_degenerate()' fixed: some typos
This commit is contained in:
Sven Schönherr
parent
70af21ca19
commit
8dc295eac0
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@ -9,9 +9,6 @@
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@! $Date$
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@! ============================================================================
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@p maximum_input_line_length = 180
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@p maximum_output_line_length = 180
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@documentclass[twoside]{article}
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@usepackage[latin1]{inputenc}
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@usepackage{a4wide2}
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@ -110,7 +107,7 @@ same manner. In general, for point sets $P$,$B$, define $me(P,B)$ as the
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smallest ellipse enclosing $P$ that has the points of $B$ on the boundary
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(if defined). Although the algorithm finally delivers a ellipse
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$me(P,\emptyset)$, it internally deals with ellipses that have a possibly
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nonempty set $B$. Here is the pseudocode of Welzl's method. To compute
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nonempty set $B$. Here is the pseudo-code of Welzl's method. To compute
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$me(P)$, it is called with the pair $(P,\emptyset)$, assuming that
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$P=\{p_1,\ldots,p_n\}$ is stored in a linked list.
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@ -193,7 +190,7 @@ First, we declare the class template \ccc{CGAL_Min_ellipse_2}.
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The actual work of the algorithm is done in the private member
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functions \ccc{me} and \ccc{compute_ellipse}. The former directly
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realizes the pseudocode of $\me(P,B)$, the latter solves the basic
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realizes the pseudo-code of $\me(P,B)$, the latter solves the basic
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case $\me(\emptyset,B)$, see Section~\ref{sec:algo}.
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\emph{Workaround:} The GNU compiler (g++ 2.7.2[.?]) does not accept types
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@ -235,7 +232,7 @@ The class interface looks as follows.
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@<Min_ellipse_2 predicates>
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private:
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// Privat member functions
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// Private member functions
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// -----------------------
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@<Min_ellipse_2 private member function `compute_ellipse'>
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@ -373,7 +370,7 @@ sequence container \ccc{list} from STL~\cite{sl-stl-95}.
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The support set $S$ of at most five support points is stored in an
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array \ccc{support_points}, the actual number of support points is
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given by \ccc{n_support_points}. During the computations, the set of
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support points coincides with the set $B$ appearing in the pseudocode
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support points coincides with the set $B$ appearing in the pseudo-code
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for $\me(P,B)$, see Section~\ref{sec:algo}.
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\emph{Workaround:} The array of support points is allocated dynamically,
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@ -388,7 +385,7 @@ array here.
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Finally, the actual ellipse is stored in a variable \ccc{ellipse}
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provided by the traits class object, by the end of computation equal to
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$me(P)$. During computation, \ccc{tco.ellipse} equals the ellipse $me$
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appearing in the pseudocode for $\me(P,B)$, see Section~\ref{sec:algo}.
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appearing in the pseudo-code for $\me(P,B)$, see Section~\ref{sec:algo}.
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@! ----------------------------------------------------------------------------
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\subsubsection{Constructors and Destructor}
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@ -439,7 +436,7 @@ $me(P)=me(P,\emptyset)$.
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else
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copy( first, last, back_inserter( points)); }
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// compute mc
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// compute me
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me( points.end(), 0);
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}
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@ -776,7 +773,7 @@ define them \ccc{inline} and put them in a separate macro.
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bool
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is_degenerate( ) const
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{
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return( number_of_support_points() < 2);
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return( number_of_support_points() < 3);
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}
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@end
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@ -820,14 +817,14 @@ corresponding predicates of class \ccc{Ellipse}.
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There is another way to build up $me(P)$, other than by supplying
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the point set $P$ at once. Namely, $me(P)$ can be built up
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incrementally, adding one point after another. If you look at the
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pseudocode in the introduction, this comes quite naturally. The
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pseudo-code in the introduction, this comes quite naturally. The
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modifying method \ccc{insert}, applied with point $p$ to a
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\ccc{CGAL_Min_ellipse_2<Traits>} object representing $me(P)$,
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computes $me(P \cup \{p\})$, where work has to be done only if $p$
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lies outside $me(P)$. In this case, $me(P \cup \{p\}) = me(P,\{p\})$
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holds, so the private member function \ccc{me} is called with
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support set $\{p\}$. After the insertion has been performed, $p$ is
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moved to the front of the point list, just like in the pseudocode in
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moved to the front of the point list, just like in the pseudo-code in
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Section~\ref{sec:algo}.
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@macro <Min_ellipse_2 modifiers> += @begin
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@ -1146,8 +1143,8 @@ noting that $|B| \leq 5$.
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This function computes the general ellipse $me(P,B)$, where $P$ contains
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the points in the range $[$\ccc{points.begin()}$,$\ccc{last}$)$ and $B$
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is given by the first \ccc{n_sp} support points in the array
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\ccc{support_points}. The function is directly modelled after the
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pseudocode above.
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\ccc{support_points}. The function is directly modeled after the
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pseudo-code above.
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@macro <Min_ellipse_2 private member function `me'> = @begin
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void
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@ -1338,7 +1335,7 @@ boundary points. They all set the variable to the smallest ellipse
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through the given points. \emph{Note:} The set function taking five
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boundary points only uses the fifth point from its input together with
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the two internally represented conics to compute the ellipse. The
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algorithm in Section~\ref{sec:algo} garantees that this set function
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algorithm in Section~\ref{sec:algo} guarantees that this set function
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is only called an ellipse that already have the first four points as
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its boundary points.
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@ -1481,7 +1478,7 @@ its boundary points.
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\subsubsection{Predicates}
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The following predicates perform in-ellipse tests and check for
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emptyness and degeneracy, resp. The way to evaluate the in-ellipse
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emptiness and degeneracy, resp. The way to evaluate the in-ellipse
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test depends on the number of boundary points and is realised by a
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case analysis. Again, the case with four points is the most difficult
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one.
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@ -2584,9 +2581,10 @@ once to ensure code coverage.
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{
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Min_ellipse me( random_points[ 1],
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random_points[ 2]);
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bool is_valid = me.is_valid( verbose);
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bool is_valid = me.is_valid( verbose);
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bool is_degenerate = me.is_degenerate();
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assert( is_valid);
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assert( me.number_of_points() == 2);
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assert( is_degenerate);
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}
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verr << endl << "three points constructor...";
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@ -2595,8 +2593,9 @@ once to ensure code coverage.
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random_points[ 4],
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random_points[ 5]);
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bool is_valid = me.is_valid( verbose);
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int num_pts = me.number_of_points();
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assert( is_valid);
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assert( me.number_of_points() == 3);
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assert( num_pts == 3);
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}
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verr << endl << "four points constructor...";
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@ -2606,8 +2605,9 @@ once to ensure code coverage.
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random_points[ 8],
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random_points[ 9]);
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bool is_valid = me.is_valid( verbose);
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int num_pts = me.number_of_points();
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assert( is_valid);
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assert( me.number_of_points() == 4);
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assert( num_pts == 4);
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}
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verr << endl << "five points constructor...";
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@ -2618,8 +2618,9 @@ once to ensure code coverage.
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random_points[ 13],
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random_points[ 14]);
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bool is_valid = me.is_valid( verbose);
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int num_pts = me.number_of_points();
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assert( is_valid);
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assert( me.number_of_points() == 5);
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assert( num_pts == 5);
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}
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verr << endl << "Point* constructor...";
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@ -2959,7 +2960,9 @@ end of each file.
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\subsection{Min\_ellipse\_2.h}
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@file <include/CGAL/Min_ellipse_2.h> = @begin
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@<Min_ellipse_2 header>("include/CGAL/Min_ellipse_2.h")
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@<file header>(
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"include/CGAL/Min_ellipse_2.h",
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"2D Smallest Enclosing Ellipse")
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#ifndef CGAL_MIN_ELLIPSE_2_H
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#define CGAL_MIN_ELLIPSE_2_H
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@ -3017,7 +3020,9 @@ end of each file.
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\subsection{Min\_ellipse\_2.C}
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@file <include/CGAL/Min_ellipse_2.C> = @begin
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@<Min_ellipse_2 header>("include/CGAL/Min_ellipse_2.C")
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@<file header>(
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"include/CGAL/Min_ellipse_2.C",
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"2D Smallest Enclosing Ellipse")
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// Class implementation (continued)
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// ================================
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@ -3035,7 +3040,9 @@ end of each file.
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\subsection{Optimisation\_ellipse\_2.h}
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@file <include/CGAL/Optimisation_ellipse_2.h> = @begin
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@<Optimisation_ellipse_2 header>("include/CGAL/Optimisation_ellipse_2.h")
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@<file header>(
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"include/CGAL/Optimisation_ellipse_2.h",
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"2D Optimisation Ellipse")
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#ifndef CGAL_OPTIMISATION_ELLIPSE_2_H
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#define CGAL_OPTIMISATION_ELLIPSE_2_H
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@ -3087,7 +3094,9 @@ end of each file.
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\subsection{Optimisation\_ellipse\_2.C}
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@file <include/CGAL/Optimisation_ellipse_2.C> = @begin
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@<Optimisation_ellipse_2 header>("include/CGAL/Optimisation_ellipse_2.C")
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@<file header>(
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"include/CGAL/Optimisation_ellipse_2.C",
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"2D Optimisation Ellipse")
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// Class implementation (continued)
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// ================================
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@ -3110,7 +3119,9 @@ end of each file.
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\subsection{Min\_ellipse\_2\_traits\_2.h}
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@file <include/CGAL/Min_ellipse_2_traits_2.h> = @begin
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@<Min_ellipse_2 header>("include/CGAL/Min_ellipse_2_traits_2.h")
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@<file header>(
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"include/CGAL/Min_ellipse_2_traits_2.h",
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"default traits class for 2D Smallest Enclosing Ellipse")
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#ifndef CGAL_MIN_ELLIPSE_2_TRAITS_2_H
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#define CGAL_MIN_ELLIPSE_2_TRAITS_2_H
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@ -3143,7 +3154,9 @@ end of each file.
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\subsection{Min\_ellipse\_2\_adapterC2.h}
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@file <include/CGAL/Min_ellipse_2_adapterC2.h> = @begin
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@<Min_ellipse_2 header>("include/CGAL/Min_ellipse_2_adapterC2.h")
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@<file header>(
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"include/CGAL/Min_ellipse_2_adapterC2.h",
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"traits class adapter for 2D Smallest Enclosing Ellipse")
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#ifndef CGAL_MIN_ELLIPSE_2_ADAPTERC2_H
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#define CGAL_MIN_ELLIPSE_2_ADAPTERC2_H
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@ -3210,7 +3223,9 @@ end of each file.
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\subsection{Min\_ellipse\_2\_adapterH2.h}
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@file <include/CGAL/Min_ellipse_2_adapterH2.h> = @begin
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@<Min_ellipse_2 header>("include/CGAL/Min_ellipse_2_adapterH2.h")
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@<file header>(
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"include/CGAL/Min_ellipse_2_adapterH2.h",
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"traits class adapter for 2D Smallest Enclosing Ellipse")
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#ifndef CGAL_MIN_ELLIPSE_2_ADAPTERH2_H
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#define CGAL_MIN_ELLIPSE_2_ADAPTERH2_H
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@ -3281,7 +3296,9 @@ end of each file.
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\subsection{test\_Min\_ellipse\_2.C}
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@file <test/Optimisation/test_Min_ellipse_2.C> = @begin
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@<Min_ellipse_2 header>("test/Optimisation/test_Min_ellipse_2.C")
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@<file header>(
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"test/optimisation/test_Min_ellipse_2.C",
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"test program for 2D Smallest Enclosing Ellipse")
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@<Min_ellipse_2 test (includes and typedefs)>
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@ -3329,23 +3346,22 @@ end of each file.
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@i ../file_header.awi
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@macro <Min_ellipse_2 header>(1) many = @begin
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@<file header>("2D Smallest Enclosing Ellipse",@1,
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"Optimisation","Optimisation/Min_ellipse_2",
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"Sven Schönherr <sven@@inf.fu-berlin.de>",
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"Bernd Gärtner",
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"ETH Zurich (Bernd Gärtner <gaertner@@inf.ethz.ch>)",
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"$Revision$","$Date$")
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And here comes the specific file header for the product files of this
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web file.
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@macro <file header>(2) many = @begin
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@<copyright notice>
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@<file name>(@1)
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@<file description>(
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"Geometric Optimisation",
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"Optimisation","Optimisation/Min_ellipse_2",
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"$Revision$","$Date$",
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"Sven Schönherr <sven@@inf.fu-berlin.de>",
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"Bernd Gärtner",
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"ETH Zürich (Bernd Gärtner <gaertner@@inf.ethz.ch>)",
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"@2")
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@end
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@macro <Optimisation_ellipse_2 header>(1) many = @begin
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@<file header>("2D Optimisation Ellipse",@1,
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"Optimisation","Optimisation/Min_ellipse_2",
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"Sven Schönherr <sven@@inf.fu-berlin.de>",
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"Bernd Gärtner",
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"ETH Zurich (Bernd Gärtner <gaertner@@inf.ethz.ch>)",
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"$Revision$","$Date$")
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@end
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@! ============================================================================
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@! Bibliography
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@! ============================================================================
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