Some fixes (thanks Michael H.)

.gitattributes

@@ -616,6 +616,8 @@ Envelope_2/doc_tex/Envelope_2/fig/ex_circle.eps -text svneol=unset#application/p
 Envelope_2/doc_tex/Envelope_2/fig/ex_circle.fig -text svneol=unset#application/octet-stream
 Envelope_2/doc_tex/Envelope_2/fig/ex_circle.gif -text svneol=unset#image/gif
 Envelope_2/doc_tex/Envelope_2/fig/ex_circle.pdf -text svneol=unset#application/pdf
+Envelope_2/doc_tex/Envelope_2/fig/lwrenv.eps -text svneol=unset#application/postscript
+Envelope_2/doc_tex/Envelope_2/fig/lwrenv.pdf -text svneol=unset#application/pdf
 Envelope_2/doc_tex/Envelope_2/fig/min_diag.eps -text svneol=unset#application/postscript
 Envelope_2/doc_tex/Envelope_2/fig/min_diag.fig -text svneol=unset#application/octet-stream
 Envelope_2/doc_tex/Envelope_2/fig/min_diag.gif -text svneol=unset#image/gif

Envelope_2/doc_tex/Envelope_2/envelope.tex

@@ -1,3 +1,19 @@
+\begin{figure}[!htp]
+\begin{center}
+\begin{ccTexOnly}
+  \includegraphics{Envelope_2/fig/lwrenv}
+\end{ccTexOnly}
+\label{fig:teaser}
+\begin{ccHtmlOnly}
+  <p><center>
+    <img src="./fig/lwrenv.gif" border=0 alt="Lower Envelope">
+  </center>
+\end{ccHtmlOnly}
+\caption{The lower envelope of a set of line segments and hyperbolic
+arc.} 
+\end{center}
+\end{figure}
+
 
 A continuous curve $C$ in $\reals^2$ is called {\em $x$-monotone}, if
 every vertical line intersects it at a single point at most. For
@@ -5,6 +21,8 @@ example, the circle $x^2 + y^2 = 1$ is {\em not} $xy$-monotone as the
 vertical line $x = 0$ intersects it at $(0, -1)$ and at $(0, 1)$;
 however, it is possible to split the circle into an upper part and a
 lower part, such that both of these parts are $x$-monotone.
+We consider vertical segments as {\em weakly} $x$-monotone, to properly
+handle inputs that contain such vertical curves.
 
 An $x$-monotone curve can be represented as a univariate function
 $y = C(x)$, defined over some continuous range $R_C \subseteq \reals$.
@@ -45,7 +63,7 @@ Given a set of $x$-monotone curves $\calC$, the {\em minimization
 diagram} of $\calC$ is a subdivision of the $x$-axis into cells,
 such that the identity of the curves that induce the lower envelope
 over a specific cell of the subdivision (an edge or a vertex) is the
-same. In non-degenerate situation, an edge --- which represents a
+same. In non-degenerate situations, an edge --- which represents a
 continuous interval on the $x$-axis --- is induced by a single
 curve (or by no curves at all, if there are no $x$-monotone curves
 defined over the interval), and a vertex is induced by a single curve
@@ -62,11 +80,9 @@ the boundary of its range of definition $R_{C_k}$ onto the $x$-axis
 and label the features it induces accordingly. Given a set
 $\hat{\calC}$ of (non necessarily $x$-monotone) curves in $\reals^2$,
 we subdivide each curve into a finite number of weakly $x$-monotone 
-curves,\footnote{We consider vertical segments as {\em weakly} 
+curves, and obtain the set $\calC$. Then, we split the set into two
-$x$-monotone, to handle degenerate inputs properly.} and obtain the set
+disjoint subsets $\calC_1$ and $\calC_2$, and we compute their envelope
-$\calC$. Then, we split the set into two disjoint subsets $\calC_1$ 
+diagrams recursively. Finally, we merge the diagrams in linear time by
-and $\calC_2$, and we compute their envelope diagrams recursively. 
-Finally, we merge the diagrams, and we do this in linear time 
 traversing both diagrams in parallel.
 
 \section{The Envelope Diagram}
@@ -75,11 +91,11 @@ traversing both diagrams in parallel.
 
 The package basically contains two sets of free functions:
 \ccc{lower_envelope_x_monotone_2 (begin, end, diag)} (similarly
- \ccc{upper_envelope_x_monotone_2()}) accepts a range of $x$-monotone
+ \ccc{upper_envelope_x_monotone_2()}) construct the envelope diagram
-curves and computes the envelope diagram; 
+for a given range of $x$-monotone curves, while 
 \ccc{lower_envelope_2 (begin, end, diag)} (similarly
-\ccc{upper_envelope_2()}) accepts a range of {\em arbitrary} (not
+\ccc{upper_envelope_2()}) construct the envelope diagram for a
-necessarily $x$-monotone) curves and computes the envelope diagram.
+range of {\em arbitrary} (not necessarily $x$-monotone) curves.
 In this section we explain more on the structure of the envelope
 diagram these functions output.
 
@@ -100,7 +116,7 @@ The minimization diagram is shown at the bottom, where
 each diagram vertex points to the point associated with it, and the
 labels of the segment that induce a diagram edge are displayed below
 this edge. Note that there exists one edge that represents an overlap
-(even if more than a single curve induces it), and there are also a 
+(i.e., there are two segments that induce it), and there are also a 
 few edges that represent empty intervals.\label{env2_fig:min_diag}}
 \end{figure}
 
@@ -116,12 +132,12 @@ Figure~\ref{env2_fig:min_diag} shows the lower envelope of a set of
 eight line segments, and sketches the structure of their minimization
 diagram. Each diagram vertex $v$ is associated with a point $p_v$ on
 the envelope, which corresponds to either a curve endpoint
-or to an intersection point of two curves (or more). The vertex is
+or to an intersection point of two (or more) curves. The vertex is
 therefore associated with a set of $x$-monotone curves that induce the
 envelope over $p_v$. Each vertex is incident to two edges, one lying
 to its left and the other to its right.
 
-An edge in the envelope diagram represent a continuous portion of the
+An edge in the envelope diagram represents a continuous portion of the
 $x$-axis, and is associated with a set of $x$-monotone curves that
 induce the envelope over this interval. Note that this set may be
 empty if no $x$-monotone curves are defined over this interval. In
@@ -144,12 +160,15 @@ Any model of the \ccRefName\ concept must define a geometric
 traits class, which in turn defines the \ccc{Point_2} and 
 \ccc{X_monotone_curve_2} types defined with the diagram features.
 The geometric traits class must be a model of the 
-\ccc{ArrangementXMonotoneTraits_2} concept in case we compute
+\ccc{ArrangementXMonotoneTraits_2} concept in case we construct
 envelopes of $x$-monotone curves. If we are interested in handling
 arbitrary (not necessarily $x$-monotone) curves, the traits class
-must be a model of the refined \ccc{ArrangementTraits_2} concept.
+must be a model of the \ccc{ArrangementTraits_2} concept. This
-In the latter case, the traits class also defines a \ccc{Curve_2}
+concepts refined the \ccc{ArrangementXMonotoneTraits_2} concept;
-type.
+a traits class that models this concepts must also defines a
+\ccc{Curve_2} type, representing an arbitrary planar curve, and
+provide a functor for subdividing such curves into $x$-monotone
+subcurves.
 
 \section{Examples}
 %=================

Envelope_2/doc_tex/Envelope_2_ref/EnvDiagEdge.tex

@@ -16,11 +16,13 @@
 An edge record in an envelope diagram, which represents a continuous portion
 of the $x$-axis. It is associated with a (possibly empty) set of curves that
 induce the envelope over this portion of the $x$-axis. Note that all curves
-in this set overlap of the interval represented by the edge.
+in this set overlap over the interval represented by the edge.
 
 \ccTypes
 %=======
 
+\ccNestedType{Size}{the size type (convertible to \ccc{size_t}).}
+
 \ccNestedType{Vertex}{the corresponding diagram-vertex type.}
 
 \ccNestedType{X_monotone_curve_2}{the $x$-monotone curve type.} 
@@ -55,10 +57,12 @@ in this set overlap of the interval represented by the edge.
     {return a past-the-end iterator for the $x$-monotone curves associated with \ccVar.}
 
 \ccMethod{Vertex_const_handle left() const;}
-    {returns the vertex lying to \ccVar's left.}
+    {returns the vertex lying to \ccVar's left.
-\ccGlue
+     \ccPrecond{\ccVar\ is not the leftmost edge in the diagram.}}
+
 \ccMethod{Vertex_const_handle right() const;}
-    {returns the vertex lying to \ccVar's right.}
+    {returns the vertex lying to \ccVar's right.
+     \ccPrecond{\ccVar\ is not the rightmost edge in the diagram.}}
 
 \ccModifiers
 %===========

Envelope_2/doc_tex/Envelope_2_ref/EnvDiagVertex.tex

@@ -14,7 +14,7 @@
 %============
 
 A vertex record in an envelope diagram. It is always associated with a point
-on the lower (upper) envelope of a set of curves. A vertex is also
+on the lower (upper) envelope of a non-empty set of curves. A vertex is also
 associated with a set of $x$-monotone curves that induce the envelope
 over this point. It is incident to two edges, one lying to its
 left and the other to its right.
@@ -22,6 +22,8 @@ left and the other to its right.
 \ccTypes
 %=======
 
+\ccNestedType{Size}{the size type (convertible to \ccc{size_t}).}
+
 \ccNestedType{Edge}{the corresponding diagram-edge type.}
 
 \ccNestedType{Point_2}{the point type associated with the vertex.}

Envelope_2/doc_tex/Envelope_2_ref/Env_default_diagram.tex

@@ -15,11 +15,14 @@
 
 The default envelope-diagram class used by the envelops functions to represent
 the lower or the upper envelope of a set of curves. It is parameterized by a
-traits class, which is a model of the \ccc{ArrangementXMontoneTraits_2}
+traits class, which is a model of the \ccc{ArrangementXMonotoneTraits_2}
 concept, in case we handle only envelopes of $x$-monotone curves, or of the
 refined \ccc{ArrangementTraits_2} concept in case we handle arbitrary planar
 curves.
 
+The space needed by this envelope-diagram class is linear in the size of
+the minimization diagram, and traversing requires linear time.
+ 
 \ccInclude{CGAL/Env_default_diagram_1.h}
 
 \ccIsModel

Envelope_2/doc_tex/Envelope_2_ref/EnvelopeDiagram.tex

@@ -13,16 +13,16 @@
 \ccDefinition
 %============
 
-This concept defines the representation of an evelope diagram of a set
+This concept defines the representation of an envelope diagram of a set
 of planar curve. The {\em envelope diagram} is a subdivision of the $x$-axis
 into $0$-dimensional cells ({\em vertices}) and $1$-dimensional cells
 ({\em edges}), such that the identity of the curves that induce the lower
 envelope (or the upper envelope) over each cell is fixed.
 
 A vertex in an envelope diagram is therefore associated with a point
-on the lower (upper) envelope, and corresponds to either a curve endpoint
+on the envelope, and corresponds to either a curve endpoint
-or to an intersection point of two curves (or more). The vertex is also
+or to an intersection point of two (or more) curves. Therefore each vertex
-associated with a set of $x$-monotone curves that induce the envelope
+is associated with a set of $x$-monotone curves that induce the envelope
 over this point. Each vertex is incident to two edges, one lying to its
 left and the other to its right.
 
@@ -51,7 +51,7 @@ traits class, which in turn defines the \ccc{Point_2} and
 \ccTypedef{typedef Traits_2::X_monotone_curve_2 X_monotone_curve_2;}
 {the $x$-monotone curve type.}
 
-\ccNestedType{Size}{the size type.}
+\ccNestedType{Size}{the size type (convertible to \ccc{size_t}).}
 
 \ccNestedType{Curve_const_iterator}
 {an iterator for the $x$-monotone curves that induce a diagram feature.
@@ -62,20 +62,25 @@ traits class, which in turn defines the \ccc{Point_2} and
 \ccNestedType{Edge}{the edge type, a model of the concept \ccc{EnvelopeDiagramEdge}.}
 
 \ccNestedType{Vertex_handle}
-{a handle to a diagram vertex.
+{a handle to a diagram vertex.}
- The \ccc{Vertex_const_handle} type is also defined.}
 \ccGlue
+\ccNestedType{Vertex_const_handle}
+{a non-mutable handle to a diagram vertex.}
+
 \ccNestedType{Edge_handle}
-{a handle to a diagram edge.
+{a handle to a diagram edge.}
- The \ccc{Edge_const_handle} type is also defined.}
+\ccGlue
+\ccNestedType{Edge_const_handle}
+{a non-mutable handle to a diagram edge.}
 
 \ccCreation
 \ccCreationVariable{diag}
 %========================
 
 \ccConstructor{EnvelopeDiagram_1();} 
-    {constructs an empty diagram containing one unbounded face,
+    {constructs an empty diagram containing one unbounded edge,
-     which corresponds to the entire plane and has no originators.}
+     which corresponds to the entire plane and has no $x$-monotone
+     curves that are associated with it.}
     
 \ccConstructor{Envelope_diagram_1 (const Self& other);}
     {copy constructor.}

Envelope_2/doc_tex/Envelope_2_ref/intro.tex

@@ -11,7 +11,7 @@
 \label{env2_ref_sec:intro}
 % ========================
 
-This package consits of functions that compute the lower (or upper)
+This package consists of functions that compute the lower (or upper)
 envelope of a set of arbitrary curves in 2D. The output is
 represented as an envelope diagram, namely a subdivision of the
 $x$-axis into intervals, such that the identity of the curves that

Envelope_2/doc_tex/Envelope_2_ref/lower_envelope_x_monotone_2.tex

@@ -21,7 +21,7 @@
     represented using the output minimization diagram \ccc{diag},
     which must be a model of the \ccc{EnvelopeDiagram_1} concept.
     \ccPrecond{The value-type of \ccc{InputIterator} is
-               \ccc{EnvelopeDiagram::Traits_2::X_monotone_curve_2}.}}
+               \ccc{EnvelopeDiagram::X_monotone_curve_2}.}}
 
 \end{ccRefFunction}
 

Envelope_2/doc_tex/Envelope_2_ref/upper_envelope_x_monotone_2.tex

@@ -21,7 +21,7 @@
     represented using the output maximization diagram \ccc{diag},
     which must be a model of the \ccc{EnvelopeDiagram_1} concept.
     \ccPrecond{The value-type of \ccc{InputIterator} is
-               \ccc{EnvelopeDiagram::Traits_2::X_monotone_curve_2}.}}
+               \ccc{EnvelopeDiagram::X_monotone_curve_2}.}}
 
 \end{ccRefFunction}
 

Envelope_2/examples/Envelope_2/ex_convex_hull.cpp

@@ -38,7 +38,7 @@ int main (int argc, char **argv)
     return (1);
   }
 
-  // Read the points from the file, and consturct their dual lines.
+  // Read the points from the file, and construct their dual lines.
   unsigned int            n;
   int                     px, py;
   std::vector<Point_2>    points;
@@ -48,7 +48,7 @@ int main (int argc, char **argv)
 
   in_file >> n;
   points.resize (n);
-  for (k = 0; k < n; k++)
+  for (k = 0; k < n; ++k)
   {
     in_file >> px >> py;
     points[k] = Point_2 (px, py);
@@ -74,9 +74,9 @@ int main (int argc, char **argv)
   upper_envelope_x_monotone_2 (dual_lines.begin(), dual_lines.end(),
                                max_diag);
 
-  // Output the points along the boundary convex hull in a counterclockwise
+  // Output the points along the boundary convex hull in counterclockwise
   // order. We start by traversing the minimization diagram from left to
-  // right, then the maximization diagram from left to right.
+  // right, then the maximization diagram from right to left.
   Diagram_1::Edge_const_handle  e = min_diag.leftmost();
 
   std::cout << "The convex hull of " << points.size() << " input points:";
@@ -87,12 +87,12 @@ int main (int argc, char **argv)
     e = e->right()->right();
   }
 
-  e = max_diag.leftmost();
+  e = max_diag.rightmost();
-  while (e != max_diag.rightmost())
+  while (e != max_diag.leftmost())
   {
     k = e->curve().data();    // The index of the dual point.
     std::cout << " (" << points[k] << ')';
-    e = e->right()->right();
+    e = e->left()->left();
   }
   std::cout << std::endl;