From 04b5f686c5786c6bcd0bdd054fc3532cf2357a86 Mon Sep 17 00:00:00 2001 From: Ron Wein Date: Wed, 8 Nov 2006 11:39:49 +0000 Subject: [PATCH] Some fixes (thanks Michael H.) --- .gitattributes | 2 + Envelope_2/doc_tex/Envelope_2/envelope.tex | 53 +- Envelope_2/doc_tex/Envelope_2/fig/lwrenv.eps | 8664 +++++++++++++++++ Envelope_2/doc_tex/Envelope_2/fig/lwrenv.pdf | Bin 0 -> 131445 bytes .../doc_tex/Envelope_2_ref/EnvDiagEdge.tex | 12 +- .../doc_tex/Envelope_2_ref/EnvDiagVertex.tex | 4 +- .../Envelope_2_ref/Env_default_diagram.tex | 5 +- .../Envelope_2_ref/EnvelopeDiagram.tex | 27 +- Envelope_2/doc_tex/Envelope_2_ref/intro.tex | 2 +- .../lower_envelope_x_monotone_2.tex | 2 +- .../upper_envelope_x_monotone_2.tex | 2 +- .../examples/Envelope_2/ex_convex_hull.cpp | 14 +- 12 files changed, 8743 insertions(+), 44 deletions(-) create mode 100755 Envelope_2/doc_tex/Envelope_2/fig/lwrenv.eps create mode 100644 Envelope_2/doc_tex/Envelope_2/fig/lwrenv.pdf diff --git a/.gitattributes b/.gitattributes index 92aa2dceba1..a338774e9f8 100644 --- a/.gitattributes +++ b/.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 diff --git a/Envelope_2/doc_tex/Envelope_2/envelope.tex b/Envelope_2/doc_tex/Envelope_2/envelope.tex index c89a48261e7..cb17aeef751 100644 --- a/Envelope_2/doc_tex/Envelope_2/envelope.tex +++ b/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} +

+ Lower Envelope +
+\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} -$x$-monotone, to handle degenerate inputs properly.} and obtain the set -$\calC$. Then, we split the set into two disjoint subsets $\calC_1$ -and $\calC_2$, and we compute their envelope diagrams recursively. -Finally, we merge the diagrams, and we do this in linear time +curves, and obtain the set $\calC$. Then, we split the set into two +disjoint subsets $\calC_1$ and $\calC_2$, and we compute their envelope +diagrams recursively. Finally, we merge the diagrams in linear time by 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 -curves and computes the envelope diagram; + \ccc{upper_envelope_x_monotone_2()}) construct 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 -necessarily $x$-monotone) curves and computes the envelope diagram. +\ccc{upper_envelope_2()}) construct the envelope diagram for a +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. -In the latter case, the traits class also defines a \ccc{Curve_2} -type. +must be a model of the \ccc{ArrangementTraits_2} concept. This +concepts refined the \ccc{ArrangementXMonotoneTraits_2} concept; +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} %================= diff --git a/Envelope_2/doc_tex/Envelope_2/fig/lwrenv.eps b/Envelope_2/doc_tex/Envelope_2/fig/lwrenv.eps new file mode 100755 index 00000000000..7e3e0ef2908 --- /dev/null +++ b/Envelope_2/doc_tex/Envelope_2/fig/lwrenv.eps @@ -0,0 +1,8664 @@ +%!PS-Adobe-2.0 EPSF-2.0 +%%Title: lwrenv.fig +%%Creator: fig2dev Version 3.2 Patchlevel 3d +%%CreationDate: Mon Jan 3 09:28:57 2005 +%%For: wein@cgal (Ron Wein,,09-576985,) +%%BoundingBox: 0 0 758 362 +%%Magnification: 1.0000 +%%EndComments +/$F2psDict 200 dict def +$F2psDict begin +$F2psDict /mtrx matrix put +/col-1 {0 setgray} bind def +/col0 {0.000 0.000 0.000 srgb} bind def +/col1 {0.000 0.000 1.000 srgb} bind def +/col2 {0.000 1.000 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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.} -\ccGlue + {returns the vertex lying to \ccVar's left. + \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 %=========== diff --git a/Envelope_2/doc_tex/Envelope_2_ref/EnvDiagVertex.tex b/Envelope_2/doc_tex/Envelope_2_ref/EnvDiagVertex.tex index f15cd2bc9ea..7ded88f8774 100644 --- a/Envelope_2/doc_tex/Envelope_2_ref/EnvDiagVertex.tex +++ b/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.} diff --git a/Envelope_2/doc_tex/Envelope_2_ref/Env_default_diagram.tex b/Envelope_2/doc_tex/Envelope_2_ref/Env_default_diagram.tex index 21917052542..308b7501ed7 100644 --- a/Envelope_2/doc_tex/Envelope_2_ref/Env_default_diagram.tex +++ b/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 diff --git a/Envelope_2/doc_tex/Envelope_2_ref/EnvelopeDiagram.tex b/Envelope_2/doc_tex/Envelope_2_ref/EnvelopeDiagram.tex index 49ad23bbee5..a97f914e589 100644 --- a/Envelope_2/doc_tex/Envelope_2_ref/EnvelopeDiagram.tex +++ b/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 -or to an intersection point of two curves (or more). The vertex is also -associated with a set of $x$-monotone curves that induce the envelope +on the envelope, and corresponds to either a curve endpoint +or to an intersection point of two (or more) curves. Therefore each vertex +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. - The \ccc{Vertex_const_handle} type is also defined.} +{a handle to a diagram vertex.} \ccGlue +\ccNestedType{Vertex_const_handle} +{a non-mutable handle to a diagram vertex.} + \ccNestedType{Edge_handle} -{a handle to a diagram edge. - The \ccc{Edge_const_handle} type is also defined.} +{a handle to a diagram edge.} +\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, - which corresponds to the entire plane and has no originators.} + {constructs an empty diagram containing one unbounded edge, + 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.} diff --git a/Envelope_2/doc_tex/Envelope_2_ref/intro.tex b/Envelope_2/doc_tex/Envelope_2_ref/intro.tex index 9a898636935..67c3dc03eba 100644 --- a/Envelope_2/doc_tex/Envelope_2_ref/intro.tex +++ b/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 diff --git a/Envelope_2/doc_tex/Envelope_2_ref/lower_envelope_x_monotone_2.tex b/Envelope_2/doc_tex/Envelope_2_ref/lower_envelope_x_monotone_2.tex index 8a272303d0b..3072afdf993 100644 --- a/Envelope_2/doc_tex/Envelope_2_ref/lower_envelope_x_monotone_2.tex +++ b/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} diff --git a/Envelope_2/doc_tex/Envelope_2_ref/upper_envelope_x_monotone_2.tex b/Envelope_2/doc_tex/Envelope_2_ref/upper_envelope_x_monotone_2.tex index fd126ab04df..c300bab52ee 100644 --- a/Envelope_2/doc_tex/Envelope_2_ref/upper_envelope_x_monotone_2.tex +++ b/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} diff --git a/Envelope_2/examples/Envelope_2/ex_convex_hull.cpp b/Envelope_2/examples/Envelope_2/ex_convex_hull.cpp index 3f6e9eaaaff..258fc47b240 100644 --- a/Envelope_2/examples/Envelope_2/ex_convex_hull.cpp +++ b/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 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(); - while (e != max_diag.rightmost()) + 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;