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Completed the User Manual.
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Ron Wein
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@ -35,13 +35,13 @@ also known as \emph{offsetting} or \emph{dilating} a polygon).
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\label{mink_sec:sum_poly}
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%====================================================
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Computing the Minkowski sum of two convex polygons $P$ ans $Q$ with
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Computing the Minkowski sum of two convex polygons $P$ and $Q$ with
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$m$ and $n$ vertices respectively is very easy, as $P \oplus Q$ is a
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convex polygon bounded by copies of the $m + n$ edges, and these edges
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are sorted by the angle they form with the $x$-axis. As the two
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input polygons are convex, their edges are already sorted by the
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angle they form with the $x$-axis. The Minkowski sum can therefore be
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computed in $O(m + n)$ time, by starting from two botommost vertices
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computed in $O(m + n)$ time, by starting from two bottommost vertices
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in $P$ and in $Q$ and performing ``merge sort'' on the edges.
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\begin{figure}[t]
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@ -78,7 +78,7 @@ We then calculate the pairwise sums $S_{ij} = P_i \oplus Q_j$ using the
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simple procedure described above, and compute the union $P \oplus Q =
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\bigcup_{ij}{S_{ij}}$.
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This approach relies on a decoposition strategy that computes the convex
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This approach relies on a decomposition strategy that computes the convex
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decomposition of the input polygons and its performance depends on the
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quality of the decomposition.
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%
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@ -91,7 +91,7 @@ order around their interiors) and compute the convolution of the two polygon
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boundaries. The {\em convolution} of these two polygons~\cite{grs-kfcg-83},
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denoted $P * Q$, is a collection of line segments of the form
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$[p_i + q_j, p_{i+1} + q_j]$,\footnote{Throughout this chapter, we increment
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or decrement an index of a vertex modulu the size of the polygon.}
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or decrement an index of a vertex modulo the size of the polygon.}
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where the vector $\overrightarrow{p_i p_{i+1}}$
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lies between $\overrightarrow{q_{j-1} q_j}$ and $\overrightarrow{q_j
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q_{j+1}}$,\footnote{We say that a vector $\vec{v}$ lies between
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@ -115,7 +115,7 @@ for an illustration.
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The number of segments in the convolution of two polygons is usually
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smaller than the number of segments that constitute the boundaries of the
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sub-sums $S_{ij}$ when using the decomposition approach. As both apporaches
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sub-sums $S_{ij}$ when using the decomposition approach. As both approaches
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construct the arrangement of these segments and extract the sum from this
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arrangement, computing Minkowski sum using the convolution approach usually
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generates a smaller intermediate arrangement, hence it is faster and
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@ -126,7 +126,7 @@ consumes less space.
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\label{mink_ssec:sum_conv}
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%------------------------------------------------------
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The function \ccc{minkowski_sum (P, Q)} accepts two simple polygons $P$
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The function \ccc{minkowski_sum_2 (P, Q)} accepts two simple polygons $P$
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and $Q$, represented using the \ccc{Polygon_2<Kernel,Container>}
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class-template and uses the convolution method in order to compute and
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return their Minkowski sum $S = P \oplus Q$.
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@ -156,12 +156,12 @@ in the example program \ccc{ex_sum_triangles.C}.}
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\label{mink_fig:sum_tri}
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\end{figure}
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The following example program constructs the Minkwoski sum of two triangles,
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The following example program constructs the Minkowski sum of two triangles,
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as depicted in Figure~\ref{mink_fig:sum_tri}. The result in this case is
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a convex hexagon. This program, as other example porgrams in this chapter,
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a convex hexagon. This program, as other example programs in this chapter,
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includes the auxiliary header file \ccc{ms_rational_nt.h} which defines
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\ccc{Number_type} as either \ccc{Gmpq} or \ccc{Quotient<MP_Float>},
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depending on whther the {\sc Gmp} library is installed or not.
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depending on whether the {\sc Gmp} library is installed or not.
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The file \ccc{print_util.h} includes auxiliary functions for printing polygons.
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\ccIncludeExampleCode{../examples/Minkowski_sum_2/ex_sum_triangles.C}
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@ -187,7 +187,7 @@ In the following program we compute the Minkowski sum of two polygons
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that are read from an input file. In this case, the sum is not simple
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and contains four holes, as illustrated in Figure~\ref{mink_fig:sum_holes}.
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Note that this example uses the predefined \cgal\ kernel with exact
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constrcutions. In general, instantiating polygons with this kernel yield
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constructions. In general, instantiating polygons with this kernel yield
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the fastest running times for Minkowski-sum computations.
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\ccIncludeExampleCode{../examples/Minkowski_sum_2/ex_sum_with_holes.C}
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@ -196,7 +196,7 @@ the fastest running times for Minkowski-sum computations.
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\label{mink_ssec:decomp}
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%------------------------------------
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In order to compute Minkwoski sums using the decomposition method, it is
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In order to compute Minkowski sums using the decomposition method, it is
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possible to call the function \ccc{minkowski_sum_2 (P, Q, decomp)}, where
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\ccc{decomp} is an instance of a class that models the concept
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\ccc{PolygonConvexDecomposition}, namely it should provide a method named
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@ -207,8 +207,8 @@ The Minkowski-sum package includes for models of the concept
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\ccc{PolygonConvexDecomposition}. The first three are classes that wrap
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the decomposition functions included in the Planar Polygon Partitioning
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package, while the fourth is an implementation of a decomposition algorithm
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introduced in~\cite{???}. The convex decompositions that it creates
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usually yield efficient running times for Minkowski sum computations:
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introduced in~\cite{cgal:afh-pdecm-02}. The convex decompositions that it
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creates usually yield efficient running times for Minkowski sum computations:
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\begin{itemize}
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\item
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The class \ccc{Optimal_convex_decomposition<Kernel>} uses the
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@ -236,7 +236,7 @@ and $O(n)$ space, and has the same approximation guarantee as
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Hertel and Mehlhorn's algorithm.
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%
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\item
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The class \ccc{Small_side_angle_bisecttor_convex_decomposition<Kernel>} uses
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The class \ccc{Small_side_angle_bisector_convex_decomposition<Kernel>} uses
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a heuristic improvement to the angle-bisector decomposition method
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suggested by Chazelle and Dobkin~\cite{cd-ocd-85}, which runs in
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$O(n^2)$ time. It starts by examining each pair of reflex vertices
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@ -314,7 +314,7 @@ sub-polygon and finally calculating the union of these sub-offsets
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the Minkowski sum of a pair of polygons, here it is also more
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efficient to compute the \emph{convolution cycle} of the polygon
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with the disc $B_r$,\footnote{As the disc is convex, it is guaranteed
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that the convolution curve is comrised of a single cycle.} which can be
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that the convolution curve is comprised of a single cycle.} which can be
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constructed by applying the process described in the previous
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paragraph. The only difference is that a circular arc induced by a
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reflex vertex $p_i$ is defined by an angle $540^{\circ} - \measuredangle
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@ -323,29 +323,43 @@ illustration. We finally compute the winding numbers of the faces of the
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arrangement induced by the convolution cycle to obtain the offset
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polygon.
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\begin{figure}[t]
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\begin{ccTexOnly}
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\begin{center}
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\includegraphics{Minkowski_sum_2/fig/ex_offset}
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\end{center}
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\end{ccTexOnly}
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\begin{ccHtmlOnly}
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<p><center>
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<img src="./fig/ex_offset.gif" border=0 alt="Offsetting a polygon">
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</center>
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\end{ccHtmlOnly}
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\caption{The offset computation performed by the example programs
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\ccc{ex_exact_offset.C} and \ccc{ex_approx_offset.C}. The input polygon
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is shaded and the boundary of its offset is drawn in a thick black line.}
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\label{mink_fig:ex_offset}
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\end{figure}
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\ccIncludeExampleCode{../examples/Minkowski_sum_2/ex_exact_offset.C}
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\subsection{Approximating the Offset with a Guaranteed Error-Bound}
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\label{mink_ssec:approx_offset}
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%------------------------------------------------------------------
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Let us assume that we are given a counterclockwise-oriented polygon
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$P = \left( p_0, \ldots, p_{n-1} \right)$, whose vertices all have rational
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coordinates (i.e., for each vertex $p_i = (x_i, y_i)$ we have
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$x_i, y_i \in {\mathbb Q}$, and we wish to compute its Minkowski sum with a
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disc of radius $r$, where $r$ is also a rational number. The boundary of this
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sum is comprised of line segments an circular arcs, where:
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\begin{itemize}
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\item
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Each circular arc is supported by a circle of radius $r$ centered at one
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of the polygon vertices. The equation of this circle is $(x - x_i)^2 +
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(y - y_i)^2 = r^2$, and has only rational coefficients.
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%
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\item
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Each line segment is supported by a line parallel to one of the polygon
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edges $p_i p_{i+1}$, which lies at distance $r$ from this edge. If we
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denote the supporting line of $p_i p_{i+1}$ by $Ax + By + C = 0$, where
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$A, B, C \in {\mathbb Q}$, then the offset edge is supported by the line
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$Ax + By + (C + \ell\cdot r) = 0$, where $\ell$ is the length of the edge
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$p_i p_{i+1}$ and is in general \emph{not} a rational number. The line segments
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that comprise the offset boundaries therefore cannot be represented as segments
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of lines with rational coefficients.
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%
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We mention that the locus of all points that lie at distance $r$ from the
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line $Ax + By + C = 0$ is given by:
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\[ \frac{(Ax + By + C)^2}{A^2 + B^2} = r^2 \ .\]
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Thus, the linear offset edges are segments of curves of an algebraic curve
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of degree 2 (a conic curve) with rational coefficients. This curve is
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actually a pair of the parallel lines $Ax + By + (C \pm \ell\cdot r) = 0$.
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In Section~\ref{mink_ssec:exact_offset} we use this representation to
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construct the offset polygon in an exact manner using the traits class for
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conic arcs.
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\end{itemize}
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\begin{figure}[t]
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\begin{ccTexOnly}
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\begin{center}
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@ -362,5 +376,112 @@ edge $p_1 p_2$ by two line segments $o'_1 q'$ and $q' o'_2$.}
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\label{mink_fig:approx_offset}
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\end{figure}
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The class-template \ccc{Gps_circle_segment_traits_2<Kernel>}, included in
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the Boolean Set-Operations package is used for representing general polygons
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whose edges are circular arcs or line segments, and for applying set operations
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(e.g. intersection, union, etc.) on such general polygon. It should be
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instantiated with a geometric kernel that employs exact rational arithmetic,
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such that the curves that comprise the polygon edges should be arcs of
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circles with rational coefficients or segments of lines with rational
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coefficients. As in our case the line segments do not satisfy this requirement,
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we apply a simple approximation scheme, such that each irrational line
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segment is approximated by two rational segments:
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\begin{enumerate}
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\item
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Consider the example depicted in Figure~\ref{mink_fig:approx_offset}, where
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the exact offset edge $o_1 o_2$ is obtained by shifting the polygon edge
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$p_1 p_2$ by a vector whose length is $r$ that form an angle $\phi$ with the
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$x$-axis. We select two points $o'_1$ and $o'_2$ with rational coordinates
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on the two circles centered at $p_1$ and $p_2$, respectively. These points
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are selected such that if we denote the angle that the vector $\vec{p_j o_j}$
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forms with the $x$-axis by $\phi'_j$ (for $j = 1, 2$), we have
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$\phi'_1 < \phi < \phi'_2$.
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%
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\item
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We construct two tangents to the two circles at $o'_1$ and $o'_2$,
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respectively. The tangent lines have rational coefficients.
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%
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\item
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We compute the intersection point of the two tangents, denoted $q'$.
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The two line segments $o'_1 q'$ and $q' o'_2$ approximate the original
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offset edge $o_1 o_2$.
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\end{enumerate}
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The function \ccc{approximated_offset_2 (P, r, eps)} accepts a polygon
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$P$, an offset radius $r$ and $\varepsilon > 0$. It constructs an
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approximation for the offset of $P$ by the radius $r$ using the procedure
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explained above. Furthermore, it is guaranteed that the approximation error,
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namely the distance of the point $q'$ from $o_1 o_2$ is bounded by
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$\varepsilon$. Using this function, we are capable of working with the
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\ccc{Gps_circle_segment_traits_2<Kernel>} class, which considerably
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speeds up the (approximate) construction of the offset polygon and the
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application of set operations on such polygons. The function returns an
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object of the nested type
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\ccc{Gps_circle_segment_traits_2<Kernel>::Polygon_with_holes_2} representing
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the approximated offset polygon (recall that if $P$ is not convex, its
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offset may not be simple an contain holes, whose boundary is also comprised
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of line segments and circular arcs).
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\begin{figure}[t]
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\begin{ccTexOnly}
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\begin{center}
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\includegraphics{Minkowski_sum_2/fig/ex_offset}
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\end{center}
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\end{ccTexOnly}
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\begin{ccHtmlOnly}
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<p><center>
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<img src="./fig/ex_offset.gif" border=0 alt="Offsetting a polygon">
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</center>
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\end{ccHtmlOnly}
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\caption{The offset computation performed by the example programs
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\ccc{ex_approx_offset.C} and \ccc{ex_exact_offset.C}. The input polygon
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is shaded and the boundary of its offset is drawn in a thick black line.}
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\label{mink_fig:ex_offset}
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\end{figure}
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The following example demonstrates the construction of an approximated
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offset of a non-convex polygon, as depicted in
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Figure~\ref{mink_fig:ex_offset}. Note that we use a geometric kernel
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parameterized with a filtered rational number-type. Using filtering
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considerably speeds up the construction of the offset.
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\ccIncludeExampleCode{../examples/Minkowski_sum_2/ex_approx_offset.C}
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\subsection{Computing the Exact Offset}
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\label{mink_ssec:exact_offset}
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%--------------------------------------
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As we previously mentioned, it is possible to represent offset polygons
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in an exact manner, if we treat their edges as arcs of conic curves with
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rational coefficients. The function \ccc{offset_polygon_2 (traits, P, r)}
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computes the offset of the polygon $P$ by a rational radius $r$ in an
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exact manner. The \ccc{traits} parameter should be an instance of an
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arrangement-traits class that is capable of handling conic arcs in an
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exact manner; using the \ccc{Arr_conic_traits_2} class with the number
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types provided by the {\sc Core} library is the preferred option.
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The function returns an object of the nested type
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\ccc{Gps_traits_2<ArrConicTraits>::Polygons_with_holes_2} (see the
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documentation of the Boolean Set-Operations package for more details
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on the traits-class adapter \ccc{Gps_traits_2}), which represented the
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exact offset polygon.
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The following example demonstrates the construction of the offset
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of the same polygon that serves as an input for the example program
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\ccc{ex_approx_offset.C}, presented in the previous subsection (see also
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Figure~\ref{mink_fig:ex_offset}). Note that the resulting polygon is
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smaller than the one generated by the approximated-offset function (recall
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that each irrational line segment in this case is approximated by two
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rational line segments), but the offset computation is considerably slower:
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\ccIncludeExampleCode{../examples/Minkowski_sum_2/ex_exact_offset.C}
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\begin{ccAdvanced}
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Both functions \ccc{approximated_offset_2()} and \ccc{offset_polygon_2()}
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also have an overloaded versions that accept a decomposition strategy
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and use the polygon-decomposition approach to compute (or approximate)
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the offset. These functions are less efficient than their counterparts
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that employ the convolution approach, and are only included in the package
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for the sake of completeness.
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\end{ccAdvanced}
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