mirror of https://github.com/CGAL/cgal
fixed figures and replaced latex macros not supported by the html converter
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Efi Fogel
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@ -23,7 +23,7 @@
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% ====================
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Given two sets $A,B \in \reals^d$, their \emph{Minkowski sum},
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denoted by $A \oplus B$, is the set $\left\{ a + b ~\vert~ a \in
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denoted by $A \oplus B$, is the set $\left\{ a + b ~|~ a \in
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A, b \in B \right\}$. Minkowski sum are used in many applications,
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such as motion planning and computer-aided design and
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manufacturing. This package contains functions for computing planar
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@ -45,19 +45,20 @@ 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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\begin{ccTexOnly}
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\begin{center}
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\begin{tabular}{c c}
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\begin{ccTexOnly}
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\begin{center}
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\begin{tabular}{c c}
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\includegraphics{Minkowski_sum_2/fig/onecyc_in} ~&~
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\includegraphics{Minkowski_sum_2/fig/onecyc_out} \\
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\end{tabular}
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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/conv_onecycle.gif" border=0 alt="Convolution cycle">
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</center>
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\end{ccHtmlOnly}
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\includegraphics{Minkowski_sum_2/fig/onecyc_out}
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\end{tabular}
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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/onecyc_in.gif" border=0 alt="Convolution cycle">
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<img src="./fig/onecyc_out.gif" border=0 alt="Convolution cycle">
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</center>
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\end{ccHtmlOnly}
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\caption{Computing the convolution of a convex polygon and a
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non-convex polygon (left). The convolution consists of a single
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self-intersecting cycle, drawn as a sequence of arrows (right).
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@ -92,16 +93,16 @@ 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 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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two vectors $\vec{u}$ and $\vec{w}$ if we reach $\vec{v}$ strictly
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before reaching $\vec{w}$ if we move all three vectors to the origin
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and rotate $\vec{u}$ counterclockwise. Note that this also covers
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the case where $\vec{u}$ has the same direction as $\vec{v}$.} and
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where the vector ${\mathbf p_i p_{i+1}}$
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lies between ${\mathbf q_{j-1} q_j}$ and ${\mathbf q_j
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q_{j+1}}$,\footnote{We say that a vector ${\mathbf v}$ lies between
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two vectors ${\mathbf u}$ and ${\mathbf w}$ if we reach ${\mathbf v}$ strictly
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before reaching ${\mathbf w}$ if we move all three vectors to the origin
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and rotate ${\mathbf u}$ counterclockwise. Note that this also covers
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the case where ${\mathbf u}$ has the same direction as ${\mathbf v}$.} and
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--- symmetrically --- of segments of the form $[p_i + q_j, p_i + q_{j+1}]$,
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where the vector $\overrightarrow{q_j q_{j+1}}$ lies between
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$\overrightarrow{p_{i-1} p_i}$ and $\overrightarrow{p_i p_{i+1}}$.
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where the vector ${\mathbf q_j q_{j+1}}$ lies between
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${\mathbf p_{i-1} p_i}$ and ${\mathbf p_i p_{i+1}}$.
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The segments of the convolution form a number of closed (not
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necessarily simple) polygonal curves called \emph{convolution
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@ -143,7 +144,7 @@ contains \ccc{S.number_of_holes()} holes in its interior).
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\begin{figure}[t]
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\begin{ccTexOnly}
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\begin{center}
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\input{Minkowski_sum_2/fig/sum_triangles.pstex_t}
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\includegraphics{Minkowski_sum_2/fig/sum_triangles}
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\end{center}
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\end{ccTexOnly}
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\begin{ccHtmlOnly}
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@ -169,7 +170,7 @@ The file \ccc{print_util.h} includes auxiliary functions for printing polygons.
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\begin{figure}[t]
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\begin{ccTexOnly}
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\begin{center}
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\input{Minkowski_sum_2/fig/tight.pstex_t}
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\includegraphics{Minkowski_sum_2/fig/tight}
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\end{center}
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\end{ccTexOnly}
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\begin{ccHtmlOnly}
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@ -280,7 +281,9 @@ is widely known as \emph{offsetting} the polygon $P$ by a radius $r$.
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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/offset.gif" border=0 alt="Computing offsets of polygons">
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<img src="./fig/convex_offset.gif" border=0 alt="Computing offsets of polygons">
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<img src="./fig/offset_decomp.gif" border=0 alt="Computing offsets of polygons">
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<img src="./fig/offset_conv.gif" border=0 alt="Computing offsets of polygons">
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</center>
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\end{ccHtmlOnly}
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\caption{(a)~Offsetting a convex polygon.
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@ -301,7 +304,7 @@ $n$ disconnected \emph{offset edges}. Each pair of adjacent offset
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edges, induced by $p_{i-1} p_i$ and $p_i p_{i+1}$, are connected
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by a circular arc of radius $r$, whose supporting circle is
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centered at $p_i$. The angle that defines such a circular arc
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equals $180^{\circ} - \measuredangle p_{i-1} p_i p_{i+1}$; see
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equals $180^{\circ} - \angle p_{i-1} p_i p_{i+1}$; see
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Figure~\ref{mink_fig:pgn_offset}(a) for an illustration. The running
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time of this simple process is of course linear with respect to
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the size of the polygon.
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@ -317,7 +320,7 @@ with the disc $B_r$,\footnote{As the disc is convex, it is guaranteed
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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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reflex vertex $p_i$ is defined by an angle $540^{\circ} - \angle
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p_{i-1} p_i p_{i+1}$; see Figure~\ref{fig:pgn_offset}(c) for an
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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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@ -363,12 +366,12 @@ conic arcs.
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\begin{figure}[t]
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\begin{ccTexOnly}
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\begin{center}
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\input{Minkowski_sum_2/fig/approx_offset.pstex_t}
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\includegraphics{Minkowski_sum_2/fig/approx_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/tight.gif" border=0 alt="Approximating an offset edge">
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<img src="./fig/approx_offset.gif" border=0 alt="Approximating an offset edge">
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</center>
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\end{ccHtmlOnly}
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\caption{Approximating the offset edge $o_1 o_2$ induced by the polygon
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@ -393,7 +396,8 @@ 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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are selected such that if we denote the angle that the vector
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${\mathbf p_j o_j}$
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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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@ -424,9 +428,9 @@ 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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\begin{center}
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\includegraphics{Minkowski_sum_2/fig/ex_offset}
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\end{center}
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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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