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fixed figures and replaced latex macros not supported by the html converter

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Efi Fogel 2006-03-15 11:41:42 +00:00
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commit f6383a5ae2
1 changed files with 36 additions and 32 deletions
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Minkowski_sum_2/doc_tex/Minkowski_sum_2/mink_sum.tex
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@ -23,7 +23,7 @@
% ==================== % ====================
Given two sets $A,B \in \reals^d$, their \emph{Minkowski sum}, Given two sets $A,B \in \reals^d$, their \emph{Minkowski sum},
denoted by $A \oplus B$, is the set $\left\{ a + b ~\vert~ a \in denoted by $A \oplus B$, is the set $\left\{ a + b ~|~ a \in
A, b \in B \right\}$. Minkowski sum are used in many applications, A, b \in B \right\}$. Minkowski sum are used in many applications,
such as motion planning and computer-aided design and such as motion planning and computer-aided design and
manufacturing. This package contains functions for computing planar manufacturing. This package contains functions for computing planar
@ -45,19 +45,20 @@ computed in $O(m + n)$ time, by starting from two bottommost vertices
in $P$ and in $Q$ and performing ``merge sort'' on the edges. in $P$ and in $Q$ and performing ``merge sort'' on the edges.
\begin{figure}[t] \begin{figure}[t]
\begin{ccTexOnly} \begin{ccTexOnly}
\begin{center} \begin{center}
\begin{tabular}{c c} \begin{tabular}{c c}
\includegraphics{Minkowski_sum_2/fig/onecyc_in} ~&~ \includegraphics{Minkowski_sum_2/fig/onecyc_in} ~&~
\includegraphics{Minkowski_sum_2/fig/onecyc_out} \\ \includegraphics{Minkowski_sum_2/fig/onecyc_out}
\end{tabular} \end{tabular}
\end{center} \end{center}
\end{ccTexOnly} \end{ccTexOnly}
\begin{ccHtmlOnly} \begin{ccHtmlOnly}
<p><center> <p><center>
<img src="./fig/conv_onecycle.gif" border=0 alt="Convolution cycle"> <img src="./fig/onecyc_in.gif" border=0 alt="Convolution cycle">
<img src="./fig/onecyc_out.gif" border=0 alt="Convolution cycle">
</center> </center>
\end{ccHtmlOnly} \end{ccHtmlOnly}
\caption{Computing the convolution of a convex polygon and a \caption{Computing the convolution of a convex polygon and a
non-convex polygon (left). The convolution consists of a single non-convex polygon (left). The convolution consists of a single
self-intersecting cycle, drawn as a sequence of arrows (right). self-intersecting cycle, drawn as a sequence of arrows (right).
@ -92,16 +93,16 @@ boundaries. The {\em convolution} of these two polygons~\cite{grs-kfcg-83},
denoted $P * Q$, is a collection of line segments of the form denoted $P * Q$, is a collection of line segments of the form
$[p_i + q_j, p_{i+1} + q_j]$,\footnote{Throughout this chapter, we increment $[p_i + q_j, p_{i+1} + q_j]$,\footnote{Throughout this chapter, we increment
or decrement an index of a vertex modulo the size of the polygon.} or decrement an index of a vertex modulo the size of the polygon.}
where the vector $\overrightarrow{p_i p_{i+1}}$ where the vector ${\mathbf p_i p_{i+1}}$
lies between $\overrightarrow{q_{j-1} q_j}$ and $\overrightarrow{q_j lies between ${\mathbf q_{j-1} q_j}$ and ${\mathbf q_j
q_{j+1}}$,\footnote{We say that a vector $\vec{v}$ lies between q_{j+1}}$,\footnote{We say that a vector ${\mathbf v}$ lies between
two vectors $\vec{u}$ and $\vec{w}$ if we reach $\vec{v}$ strictly two vectors ${\mathbf u}$ and ${\mathbf w}$ if we reach ${\mathbf v}$ strictly
before reaching $\vec{w}$ if we move all three vectors to the origin before reaching ${\mathbf w}$ if we move all three vectors to the origin
and rotate $\vec{u}$ counterclockwise. Note that this also covers and rotate ${\mathbf u}$ counterclockwise. Note that this also covers
the case where $\vec{u}$ has the same direction as $\vec{v}$.} and the case where ${\mathbf u}$ has the same direction as ${\mathbf v}$.} and
--- symmetrically --- of segments of the form $[p_i + q_j, p_i + q_{j+1}]$, --- symmetrically --- of segments of the form $[p_i + q_j, p_i + q_{j+1}]$,
where the vector $\overrightarrow{q_j q_{j+1}}$ lies between where the vector ${\mathbf q_j q_{j+1}}$ lies between
$\overrightarrow{p_{i-1} p_i}$ and $\overrightarrow{p_i p_{i+1}}$. ${\mathbf p_{i-1} p_i}$ and ${\mathbf p_i p_{i+1}}$.
The segments of the convolution form a number of closed (not The segments of the convolution form a number of closed (not
necessarily simple) polygonal curves called \emph{convolution necessarily simple) polygonal curves called \emph{convolution
@ -143,7 +144,7 @@ contains \ccc{S.number_of_holes()} holes in its interior).
\begin{figure}[t] \begin{figure}[t]
\begin{ccTexOnly} \begin{ccTexOnly}
\begin{center} \begin{center}
\input{Minkowski_sum_2/fig/sum_triangles.pstex_t} \includegraphics{Minkowski_sum_2/fig/sum_triangles}
\end{center} \end{center}
\end{ccTexOnly} \end{ccTexOnly}
\begin{ccHtmlOnly} \begin{ccHtmlOnly}
@ -169,7 +170,7 @@ The file \ccc{print_util.h} includes auxiliary functions for printing polygons.
\begin{figure}[t] \begin{figure}[t]
\begin{ccTexOnly} \begin{ccTexOnly}
\begin{center} \begin{center}
\input{Minkowski_sum_2/fig/tight.pstex_t} \includegraphics{Minkowski_sum_2/fig/tight}
\end{center} \end{center}
\end{ccTexOnly} \end{ccTexOnly}
\begin{ccHtmlOnly} \begin{ccHtmlOnly}
@ -280,7 +281,9 @@ is widely known as \emph{offsetting} the polygon $P$ by a radius $r$.
\end{ccTexOnly} \end{ccTexOnly}
\begin{ccHtmlOnly} \begin{ccHtmlOnly}
<p><center> <p><center>
<img src="./fig/offset.gif" border=0 alt="Computing offsets of polygons"> <img src="./fig/convex_offset.gif" border=0 alt="Computing offsets of polygons">
<img src="./fig/offset_decomp.gif" border=0 alt="Computing offsets of polygons">
<img src="./fig/offset_conv.gif" border=0 alt="Computing offsets of polygons">
</center> </center>
\end{ccHtmlOnly} \end{ccHtmlOnly}
\caption{(a)~Offsetting a convex polygon. \caption{(a)~Offsetting a convex polygon.
@ -301,7 +304,7 @@ $n$ disconnected \emph{offset edges}. Each pair of adjacent offset
edges, induced by $p_{i-1} p_i$ and $p_i p_{i+1}$, are connected edges, induced by $p_{i-1} p_i$ and $p_i p_{i+1}$, are connected
by a circular arc of radius $r$, whose supporting circle is by a circular arc of radius $r$, whose supporting circle is
centered at $p_i$. The angle that defines such a circular arc centered at $p_i$. The angle that defines such a circular arc
equals $180^{\circ} - \measuredangle p_{i-1} p_i p_{i+1}$; see equals $180^{\circ} - \angle p_{i-1} p_i p_{i+1}$; see
Figure~\ref{mink_fig:pgn_offset}(a) for an illustration. The running Figure~\ref{mink_fig:pgn_offset}(a) for an illustration. The running
time of this simple process is of course linear with respect to time of this simple process is of course linear with respect to
the size of the polygon. the size of the polygon.
@ -317,7 +320,7 @@ with the disc $B_r$,\footnote{As the disc is convex, it is guaranteed
that the convolution curve is comprised of a single cycle.} which can be that the convolution curve is comprised of a single cycle.} which can be
constructed by applying the process described in the previous constructed by applying the process described in the previous
paragraph. The only difference is that a circular arc induced by a paragraph. The only difference is that a circular arc induced by a
reflex vertex $p_i$ is defined by an angle $540^{\circ} - \measuredangle reflex vertex $p_i$ is defined by an angle $540^{\circ} - \angle
p_{i-1} p_i p_{i+1}$; see Figure~\ref{fig:pgn_offset}(c) for an p_{i-1} p_i p_{i+1}$; see Figure~\ref{fig:pgn_offset}(c) for an
illustration. We finally compute the winding numbers of the faces of the illustration. We finally compute the winding numbers of the faces of the
arrangement induced by the convolution cycle to obtain the offset arrangement induced by the convolution cycle to obtain the offset
@ -363,12 +366,12 @@ conic arcs.
\begin{figure}[t] \begin{figure}[t]
\begin{ccTexOnly} \begin{ccTexOnly}
\begin{center} \begin{center}
\input{Minkowski_sum_2/fig/approx_offset.pstex_t} \includegraphics{Minkowski_sum_2/fig/approx_offset}
\end{center} \end{center}
\end{ccTexOnly} \end{ccTexOnly}
\begin{ccHtmlOnly} \begin{ccHtmlOnly}
<p><center> <p><center>
<img src="./fig/tight.gif" border=0 alt="Approximating an offset edge"> <img src="./fig/approx_offset.gif" border=0 alt="Approximating an offset edge">
</center> </center>
\end{ccHtmlOnly} \end{ccHtmlOnly}
\caption{Approximating the offset edge $o_1 o_2$ induced by the polygon \caption{Approximating the offset edge $o_1 o_2$ induced by the polygon
@ -393,7 +396,8 @@ the exact offset edge $o_1 o_2$ is obtained by shifting the polygon edge
$p_1 p_2$ by a vector whose length is $r$ that form an angle $\phi$ with the $p_1 p_2$ by a vector whose length is $r$ that form an angle $\phi$ with the
$x$-axis. We select two points $o'_1$ and $o'_2$ with rational coordinates $x$-axis. We select two points $o'_1$ and $o'_2$ with rational coordinates
on the two circles centered at $p_1$ and $p_2$, respectively. These points on the two circles centered at $p_1$ and $p_2$, respectively. These points
are selected such that if we denote the angle that the vector $\vec{p_j o_j}$ are selected such that if we denote the angle that the vector
${\mathbf p_j o_j}$
forms with the $x$-axis by $\phi'_j$ (for $j = 1, 2$), we have forms with the $x$-axis by $\phi'_j$ (for $j = 1, 2$), we have
$\phi'_1 < \phi < \phi'_2$. $\phi'_1 < \phi < \phi'_2$.
% %
@ -424,9 +428,9 @@ of line segments and circular arcs).
\begin{figure}[t] \begin{figure}[t]
\begin{ccTexOnly} \begin{ccTexOnly}
\begin{center} \begin{center}
\includegraphics{Minkowski_sum_2/fig/ex_offset} \includegraphics{Minkowski_sum_2/fig/ex_offset}
\end{center} \end{center}
\end{ccTexOnly} \end{ccTexOnly}
\begin{ccHtmlOnly} \begin{ccHtmlOnly}
<p><center> <p><center>