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\begin{document}

\centerline{\bf ME 115(b): Homework $\#3$}
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\centerline{(Due Friday, May 2, 2008)}
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Consider the convex polygonal robot, $\As$, and obstacle, $\Bs$, shown in
Figure \ref{fig:workspace}.  The obstacle is an equilaterial triangle with
side dimension of $10$ units, whose center is coincident with the origin of
the fixed workspace observing reference frame (whose axes are denoted by $X_R$
and $Y_R$).  One triangle face is parallel to the $x$-axis of the workspace
reference frame. The robot is an isoceles triangle whose base dimension is 4
and whose height is 6.  Its body fixed reference frame is located so that its
$x$-axis is aligned with the triangle's centerline, and its origin is located
at the vertex bounded by the two equal sides.

\begin{figure}[h]
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\centerline{\psfig{figure=WorkspaceGeometry.ps,width=0.875\hsize}}
\label{fig:workspace}
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\end{figure}

{\bf Problem 1:} Write a Mathematica (or other programming language) function
to create the outline of the c-obstacle for a fixed orientation of $\As$.
Create the c-obstacle outline for the case of $\theta=45^o$, where $\theta$
is the orientation of $\As$.

{\bf Problem 2:}  Using the function from Problem 1, create an visualization of
the c-obstacle by superimposing on a single 3-dimensional view the constant orientation
c-obtacle boundaries for orientations of $\As$ in $10^o$ increments (in the range
$\theta\in[0^o,360^o]$).  That is, plot 36 constant orientation slices (with each
orientation differing by $10^o$) on a single 3-dimensional view (with the axes
being $x$, $y$, and $\theta$).

{\bf Problem 3:} Create the function that describes the surface boundary
``patch'' of the c-obstacle associated with Type EV contact between robot edge
$E^{\As}_1$ (which connects vertices $a_1$ and $a_2$) and obstacle vertex
$o_1$.  Also determine the boundaries of this patch.  Plot this patch using
Mathematica, Matlab, or anoother approach.


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