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\title{14.  The Oriented Plane}
\subtitle{ Mechanics of Manipulation}
\author{Matt Mason}
\institution{Carnegie Mellon}
\email{matt.mason@cs.cmu.edu \\
http://www.cs.cmu.edu/\symbol{"7E}mason}
\slideCaption{Lecture 14. \hfill Mechanics of Manipulation }

\begin{document}

\maketitle

% ============
%
%  Slide 1.5
%


\begin{slide}[R]{Exhortations}

``Hands are levers of influence on the world that made intelligence
worth having.  Precision hands and precision intelligence coevolved in
the human lineage, and the fossil record shows that hands led the
way.''---Steven Pinker, in \emph{How the Mind Works}

\end{slide}


%=======================================================================
%
% SLIDE 2
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\begin{slide}[R]{Lecture 14.  The Oriented Plane.}
\vspace{-0.5in}
\includegraphics[height=3.0in]{TOC14}
\end{slide}

%=======================================================================
%
% SLIDE 3
%
%

\begin{slide}[R]{Outline.}

\begin{trivlist}
\item Reuleaux's space$\ldots$what is it?
\item Formal definitions.
\item Central projection.
\item Convexity in the oriented plane.
\item Relation to polyhedral convex cones.
\item Rotation centers and the oriented plane.
\end{trivlist}

\end{slide}


%=======================================================================
%
% SLIDE 4
%
%
\begin{slide}[R]{Reuleaux's space}
\begin{minipage}[t]{6cm}
\begin{trivlist}
\item[]Reuleaux represents PCC's in planar diff'l twist space.
\begin{itemize}
\item[]3D twist space is 6D.
\item[]planar twist space is 3D.
\item[]Reuleaux does it in 2D!
\end{itemize}
\item[]What space do signed rotation centers live in?
\item[]Plane with $+$ and/or $-$ label, 
\item[]and points at infinity?
\item[]Stolfi and Guibas:  the \emph{oriented plane}.
\end{trivlist}
\end{minipage}
\hfill
\begin{minipage}[t]{4cm}
\begin{figure}[h!]
\includegraphics[width=4cm]{unilat}
%\includegraphics[height=3cm]{*file*}
\end{figure}\end{minipage}
\end{slide}

%=======================================================================
%
% SLIDE 5
%
%
\begin{slide}[R]{Formal definition}
\begin{trivlist}
\item[]
Consider homogeneous coordinates $(x, y, w) \neq (0,0,0)$.

\item[]A \defterm{point in the oriented plane} is a ray of triples:
\begin{equation}
        \{ (kx, ky, kw) \; | \;  k > 0 \}
\end{equation}

\item[]

Three cases:
\begin{description}
\item[$w > 0:$]Signed Euclidean point $(x/w, y/w, \oplus)$.

\item[$w < 0:$]
Signed Euclidean point $(x/w, y/w, \ominus)$ .

\item[$w = 0:$]
Ideal point:  point at infinity.

\end{description}
\end{trivlist}

\end{slide}

%=======================================================================
%
% SLIDE 6
%
%
\begin{slide}[R]{Central projection}
\begin{minipage}[t]{4cm}
\begin{trivlist}
\item[]Project each point $(x,y,w)$ to $w=1$ plane.
\item[]Attach appropriate sign.
\item[]Ideal points miss the plane but hit the equator.
\end{trivlist}
\end{minipage}
\hfill
\begin{minipage}[t]{6cm}
\begin{figure}[h!]
\includegraphics[width=6cm]{oplane}
%\includegraphics[height=3cm]{*file*}
\end{figure}\end{minipage}
\end{slide}

%=======================================================================
%
% SLIDE 7
%
%
\begin{slide}[R]{Relation to projective plane}
\begin{minipage}[t]{6cm}
\begin{trivlist}
\item[]Maybe we should call it the \emph{Oriented Projective Plane}.
\item[]Projective plane is set of lines through the origin of
  $\mathbb{E}^3$.
\item[]Oriented plane is set of \emph{directed} lines through the
  origin of $\mathbb{E}^3$.
\item[]The projective plane is the sphere $\mathbb{S}(2)$ with
  antipodes identified.
\item[]The oriented plane is the sphere $\mathbb{S}(2)$.
\begin{itemize}
\item[]Northern hemisphere is the positive plane,
\item[]Southern hemisphere is the negative plane,
\item[]Equator is the ideal line.
\end{itemize}
\end{trivlist}
\end{minipage}
\hfill
\begin{minipage}[t]{4cm}
\begin{figure}[h!]
%\includegraphics[width=4cm]{*file*}
\includegraphics[height=3cm]{oplane}
\end{figure}
\begin{figure}[h!]
%\includegraphics[width=4cm]{*file*}
\includegraphics[height=3cm]{inftyproj}
\end{figure}\end{minipage}
\end{slide}

%=======================================================================
%
% SLIDE 7
%
%
\begin{slide}[R]{Why do we need it?}
\begin{trivlist}
\item[]Because, e.g. Reuleaux's method is easier than analyzing
  polyhedral convex cones in planar differential twist space.
\begin{itemize}
\item[] (or rather, Reuleaux's method is an easy way to analyze PCCs
  in planar differential twist space.)
\end{itemize}
\item[]In practice, we work directly in the oriented plane, as
  Reuleaux did, not worrying too much about the projection.
\item[]But to answer deep questions, refer back to the projection.
\end{trivlist}

\end{slide}

%=======================================================================
%
% SLIDE 8
%
%
\overlays{3}{
\begin{slide}[R]{Geometry}
  \begin{trivlist}
  \item[]An example question.  We know what points are in the oriented
    plane.  Are there lines?  \fromSlide{2}{
  \item[]Transform to a question about spheres.  In spherical
      geometry, are there lines?  How are they defined?}
 \fromSlide{3}{
\item[]So $\ldots$
\begin{itemize}
\item Two non-antipodal points determine a line.
\begin{itemize}
\item[] (The ideal line, or a Euclidean line labeled ``$\pm$''.)
\end{itemize}
\item Every pair of lines intersects in \emph{two} antipodal points.
\end{itemize}}
\end{trivlist}
\end{slide}}

%=======================================================================
%
% SLIDE 9
%
%

\overlays{4}{
\begin{slide}[R]{Convexity}
\begin{trivlist}
\item[]Another ``deep'' question:  How does convexity work in the
oriented plane? 
\fromSlide{2}{
\item[]How is convexity defined for Euclidean space?}
\fromSlide{3}{
\begin{itemize}
\item[]If $a$ and $b$ are in the figure, then so is $\overline{a b}$.
\end{itemize}}
\fromSlide{4}{
\item[]What is $\overline{a b}$ in the oriented plane?  What is $\conv(\{a,b\})$?}
\end{trivlist}

\end{slide}}

%=======================================================================
%
% SLIDE 10
%
%
\begin{slide}[R]{Convexity:  plus and plus}
\begin{minipage}[t]{6cm}
\begin{trivlist}
\item[]Consider the polyhedral convex cone determined by the two rays.
\item[]Project it to the plane.
\item[]So, if $a$ and $b$ have the same sign then $\conv(\{a,b\})$ is
the line segment between them with the same sign.
\end{trivlist}
\end{minipage}
\hfill
\begin{minipage}[t]{4cm}
\begin{figure}[h!]
\includegraphics[width=4cm]{opproj++.eps}
%\includegraphics[height=3cm]{*file*}
\end{figure}\end{minipage}
\end{slide}

%=======================================================================
%
% SLIDE11
%
%
\begin{slide}[R]{Convexity:  plus and minus}
\begin{minipage}[t]{6cm}
\begin{trivlist}
\item[]Consider the PCC $\ldots$
\item[]Define the \emph{external} line segment.
\end{trivlist}
\end{minipage}
\hfill
\begin{minipage}[t]{4cm}
\vspace{-2cm}
\begin{figure}[h!]
\includegraphics[width=4cm]{opproj+-.eps}
%\includegraphics[height=3cm]{*file*}
\end{figure}\end{minipage}
\end{slide}

%=======================================================================
%
% SLIDE 12
%
%
\begin{slide}[R]{Problem Set 4}
\begin{trivlist}
\item[]Let's do PS4!
\item[]How do you construct the convex hull of some set of points?
\begin{enumerate}
\item Pick two points, and construct their convex hull.
\item Repeat step 1, ad nauseum.
\end{enumerate}
\end{trivlist}

\end{slide}

%=======================================================================
%
% SLIDE 13
%
%
\begin{slide}[R]{Reuleaux's method}
\begin{trivlist}
\item[]Given a differential velocity twist $(v_{0x}, v_{0y},
\omega_z)$, what is the rotation center? 
\item[]Velocity at a point $\vv p = (x,y)$ is 
\begin{align*}
  & \vv v_0 + \omega \times \vv p\\
 = & \left(\begin{array}{c} v_{0x} - \omega y \\ v_{0y} + \omega x
 \end{array}\right) 
\end{align*}
\item[]Set equal to $(0,0)$ and solve for $(x,y) \ldots$
\item[]Rotation center is at
\[ \left(\begin{array}{c}-v_{0y} / \omega \\ v_{0x} / \omega
\end{array}\right) \] 
\item[]Central projection, and rotation of coordinates.
\end{trivlist}

\end{slide}

%=======================================================================
%
% SLIDE last
%

\begin{slide}[R]{Next:  Moment labeling.}
\vspace{-0.5in}
\includegraphics[height=3.0in]{TOC15}
\end{slide}
\end{document}



\end{document}
