3_wang.tex

\documentclass{article}
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\usepackage{tikz}
\usepackage{algorithm,algorithmic}

\newcommand{\question}[1]{\vspace{10pt}\noindent $\mathbf{#1}$}
\newcommand{\R}{\mathbb{R}}
\newcommand{\vect}[2]{\bigl[\begin{smallmatrix}#1\\#2\end{smallmatrix}\bigr]}
\newcommand{\vectt}[3]{\bigl[\begin{smallmatrix}#1\\#2\\#3\end{smallmatrix}\bigr]}
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\newcommand{\vor}{\mathrm{Vor}}
\newcommand{\CC}{\mathrm{conv}}

\title{Computational Geometry Homework 3}
\author{}

\begin{document}
  \maketitle

  \section*{Administration}
    Your answers should be typeset in \LaTeX\ or some equivalent and submitted as a \textbf{pdf}.
    The \LaTeX\ source of these questions may be found on the course website under ``homework''.
    Name your files as ``2\_\emph{your\_last\_name}.pdf'', all lowercase letters.
    For example, I would call mine \textbf{3\_sheehy.pdf}.

    Submission:  Use the same repository that you used for your last submission.  If you did not submit the last homework:
    Create a \textbf{private} repository on github.uconn.edu and add me (userid: she13001) as a collaborator.  Name your repository ``your\_last\_name\_compgeom'' (all lowercase).

    \textbf{Checkin by noon, Nov 30: Add the pdf to the repository (but include the tex source too).  I will pull your changes and check in my comments directly on your pdf file.}

  \section*{Projective Duality}

  \question{1}
  Let $a,b\in\R^2$.
  Consider the lifted points $\bar a = \vect{a}{\|a\|^2}$ and $\bar b = \vect{b}{\|b\|^2}$ in $\R^3$.
  Prove that the projection of $\bar{a}^*\cap \bar{b}^*$ is the perpendicular bisector between $a$ and $b$.

  \vspace{1em}
  If you are feeling ambitious, prove that for $a,b\in\R^d$, the same holds, where the perpendicular bisector of $2$ points in $\R^d$ is a hyperplane.\\
  For point p={$P_x$,$P_y$,$P_z$} in $\R^3$. Its duality is $Z=2P_x$X+2$P_y$Y-$P_z$.\\
  Thus, for point a and b. its duality is a hyperplane.\\
  For point a, its duality a* is the hyperplane which Z=2$P^T$a-$\|a\|^2$.And the duality of point b is Z=2$P^T$b-$\|b\|^2$.\\
  Thus, the area of $\bar{a}^*\cap \bar{b}^*$ is the result of the equation
\[
\left\{
\begin{aligned}
&Z=2P^Ta-\|a\|^2\\
&Z=2P^Tb-\|b\|^2
\end{aligned}
\right.
\]
we  delete  Z in these two equation and got that $2P^Ta-\|a\|^2=2P^Tb-\|b\|^2$.Thus $2P^T(b-a)+\|b\|^2-\|a\|^2=0$.\\
For any point P in $\bar{a}^*\cap \bar{b}^*$.we calculate the difference of two distance $\|P-a\|^2-\|P-b\|^2$\\
$\|P-a\|^2-\|P-b\|^2$=$2P^T(b-a)+\|b\|^2-\|a\|^2=0$.Thus, the the projection of $\bar{a}^*\cap \bar{b}^*$ is the perpendicular bisector between $a$ and $b$.\\
  \question{2}
  Let $a,b,c\in \R^2$, not all collinear.
  As in the previous question, we lift these points to $\bar a = \vect{a}{\|a\|^2}$, $\bar b = \vect{b}{\|b\|^2}$, and $\bar c = \vect{c}{\|c\|^2}$ in $\R^3$.
  Let $h = \{\bar{x} = \vect{x}{x_z} \mid x_z = 2n^{\top}x - n_z\}$ for $n\in \R^2$ and $n_z\in \R$ be the plane passing thru $\bar{a}$, $\bar{b}$, and $\bar{c}$.
  Prove that $n$ is the circumcenter of $a$, $b$, and $c$.\\

  \vspace{1em}
  \emph{Hint: dualize h, use incidence preservation, and recall that $2n^{\top} a - \|a\|^2 = \|n\|^2 - \|n-a\|^2$.}
  \vspace{1em}

  If you are feeling ambitious, consider the following high-dimensional version.
  Let $a_0,\ldots a_d \in \R^d$ and the corresponding lifted points $a_i = \vect{a}{\|a\|^2}$ for $i=0\ldots d$.
  Let $h = \{\bar{x} = \vect{x}{x_z} \mid x_z = 2n^{\top}x - n_z\}$ for $n\in \R^d$ and $n_z\in \R$ be the plane passing thru $\bar{a}$, $\bar{b}$, and $\bar{c}$.
  Prove that $n$ is the circumcenter of $a_0,\ldots, a_d$.\\
  Consider the dual h, since $h = \{\bar{x} = \vect{x}{x_z} \mid x_z = 2n^{\top}x - n_z\}$, the dual of h, h* is the point which is $\vect{n}{n_Z} $\\
  Consider the dual of $\bar a$,$\bar b$ and $\bar c$, each point correspond to a hyperplane,and from the question above,the intersection of their dual is the the perpendicular bisector.\\
  From the original graph, point a and b and c are all in the plane h. In the dual graph, h* is in the intersection of a* b* and c*. Thus n is in the bisector of (a,b) (a,c) and(b,c),so n is the circumcenter of $a,b,c$\\

  \section*{Project Checkpoint} % (fold)
  \label{sec:project_checkpoint}

    For your projects, create a public git repository (one per group) and add me as a user.

    Add a \texttt{Readme} file explaining three things: 1.\ The concept your program will illustrate, 2.\ the user interaction involved, and 3.\ a description of what it will look like.

    You project repository should also include a simple processing project to show that you can download and run the software.
    This is just to make sure you don't get caught on technical hurdles late.
    The program should run successfully in javascript mode.
    It could just draw a couple figures (a HelloWorld equivalent).

  % section project_checkpoint (end)


\end{document}