3_Chou.tex

\documentclass{article}\usepackage{graphicx,amsmath,amsthm,amssymb}\usepackage{fullpage}\usepackage{eucal}\usepackage{graphicx}\usepackage{color}\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]}\newcommand{\gap}{{~~~~~~~~~~~~~~~~~~~~~}}\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.\\
Answer: We assume that $a(a_x,a_y)$,$b(b_x,b_y)$, we dualize $\bar{a}$ and $\bar{b}$, then we can get $2n^{\top}a-\|a\|^2$, $2n^{\top}b-\|b\|^2$, then, we can get $2n^{\top}a-\|a\|^2$=$\|n\|^2-\|n-a\|^2$,$2n^{\top}b-\|b\|^2$=$\|n\|^2-\|n-b\|^2$, we get\\
\[
\left\{
\begin{aligned}
&Z=2n^Ta-\|a\|^2\\
&Z=2n^Tb-\|b\|^2
\end{aligned}
\right.
\]
Now, we need to know  $\bar{a}^*\cap \bar{b}^*$ , from the function above, we replace Z, thus , we get $2n^Ta-\|a\|^2=2n^Tb-\|b\|^2$, then, $2n^T(a-b)-\|a\|^2+\|b\|^2=0$, let's calculate the distance between $a,b$, $\|n-a\|^2-\|n-b\|^2$ is the difference between the distance of $a,b$, after simplification, we get $2n^T(a-b)-\|a\|^2+\|b\|^2$, and this equals to $0$, which means we prove it.\\
\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 $a_0,\ldots, a_d$.

Answer: From the first question, we know that we dualize $h = \{\bar{x} = \vect{x}{x_z} \mid x_z = 2n^{\top}x - n_z\}$, we get $h^*$, every dual of a point in $\R^3$ is a hyperplane, thus, these hyperplane intersect, and each of them is a perpendicular bisector,  from the question, we know that $a,b,c$ are all in plane h, so $\bar{a},\bar{b},\bar{c}$ are all in $h^*$, which means they are all in the intersection, therefore, for any point n in $\R^3$, $n_z$ is the plane who passes through $\bar{a}$, $\bar{b}$, and $\bar{c}$, $n$ is the circumcenter.

  \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}
	\documentclass{article}\usepackage{graphicx,amsmath,amsthm,amssymb}\usepackage{fullpage}\usepackage{eucal}\usepackage{graphicx}\usepackage{color}\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]}\newcommand{\gap}{{~~~~~~~~~~~~~~~~~~~~~}}\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.\\
	Answer: We assume that $a(a_x,a_y)$,$b(b_x,b_y)$, we dualize $\bar{a}$ and $\bar{b}$, then we can get $2n^{\top}a-\\|a\\|^2$, $2n^{\top}b-\\|b\\|^2$, then, we can get $2n^{\top}a-\\|a\\|^2$=$\\|n\\|^2-\\|n-a\\|^2$,$2n^{\top}b-\\|b\\|^2$=$\\|n\\|^2-\\|n-b\\|^2$, we get\\
	\[
	\left\{
	\begin{aligned}
	&Z=2n^Ta-\\|a\\|^2\\
	&Z=2n^Tb-\\|b\\|^2
	\end{aligned}
	\right.
	\]
	Now, we need to know $\bar{a}^\cap \bar{b}^$ , from the function above, we replace Z, thus , we get $2n^Ta-\\|a\\|^2=2n^Tb-\\|b\\|^2$, then, $2n^T(a-b)-\\|a\\|^2+\\|b\\|^2=0$, let's calculate the distance between $a,b$, $\\|n-a\\|^2-\\|n-b\\|^2$ is the difference between the distance of $a,b$, after simplification, we get $2n^T(a-b)-\\|a\\|^2+\\|b\\|^2$, and this equals to $0$, which means we prove it.\\
	\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 $a_0,\ldots, a_d$.

	Answer: From the first question, we know that we dualize $h = \{\bar{x} = \vect{x}{x_z} \mid x_z = 2n^{\top}x - n_z\}$, we get $h^$, every dual of a point in $\R^3$ is a hyperplane, thus, these hyperplane intersect, and each of them is a perpendicular bisector, from the question, we know that $a,b,c$ are all in plane h, so $\bar{a},\bar{b},\bar{c}$ are all in $h^$, which means they are all in the intersection, therefore, for any point n in $\R^3$, $n_z$ is the plane who passes through $\bar{a}$, $\bar{b}$, and $\bar{c}$, $n$ is the circumcenter.

	\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}