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'intersection of sphere and plane'
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| Title of object: |
intersection of sphere and plane |
| Canonical Name: |
IntersectionOfSphereAndPlane |
| Type: |
Theorem |
| Created on: |
2008-08-11 11:49:58 |
| Modified on: |
2008-08-13 11:06:12 |
| Classification: |
msc:51M05 |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amsthm}
\usepackage{mathrsfs}
\usepackage{pstricks}
\usepackage{pst-plot}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\newcommand{\sR}[0]{\mathbb{R}}
\newcommand{\sC}[0]{\mathbb{C}}
\newcommand{\sN}[0]{\mathbb{N}}
\newcommand{\sZ}[0]{\mathbb{Z}}
\usepackage{bbm}
\newcommand{\Z}{\mathbbmss{Z}}
\newcommand{\C}{\mathbbmss{C}}
\newcommand{\F}{\mathbbmss{F}}
\newcommand{\R}{\mathbbmss{R}}
\newcommand{\Q}{\mathbbmss{Q}}
\newcommand*{\norm}[1]{\lVert #1 \rVert}
\newcommand*{\abs}[1]{| #1 |}
\newtheorem{thm}{Theorem}
\newtheorem{defn}{Definition}
\newtheorem{prop}{Proposition}
\newtheorem{lemma}{Lemma}
\newtheorem{cor}{Corollary} |
Content:
\textbf{Theorem.}\, The intersection curve of a sphere and a plane is a circle.
{\em Proof.}\, We prove the theorem without the equation of the sphere.\, Let $c$ be the intersection curve, $r$ the radius of the sphere and $OQ$ be the distance of the centre $O$ of the sphere and the plane.\, If $P$ is an arbitrary point of $c$, then $OPQ$ is a right triangle.\, By the Pythagorean theorem,
$$PQ = \varrho = \sqrt{r^2\!-\!OQ^2} = \mbox{\;constant}.$$
Thus any point of the curve $c$ is in the plane at a \PMlinkescapetext{constant} distance $\varrho$ from the point $Q$, whence $c$ is a circle.
\begin{center}
\begin{pspicture}(-3.2,-3.5)(3.5,3.5)
\psdot[linecolor=white](-3.2,-3.5)
\psdot[linecolor=white](-3.2,3.5)
\psdot[linecolor=white](3.5,3.5)
\psdot[linecolor=white](3.5,-3.5)
\psdots(0,0)(0,1.23)
\psdot[linecolor=blue](-2.41,0.95)
\rput(0.3,0){$O$}
\rput(2,0.68){$c$}
\rput(-2.45,0.68){$P$}
\rput(0.3,1.23){$Q$}
\rput(-1.1,0.22){$r$}
\rput(-1.1,1.28){$\varrho$}
\psline[linestyle=dashed](0,0)(0,0.6)
\psline[linestyle=dotted](0,0.75)(0,1.23)
\psline[linestyle=dashed](0,1.23)(-2.41,0.95)
\psline[linestyle=dashed](0,0)(-2.41,0.95)
\psline(0,1.09)(-0.12,1.07)
\psline(-0.12,1.07)(-0.12,1.22)
\psellipse[linecolor=blue](0,1.23)(2.7,0.6)
\pscircle[linecolor=blue](0,0){3}
\end{pspicture}
\end{center}
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