catacausticCatacaustic2009-04-03 20:06:442009-09-23 15:57:22Definitiondetermining envelope% 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}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\theoremstyle{definition}
\newtheorem*{thmplain}{Theorem}
Given a plane curve $\gamma$, its \emph{catacaustic} (Greek $\varkappa\alpha\tau\acute{\alpha}\, \varkappa\alpha\upsilon\sigma\tau\iota\varkappa \acute{o}\varsigma$ `burning along') is the envelope of a family of rays reflected from $\gamma$ after having emanated from a \PMlinkescapetext{fixed} point (which may be infinitely far, in which case the rays are initially parallel).
For example, the catacaustic of a logarithmic spiral reflecting the rays emanating from the origin is a congruent spiral.\; The catacaustic of the \PMlinkname{exponential curve}{ExponentialFunction} \,$y = e^x$\, reflecting the vertical rays \,$x = t$\, is the catenary \,$y = \cosh(x\!+\!1)$.