Talk:PlanetPhysics/Catacaustic

Original TeX Content from PlanetPhysics Archive

%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: catacaustic %%% Primary Category Code: 42.15.-i %%% Filename: Catacaustic.tex %%% Version: 1 %%% Owner: pahio %%% Author(s): pahio %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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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 light rays reflected from $\gamma$ after having emanated from a 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 exponential curve \,$y = e^x$\, reflecting the vertical rays \,$x = t$\, is the \htmladdnormallink{catenary}{http://planetphysics.us/encyclopedia/Catenary.html} \,$y = \cosh(x\!+\!1)$.

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