catacaustic

Given a plane curve $ \gamma$ , its 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 catenary $ y = \cosh(x\!+\!1)$ .



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