The Theory of Catastrophes Before Poincaré
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In 1654 Huygens (1673) constructed the theory of evolutes and involutes of plane curves. He noted that an involute or an evolute of a smooth curve has cusps of semi-cubical type. The evolute of a smooth curve is a caustic (the envelope of the family of rays orthogonal to the curve); see Fig. 6. (Leonardo da Vinci (see Bennequin (1986)) had already investigated caustics; the term was apparently introduced by Tschirnhaus (1682)). The family of involutes of the caustics in Fig. 6 is shown in Fig. 7. It is the family of wave fronts corresponding to the same family of rays. (All of this was also discovered by Huygens.) Thus, Huygens essentially discovered the stability of cusps on caustics and wave fronts. Nowadays these singularities are connected with the pleats of the corresponding smooth mappings, and they belong to the most important applications of catostrope theory. Fig. 7 was discussed by Cayley (1868b).
KeywordsWave Front Morse Theory Bifurcation Curve Catastrophe Theory Umbilic Point
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- 1.The word metamorphosis was used as a translation of the Russian “perestroika” until recently. In Russian “Morse surgery” was always called “perestroika of Morse”. Nowadays, when we have an international word “perestroika”, we don’t need to substitute “metamorphosis” for it.Google Scholar