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Iterating Evolutes and Involutes Iterating evolutes and involutes Work in progress with M. Arnold, D. Fuchs, I. Izmestiev, and E. Tsukerman Integrability in Mechanics and Geometry: Theory and Computations June 2015 1 2 Evolutes and involutes Evolute: the envelope of the normals, the locus of the centers of curvature; free from inflections and has zero algebraic length. Involute: string construction; come in 1-parameter families. 3 Hedgehogs Wave fronts without inflection and total rotation 2π, given by their support function p(α). ....................................... ....................................... ...................................... .................. .................. π ........... .......... ........ ........ .............. α + ...... .... ...... ......... ..... .... ..... ...... .... .... ........ 2 .... .... ....... .... .... .... ...... .... .. .... ....... .. .. ........... .. ..... ......... ....... p!(α).......... .... ....... .. ........... .... .. ...... ... .......... .... ... ....... .... .......... .... .... ....... .... ......... .... .......... ..... ........ .... ......... ..... O ..... .... ......... ....... γ ..... .......... ........ ..... ........... .... ........... •......p(α) ................ .... ..................... ...... ........................... .... ....................................................................................................... ....... .... ...... ....... .... ............ .... .......... .... ....... ..... ..... ....... ..... .α The evolute map: p(α) 7! p0(α − π=2), invertible on functions with zero average. 4 Steiner point: the center of mass of the curve with the density equal to the curvature: 1 Z 2π ! p(α) (cos α, sin α) dα π 0 Choosing the origin there \kills" the first harmonics of p(α). Theorem: the iterated involutes converge to the Steiner point, the limiting shape being a hypocycloid, generically, of order two. .... .. .. ..... .... .... .. ... .... ... .... .... .. .. ... .. .. ... .... .... ... .. ........ .. .. ........ .... .... .. ... .......... .. .. .......... .... .... ... ... ............ .. .. ........... ..... ..... ... ... .... ........... .. .. .......... .... ..... ..... ... .... .... ............ ............ .... ..... ..... .... .... .... ............................ .... ....... ...... .... .... ... .................................. ... .......... .......... .... .... ................... ................... ............ ........... .... .... ............................. .. ........................... ................. ................. .... .... .................................. ..... ..... .................................. ................ ............... .... .... ................................ ...... ....... ................................ ............ ........... .... .... ..................... ... ... ..................... .......... • ......... ..... • ..... ....................... • ...................... ...... ...... .... ..... .... ............................ .... ..... ..... .... ....................................................................... .... .... ........... ............ .... ..... ..... ................................. ................................... .... ........... ... .. ........... .... ..... ..... .............. .............. .... ......... .. .. ......... .... .... .... .......... .. .. ............ .... .... ......... .. .. ......... .... .... ....... .. .. ....... ... ... .... ...... .... .... .. .. 5 Corollary: if a hedgehog is similar to its evolute then it is a hypocycloid. .. .. .. .. .... .. .. .. .. .... ... .. .. .. .. ... ... .. .. .. .. ... .... ... .. .... .. ... ... ..... ..... ... ..... .... .... .... .... ... ... .. ... ..... .. .. ..... 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