Iterated evolutoidal transformations

Ogandzhanyants, Oleg (2013) Iterated evolutoidal transformations. Masters thesis, Memorial University of Newfoundland.

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Abstract

Evolutoids of a plane curve are a generalization of an evolute. They form a one parameter family of curves whose tangents cut the given curve under a fixed angle, which in the case of the evolute is the right angle. The evolutoidal transformation is a point transformation determined by the inclination of t he said tangents and the radius of curvature of the original curve at the given point. Iterated evolutoidal transformations depend on the derivatives of the radius of curvature. The main results of this thesis concern the geometry and structure of the sets consisting of all images of a certain point on the original curve (image-sets) under iterated evolutoidal transformations with varying inclinations of tangents. To my knowledge, a systematic study of such sets has not been undertaken before and several results presented in this thesis are new. A number of special curves, such as sinusoidal spirals and epi- and hypocycloids, appear frequently in this study, in particular, as boundaries of the image-sets. A geometrical construction of the 2nd and 3rd iterations as well as some particular cases of iterations of higher orders reveal interesting connections to elegant theorems of Euclidean geometry. Some of them do appear in old literature, but here they are reinterpreted and proved in a different way.

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