Closed plane curves described by finite and infinite sums of rotating vectors |
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| Authors: | MG Pawley |
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| Institution: | 3686 San Simeon Way, Riverside, CA 92506 USA |
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| Abstract: | A large group of closed plane curves may be classified as roulettes, including epicycloids, hypocycloids, and related epitrochoids and hypotrochoids. The equation for the roulette in complex polar form shows that any roulette may be described by the vector sum of two vectors of specified constant magnitudes rotating with constant angular velocities. Methods for plotting, and for electronic display of roulettes are described. An equation is derived for a roulette approximation for an N-sided regular polygon. In particular, an application to two-dimensional potential theory is described and illustrated by consideration of the roulette approximation for a square as an equipotential curve, with derivation of equations for equipotential curves in the field surrounding the square. General equations are derived for given closed plane curves with points whose x and y coordinates may separately be expanded in Fourier series as functions of the polar angle, assuming these expansions are valid. It is shown that, in general, a closed plane curve may be considered as being described by an infinite sum of vectors, each rotating in a circle. Simplifying effects of symmetry about a polar axis and/or about the origin are discussed, and methods for harmonic analysis of a given closed plane curve with aid of an electronic calculator are described. |
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