A remarkable property of cycloidal curves

• Mikhail Vladimirovich Taldykin Institute of Mechanics and Engineering - Subdivision of the Federal State Budgetary Institution of Science “Kazan Scientific Center of the Russian Academy of Sciences” http://orcid.org/0000-0001-9977-1224

DOI: https://doi.org/10.46925//rdluz.33.02

Keywords: angle of rotation of the generating circle; cycloid; cycloidal curves; generating circle; mechanisms for drawing curves; normal to sinusoid; shortened cycloid; sinusoid; tangent to a sinusoid

Abstract

The purpose of this theoretical work is to establish a connection between the most important properties of plane curves: cycloids and sinusoids. For this, a drawing mechanism is considered, which simultaneously draws a sinusoid and two cycloids. Based on the results obtained using this mechanical method of obtaining curves, the following important, previously unknown, theoretical facts are established. Firstly, new in theoretical terms is that the sinusoid is not represented as a graph of a trigonometric function, but as a locus of points equidistant from the current points of two cycloids: an ordinary and another cycloid congruent to the original one, inverted and shifted along the axis by half a period. Secondly, the line passing through the current points of these cycloids is nothing like a normal to the resulting sinusoid. This property greatly simplifies the graphical construction of such a normal. And, finally, a simple trigonometric relationship was established between the angle of rotation of the generating circle and the angle of deviation of the normal from the vertical.

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