ANALYSIS OF THE GEOMETRY FEATURES OF VARIOUS SPIRAL SHAPES

This work is devoted to the topical issue of the analysis of geometric features and the clarification of the mathematical description of the Cornu spiral (clothoid) and some types of sinusoidal spirals, namely cardioids, Bernoulli lemniscates, and Galilean spirals. Their theoretical and practical aspects are considered, the mathematical description in the form of equations in parametric form and in various aspects of presentation for the convenience of conducting research is highlighted. Geometric models of the specified types of spirals are constructed, their geometric properties are analyzed, and the spheres and directions of application in practical human activity are highlighted. Geometric modeling of the shape of the Bernoulli lemniscate was carried out by introducing additional parameters into the polar equation, which allow transforming the shape of the curve, obtaining a set of geometric solutions, expanding the capabilities of the classical form. It is shown that due to their geometric features, spirals are widely used for modeling lines and surfaces of an s-shaped shape, as well as for forming transitional curves of railway tracks, highways, various types of attractions, etc. It has been geometrically proven that the shapes of the bodies, curls and cutouts in musical string instruments are performed according to the clothoid, which allows you to get a strange soft, singing sound that fascinates with its melody. The article simulates the trajectory of a stone falling from a height in Galileo's experiment, which proves that the trajectory of a body's movement under the action of its own weight is a part of the Galilean spiral. It is shown that one of the practical areas of use of cardioid is microphones used in vocal recording in recording studios. In addition, it is noted that sinusoidal curves can be used in mechanics, geodesy, and in the study of caustics. It is noted that spiral shapes are often used in mechanical engineering, in automotive and aviation aerodynamics, as well as in computer modeling for smooth connection of trajectories with variable curvature. In many branches of the mechanical engineering industry, Galilean spirals are used to design mechanisms where a change in speed is required.