SOME ASPECTS OF MODIFICATION OF BERNULLI'S LEMNISCATE

Abstract

The article is devoted to the study of modified lemniscates. The lemniscate equation written in the polar coordinate system is modified. The purpose of the modification is to provide the necessary angles of inclination of the tangents at the start and end points of the half of the lemniscate petal located in the region of positive values of the abscissas and ordinates of the orthogonal coordinate system. An ordinary lemniscate has a tangent inclination angle of 45° at the origin of this coordinate system. At the point of intersection of the petals of the lemniscate with the abscissa axis of the orthogonal coordinates, the tangent to it is perpendicular to this axis. To modify the lemniscate, two parameters are introduced into its polar equation, one of which is the power of the root, and the second is some rational positive or negative number, but one that does not lead to a negative cosine value under the root sign. The required value of the angle of inclination of the tangent at the origin of the orthogonal coordinate system is achieved by the appropriate choice of the value of the coefficient at the polar angle of the cosine function of the lemniscate equation. The change in the angle of inclination of the tangent at the starting point of the petal of the lemniscate is implemented by introducing an additional component under the root sign, which is a trigonometric function of the desired angle of inclination of the tangent. A method is developed for drawing the arc of the modified lemniscate through a certain point specified in the plane of the lemniscate with arbitrary angles of inclination of the tangents at the initial and end points of the modeled arc of the modified lemniscate. The proposed method for modifying the lemniscate is implemented in the form of a computer code that allows, in addition to numerical results, to obtain graphic images of the simulated curves on the computer monitor screen. The method can be applied when constructing profiles of turbine blades and transition curves of railway tracks, as well as in other practical applications where it is required to build a smooth curve, provided that the given angles of inclination of the tangents and some intermediate point through which the modeled line must pass.

Key words: lemniscate, modification, tangent, curvature.