Using Exponential Complex Polynomials for Constructing Closed Curves with Given Properties

In this paper, we present a method to compute the coefficients of a complex exponential polynomial of real argument that, while being decomposed into real and imaginary parts by Euler’s formula, obtains required interpolating and differential properties at any given points of its real graph. Moreover, imaginary components in their nodes of interpolation and differentiation serve as additional control tools that shape the polynomial appearance. Although the impact of these components is not yet studied extensively, we can still use them to achieve useful properties, e. g. we can minimize the total height of the polynomial graph. From the geometry standpoint, having these properties implies that the parametric curves constructed with such polynomials can go through given points, have predetermined tangent vectors in those or other points, and retain enough variability to have additional useful properties, for instance, the total length of these curves, or their maximal curvature can also be minimized within limits.