Extended graphical representation of polynomials with applications to cybernetics

Abstract

In this paper, the polynomial of a complex variables(≡x+iy) with real coefficients

$$K = a_0 s^n + a_1 s^{n - 1} + \cdots \cdots + a_{n - 1} s + a_n $$

is graphically represented by three plane curves which are the projections of a space curve on three coordinate planes of the coordinate system (x, iy. K) in whichK is confined to be real. The projection on (x, iy) plane is just the root locus of the polynomial withK as a real parameter. It is remarkable that the equation of the root-locus ism-th degree iny ², whethern=2m+1 orn=2m+2. In addition to the real curveK _r =f(x) in the figure (K, x) there exists another curveK _c which is plotted by the real parts of all complex roots againstK. The (K, x) curve is particularly important to determine the absolute as well as the relative stable interval ofK for linear systems. For cybernetics, the (K, iy) curve can be used to show the relation between the nature frequency ω and the gainK. Such three figures are useful for studying the theory of equation and cybernetics.