Extended graphical representation of polynomials with applications to cybernetics

In this paper, the polynomial of a complex variable s(≡x+iy) with real coefficientsis 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 which K is confined to be real. The projection on (x, iy) plane is just the root locus of the polynomial with K as a real parameter. It is remarkable that the equation of the root-locus is m-th degree in y², whether n=2m+1 or n=2m+2. In addition to the real curve K_r = ?(x) in the figure (K, x) there exists another curve K_t which is plotted by the real parts of all complex roots against K. The (K, x) curve is particularly important to determine the absolute as well as the relative stable interval of K for linear systems. For cybernetics, the (K,iy) curve can be used to show the relation between the nature frequency and the gain K. Such three figures are useful for studying the theory of equation and cybernetics.