Extended graphical representation of polynomials with applications to cybernetics (Q1085567)

The polynomial of a complex variable \(s(\equiv x+iy)\) with real coefficients \(K=a_ 0s^ n+a_ 1s^{n-1}+......+a_{n-1}s+a_ n\) is represented graphically 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 \(y^ 2\), whether \(n=2^ m+1\) or \(n=2^ m+2\). In addition to the real curve \(K_ r=f(x)\) in the figure (K,x) there exists another curve \(K_ c\) 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 stability interval of K for linear systems. For cybernetics, the (K,iy) curve can be used to show the relation between the nature frequency \(\omega\) and the gain K. Such three figures are useful for studying the theory of equation and cybernetics.