Graphically including gain K into conventional root locus

The root locus method in control systems shows damping a and natural frequency w, but somewhat surprisingly, does not include graphically the gain K. This paper proposes the natural continuation of root locus, that K be shown graphically with the s-plane. To illustrate the principle, graphs were drawn with Matlab for transfer functions ranging from first order to second order with multiple zeros. Far from the simple curves of the conventional root locus, a wide range of complex curves and artistic shapes were observed. The three dimensional plot of the real K axis with the complex s plane raises some fundamental mathematical questions. How much is the validity and usefulness of real axis perpendicular to a complex plane. The graphical incorporation of K into the root locus has become easier than ever with the availability of softwares such as Matlab.