Projective Root-Locus: An Extension of Root-Locus Plot to the Projective Plane

TL;DR

This paper introduces the Projective Root-Locus (PjRL), extending the classical Root-Locus method to the projective plane using gnomonic projection, enabling analysis of points at infinity and complementary plots.

Contribution

It extends the Root-Locus method to the projective plane, allowing for the computation of points at infinity and new visualization perspectives.

Findings

PjRL reduces to classical RL in the affine XY plane.

PjRL allows plotting RL in other affine components like ZY plane.

Points at infinity are obtained as solutions of algebraic equations.

Abstract

In this paper we present an extension of the classical Root-Locus (RL) method where the points are calculated in the real projective plane instead of the conventional affine real plane; we denominate this extension of the Root-Locus as "Projective Root-Locus (PjRL)". To plot the PjRL we use the concept of "Gnomonic Projection" in order to have a representation of the projective real plane as a simi-sphere of radius one in ${\mathbb R}^3$. We will see that the PjRL reduces to the RL in the affine $XY$ plane, but also we can plot the RL onto another affine component of the projective plane, like $ZY$ affine plane for instance, to obtain what we denominate complementary plots of the conventional RL. We also show that with the PjRL the points at infinity of the RL can be computed as solutions of a set algebraic equations.