M\"obius Transformations and the Analytic--Geometric Reconstruction of the Induction--Machine Circle Diagram

The Heyland circle diagram is a classical graphical method for representing the steady--state behavior of induction machines using no--load and blocked--rotor test data. Despite its long pedagogical history, the traditional geometric construction has not been formalized within a closed analytic framework. This note develops a complete Euclidean reconstruction of the diagram using only the two measured phasors and elementary geometric operations, yielding a unique circle, a torque chord, a slip scale, and a maximum--torque point. We prove that this constructed circle coincides precisely with the analytic steady--state current locus obtained from the per--phase equivalent circuit. A M\"obius transformation interpretation reveals the complex--analytic origin of the diagram's circularity and offers a compact explanation of its geometric structure.