Plane-Geometric Investigation of a Proof of the Pohlke’s Fundamental Theorem of Axonometry

Abstract

Consider a bundle of three given coplanar line segments (radii) where only two of them are permitted to coincide. Each pair of these radii can be considered as a pair of two conjugate semidiameters of an ellipse. Thus, three concentric ellipses E _i, i = 1, 2, 3, are then formed. In a proof by G.A. Peschka of Karl Pohlke’s fundamental theorem of axonometry, a parallel projection of a sphere onto a plane, say, $\mathbb E$, is adopted to show that a new concentric (to E _i) ellipse E exists, “circumscribing” all E _i, i.e., E is simultaneously tangent to all $E_i\subset \mathbb E$, i = 1, 2, 3. Motivated by the above statement, this paper investigates the problem of determining the form and properties of the circumscribing ellipse E of E _i, i = 1, 2, 3, exclusively from the analytic plane geometry’s point of view (unlike the sphere’s parallel projection that requires the adoption of a three-dimensional space). All the results are demonstrated by the actual corresponding figures as well as with the calculations given in various examples.