Double Inheritance of the Focal Property under Offset Rotation: A Named Design Primitive of Geometric Wave Engineering

We isolate and physically explain a design primitive of the research program Geometric Wave Engineering (GWE). The primitive, which we name double inheritance of the focal property, governs the transfer of the planar focal law of a classical conic to the three-dimensional surface obtained by rotating that conic about an axis parallel to, but offset from, its own symmetry axis. For the hyperbolic generator the resulting surface — the pseudohyperboloid of order two in the terminology of the GWE program — exhibits a structured ray behavior that is neither point-focusing (as for the standard quadrics of revolution) nor ergodic (as for dispersing billiards), but is segmented by the conserved azimuthal invariant J = ρ vφ. We give a step-by-step physical derivation of this behavior from the elementary fact of focal transfer: (i) the hyperbolic focal law of Apollonius survives as the local reflection rule in every meridional half-plane; (ii) the two point foci, displaced from the rotation axis by R > 0, generate, under rotation, two focal rings that act as virtual external references never reached by interior rays; (iii) the planar two-focus reflection law lifts to a one-parameter family of three-dimensional reflection laws indexed by J, producing two topologically distinct coexisting concentration structures (axial and toroidal). We assert that this construction — double inheritance plus J-stratification — is a named physical primitive of the GWE program. All statements are kept in the geometric-optics limit λ → 0; quantitative ray verification on the same geometry is the subject of a companion report.