Geometric Wave Engineering: Comparative Ray-Level Verification of Second-Order Pseudohyperboloids of the Vertical and Horizontal Types

This preprint presents a comparative ray-level verification of second-order vertical- and horizontal-type pseudohyperboloids within the framework of Geometric Wave Engineering. The calculation is strictly ray-based: a point ray propagates along straight segments and reflects specularly from the boundary. Diffraction, interference, modal structure, losses, and full-wave field solutions are not included at this stage. Within this model, the results show that the vertical and horizontal forms are not ray-equivalent. A central element of the analysis is the introduction of three J classes: low-J, mid-J, and high-J. These classes separate rays by the normalized axial angular-momentum content of the initial trajectory. Without this separation, averaging over all rays would mix physically distinct trajectory families and obscure the main result. The vertical type is dominated by high-J returns to the equatorial focal ring, whereas the horizontal type is dominated by low-J returns to the two throat focal disks. The geometric figures show only the external 2D and 3D forms of the two types together with the focal zones used in the metric definitions. The main quantitative result is that the highest corrected strict-sequence density is obtained for the vertical closed-surface configuration G2 at h/a = 0.30 in the high-J class, where the metric reaches 1.229%.