Geometric Wave Engineering: Pseudo-Surfaces of Variable Negative Gaussian Curvature as a Geometric Basis for Programmable Wave Control

This paper presents Geometric Wave Engineering (GWE) as a proposed design discipline in which the macroscopic geometry of a wave domain is treated as an independent control variable alongside material, boundary conditions, and excitation. The core geometric objects are pseudo-surfaces of variable negative Gaussian curvature generated from hyperbolic, parabolic, and elliptic profiles. The most developed construction is the higher-order pseudo-hyperboloid built by a recursive interval rule and a Merge operation that forms a common internal volume without artificial connecting segments. Pseudo-paraboloids and pseudo-ellipsoids generalize the same constructive logic to other generators. In addition to the geometric and computational construction, numerous stochastic ray-tracing studies have already been carried out for all three pseudo-surface families. Without entering into detailed numerical data here, these studies qualitatively support many of the criteria C1–C8 and indicate remarkable possibilities for controlling waves of different physical nature and across broad frequency ranges within the proposed GWE framework. The present preprint focuses on definitions, construction rules, representative figures, functional zones, and a compact verification roadmap.