Pre-Rationalization Approach for the Low-Tech Construction of Doubly Curved Surfaces. Part I: Conjugate Patches with One Direction of Planar Curves and the Other Generating Developable Strips

This two-parts paper is dedicated to the development of a pre-rationalization approach that provides favorable constructive properties for the low-tech construction of doubly curved structures. In part I, we provide methods for generating geometric constructs (patches) with two essential properties: One family of coordinate curves are planar (PU-property), while the other generating singly curved strips (develpable V-strips). The geometric constructs are done on three levels of geometric properties, of being: conjugate, orthogonal-conjugate (principal), and orthogonal-conjugate with negative constant Gaussian curvature (principal Tchebyshef nCGC). The main contribution of part I, is the articulation of a space of design variants within the pre-rationalization of the three levels, by introducing transforms leaving the geometric properties invariant, namely, Projective, Mo ̈bius, and Ba ̈cklund transforms.