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On the discrete differential geometry of surfaces in S4
The Grassmannian space [special characters omitted](2, 4) embedded in [special characters omitted] as the Klein quadric of twistor theory has a natural interpretation in terms of the geometry of “round” 2-spheres in S 4. The incidence of two lines in [special characters omitted] corresponds to the contact properties of two 2-spheres, where contact is generalized from tangency to include “half-tangency:” 2-spheres may be in contact at two isolated points. There is a connection between the contact properties of 2-spheres and soliton geometry through the classical Ribaucour and Darboux transformations. The transformation theory of surfaces in S4 is investigated using the recently developed theory of “Discrete Differential Geometry” with results leading to the conclusion that the discrete conformal maps into [special characters omitted] of Hertrich-Jeromin, McIntosh, Norman and Pedit may be defined in terms a discrete integrable system employing half-tangency in S 4.
Shapiro, George, "On the discrete differential geometry of surfaces in S4" (2009). Doctoral Dissertations Available from Proquest. AAI3380019.