A surface of Voss was originally defined by Aurel Voss in the context of smooth differential geometry as a regular surface that admits a parameterization whose coordinate curves are geodesics and form a conjugate system [1]. What makes these surfaces interesting is their relatively simple isometric deformation. In the smooth scenario, this deformation can be simply demonstrated at the level of the second fundamental form, although deriving the immersion remains a challenge. However, this apparent simplicity comes with the complexity of their construction, as creating the V-surface patches necessitates solving multiple systems of partial differential equations.

Years later, Robert Sauer significantly eased the construction of these surfaces by introducing their discrete counterparts. Sauer showed how these surfaces become remarkably simpler when examined within a discrete framework. Here, they are merely quad surfaces, each vertex bearing equal opposite angles. This streamlined creation process naturally predisposed these surfaces for design applications. Furthermore, their capability to undergo a 1-parametric isometric deformation rendered them fit for transformable designs [2].

The goal of this talk is to construct these surfaces in the discrete and smooth scenarios and then expand the current knowledge to the semi-discrete world. In doing so, we build an interesting relation between the surfaces of Voss, asymptotic line parametrization of pseudospherical surfaces (K-nets) and their common Gauss maps which happen to be spherical Chebyshev nets. We explore various construction methods, highlighting the challenges encountered in the smooth scenario where systems of partial differential equations (PDEs) emerge. Our approach illustrates that the semi-discrete scenario, which requires dealing with systems of ordinary differential equations (ODEs), is slightly more complex than the discrete one that involves only difference equations. However, it is notably less complex than solving the PDE systems that arise in the smooth case.

Finally, in the second part of the talk, we will describe the 1-parameter family of isometric deformations of surfaces of Voss, linking them with spectral deformations and squeeze deformation of the corresponding K-nets and spherical Chebyshev nets respectively.