Certain types of freeform shells can be fabricated by bending originally flat and straight slats into curved structuralelements. In their final position, one obtains a grid of surface strips forming the basis of an architectural structure. Wepresent recent work on structures which are formed by three or four families of strips, arranged in a web.In their final curved position, the strips can be modeled as rectifying developable surfaces of their boundaries: Eachtangent plane of a strip is orthogonal to the osculating plane at the corresponding boundary curve point. Thus, if a stripis placed orthogonal to an underlying reference surfaceS, it has to follow an asymptotic curve ofS. If it is arrangedtangentially toS, if follows a geodesic curve onS. Hence, these gridshells are designed from hybrid webs ofasymptotic(A) andgeodesic(G) curves on freeform surfaces.Previous work focused mainly on a quadrilateral grid arrangement. If both families of strips are placed orthogonaltoS, one obtains the asymptotic gridshells (AA) of Eike Schling. Under the additional constraint of orthogonal nodeangles, the underlying surface is a minimal surface. The case (GG) of two tangential families of strips has recently beenstudied from the perpective of deployment from an arrangement of planar straight strips.