Smooth T-surfaces can be thought of as a generalization of surfaces of revolution in such a way that the axis of rotation is not fixed at one point but rather traces a smooth path on the base plane. Furthermore, the action, by which the aforementioned surface is obtained does not need to be merely rotation but any "suitable" planar equiform transformation (stretch-rotation and translation) applied to the points of a planar smooth profile curve. In analogy to the smooth setting, if the axis foot points sweep a polyline on the base plane and if the profile curve is discretely chosen then a discrete T-surface or T-hydron with planar trapezoidal faces is obtained.

From an applied point of view, the straightforwardness of the generation of these surfaces predestines them for building and design processes. In fact, one can find many built objects belonging to different classes of the T-surfaces such as surfaces of revolution, translational surfaces and moulding surfaces. Furthermore, in the discrete version, possessing the flat trapezoidal faces, paves the way for steel/glass construction in industry.

The main goal of this talk is to give a rigorous definition and classification of the T-surface/T-hydron along with suitable parametrizations. Additionally, we present those isometric deformations which preserve the class of T-surfaces (e.g. the isometries that map a surface of revolution onto a surface of revolution) in both smooth and discrete settings.