Updated: Jun 26
A developable surface in differential geometry is a surface with zero Gaussian curvature. Such a surface can be superimposed on a plane by bending. Conversely, a unfolding surface can be obtained by transforming the plane (for example, bending, folding, gluing). From a practical engineering point of view, developable surfaces are the surfaces of cylinders and cones. NURBS can be used to model developable surfaces. For this, the forming of the cone surface must coincide with the line of the equal parameter NURBS. The second prerequisite for developable surfaces is the condition that the normal vectors to surface are equal to the surface along the guide line of an equal parameter. Many people think that for modeling the unfolding surface, it is enough to make the formings straight and do not think about the second condition. You can also check the condition for surface unfolding by visualizing the Gaussian curvature. In this case, it is important to set acceptable upper and lower limits for the visualization of curvature. Otherwise, all surfaces will appear developable, even those that are not.
Developable surfaces are very common in shipbuilding, especially in the design and construction of small vessels. Shall plates built on such surfaces usually do not require special bending and can be easily mounted on the hull. In some cases, for example, when using plywood or other composite materials for manufacturing the hull, plastic deformation of the stretching shell plates is physically impossible. Therefore, when modeling the hulls of such vessels, it is simply impossible to use any other types of surfaces. Errors in the design of the developing surfaces of such vessels can create gaps between the shell and the inner structure of the vessel. Despite the external simplicity, modeling such surfaces can cause certain difficulties. It is important not only to make the surface developable, but also to achieve the required shape of the hull.
Shape Maker does not have special functions for constructing unfolding surfaces, but the ability to build triangle patches of the surface and the topological connection between the surface sections allow you to build a fairly flexible model entirely consisting of developable surfaces - cylinders and cones.
The cone is easily modeled by a triangular surface patch. It is enough to straighten the formings coming to the top of the cone.
Thanks to topological connections, you can create a set of interconnected cones, based on a common guide line. You can change the shape of such a surface by changing the shape of the guide line or by changing the position of the vertices of the cones. An additional degree of freedom is given by the vertices of the cones hung on the guide line of the boundary of the adjacent cone. If the modification is performed in the mode of changing all control points “All”, then all surfaces will remain unfolded.
I tried to make a model of a simple sharp bilge boat entirely consisting of developable surfaces. That's what came out of it.
The bottom and side are represented by a set of cones resting on the bilge line. All excess is cut off by the cylindrical surfaces of the deck and transom and the surface of the center plane. The shape of the hull surface changes with a change in the shape of the bilge line, the position of the tops of the cones and the geometry of the deck and transom. At the same time, the surfaces remain developable to the plane.
A similar method of design developable surfaces is used for manual surface design and is well known to specialists in the design of boats and yachts.
Example form this article you can find here.