Must a Developable Surface be Tangent Developable or a Generalised Cone/Cylinder?

Solution 1:

Developable surfaces are surfaces with zero Gaussian curvature. Since there exist formulas for the Gaussian curvature in terms of a particular parametrization, the equation $K=0$ is a non linear differential equation which in general is untractable. As you pointed out cylinders and cones satisfy the equation $K=0$ as well as planes. Other classes of surfaces can be developable. This is for instance the case of non cylindrical ruled surfaces when the condition $(\overrightarrow{\alpha}^{'} \times\overrightarrow{w}) \cdot \overrightarrow{w}^{'}=0$, where the curve $\overrightarrow{\alpha}(t)$ is the directrix of the surface, while $\overrightarrow{w}(t)$ is the vector giving the direction. A general developable surface is in some sense the union of tangent developables. In other word a developable surface is the union of cylinders, cones, and tangent developables.