Holditch's Theorem in 3D Space

Holditch's theorem is an old result on the area generated by a moving chord for closed planar curves. Some generalizations of this result have been given before, but none of these follows the same natural construction of the plane but done in the space. In this work, the notion of Holditch surface is defined, some properties of these surfaces are proved and they are used to generalize Holditch's theorem for closed space curves naturally. Moreover, an approximation for the area of interest is given. Finally, it is showed that the only minimal non-planar Holditch surface is the helicoid.