Developable Surfaces

Abstract

Developable surfaces are surfaces in Euclidean space which ‘can be made of a piece of paper’, i.e., are isometric to part of the Euclidean plane, at least locally. If we do not assume sufficient smoothness, the class of such surfaces is too large to be useful — if includes all possible aways of arranging crumpled paper in space. For C 2 surfaces, however, developability is characterized by vanishing Gaussian curvature, and by being made of pieces of torsal ruled surfaces. We will here use ‘developable’ and ‘torsal ruled’ as synonyms, because we are most interested in the ruled surface which carries a developable surface patch. We first have a look at the Euclidean differential geometry of developable surfaces, and then study developables as envelopes of their tangent planes. This view-point identifies the curves in dual projective space with the torsal ruled surfaces. We describe some fields of applications where the concept of developable surface arises naturally and knowledge of the theory leads to geometric insights. These include developables of constant slope, the cyclographic mapping, medial axis computations, geometrical optics, rational curves with rational offsets, geometric tolerancing, and the use of developable surfaces in industrial processes.