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Journal for Geometry and Graphics 20 (2016), No. 2, 225--241
Copyright Heldermann Verlag 2016

Two Examples of Solids Constructed From Given Developments

Hellmuth Stachel
Institute of Discrete Mathematics and Geometry, Vienna University of Technology, Wiedner Hauptstr. 8-10 / 104, 1040 Vienna, Austria

The paper provides two examples, where the bending of a planar area Φ0 with boundary c0 generates a developable surface patch Φ bounded by a particular spatial curve c. There are various ways to restrict such bendings. In the first example the surface Φ is a cylinder with given rulings, and the spatial counterpart c of the boundary c0 is planar. In the second example the rulings are unknown. Instead of this constraint, the closed boundary c0 is subdivided into two subarcs which are glued together while Φ0 is bent. In both examples we obtain solids enclosed by torses with geodesic circles c as curved edges.

Keywords: Surfaces of constant curvature, curved folding, developable surfaces, geodesic circle.

MSC: 53A03; 51M04, 51N05, 68U07

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